Pressure gradient

The acceleration of the flow is directly related to the pressure gradient. If the pressure gradi- ent is insufficiently steep, the flow does not become sonic. Thus, there is a minimum pressure difference across thevena contracta required for achieving sonic flow. For a polytropic gas with the adiabatic index γ, the ratio of the stagnation pressure pst upstream of the vena contracta to the critical downstream pressure p∗₀ required for achieving sonic flow is:

p_st p∗0 = γ+ 1 2 _γ−γ₁ (6.2) (6.2) is exact only if the pressure difference occurs in the immediate vicinity of the vena contracta. In such case, the critical pressure ratio is p_st/p∗₀ = 2.053 for a monatomic ideal gas withγ = 5/3, and p_st/p₀∗ = 1.893for a diatomic ideal gas with γ = 7/5. The critical pressure ratio varies almost linearly with γ, and is of the same order of magnitude for all

gases (pst/p ∗

0 ∼2). When (6.2) is satisfied, the flow velocity attainscsat thevena contracta. Further decreasing the downstream pressure does not cause the flow velocity at the vena contracta to exceed c_s, since Ma = 1, if it occurs, must be at the vena contracta. This

phenomenon is known as flow velocity choking. When the flow is choked, the flow conditions upstream of thevena contracta become insensitive to the flow conditions downstream.

In reality, the pressure drop is not immediate but instead manifests over a nonzero dis- tance. Figure6.2plots the static pressurepalong thez-axis inPR-Ofor the CFD simulations performed withm˙ = 100SCCM of Ar, using p₀ = 0.349Torr (blue line) and p₀ = 0Torr (blue line). Since p falls continuously along the entire length of the discharge chamber, p_st/p∗₀ (6.2) cannot be used to determine if the flow velocity choking condition is met. Non-

etheless, the close proximity of thep profiles (blue and lines) suggests that the flow

conditions upstream are very similar despite the difference inp₀ downstream, and therefore

is indicative of some degree of of flow velocity choking inPR-O.

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Figure 6.2: Static pressurep (blue) and Knudsen number Kn (red) along z-axis in PR-Ofor CFD simulations performed with p₀ = 0.349Torr ( lines) and p₀ = 0Torr ( lines). p

falls continuously along the entire length of the discharge chamber, and exhibits deviation between the p0 = 0.349Torr andp0 = 0Torr cases.

PR-Con the other hand demonstrates a well defined choked flow behaviour. Thepprofiles

for both the p₀ = 0.349Torr (blue line) and p₀ = 0Torr (blue line) cases shown in Figure 6.3 drop sharply in the vicinity of the constricted nozzle throat at z = −3.0mm. Furthermore, thep profiles in both cases coincide along the length of PR-C up to very near the discharge chamber exit, exhibiting the characteristic behaviour of choked flow, whereby flow conditions upstream of the constricted nozzle throat are identical despite the significant difference in the downstream pressurep0. This is in contrast to thepprofiles inPR-O(Figure

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Figure 6.3: Static pressurep (blue) and Knudsen number Kn (red) alongz-axis inPR-C for CFD simulations performed withp0= 0.349Torr ( lines) andp0 = 0Torr ( lines). The pressure gradient is slight in the plenum and discharge chamber, andpis mostly dropped at

the constricted nozzle throat. Thepprofiles in both cases coincide, exhibiting characteristic

choked flow behaviour.

The stagnation pressure in the PR-Cplenum for thep₀ = 0Torr case is p_st= 2.745Torr, which is about twice that of the PR-O, p0 = 0Torr case where pst = 1.336Torr. Due to the constricted nozzle throat, the discharge chamber upstream of the convergent section (z≤ −6.5mm) is held at a high static pressure with only a slight gradient along the z-axis,

in contrast with the continuously fallingp in PR-O. Since most of the pressure difference is dropped over a short distance in the constricted nozzle, (6.2) is somewhat applicable forPR-C. Using the value ofpstquoted above, the critical downstream pressure isp

∗

0 .1.337Torr. This is another confirmation that the flow in PR-C is choked for both the p₀ = 0.349Torr and

p₀= 0Torr cases.

Also plotted in Figures 6.2 and 6.3 is the Knudsen number Kn (3.2) in the plenum, discharge chamber, and downstream regions (red and lines lines). Since Kn ∝1/p,

a decrease inpresults in an increase in Kn. In the discharge chamber, Kn is highest at the

exit (z= 0mm), but remains in the slip regime (Kn.0.1) within which the fluid numerical method is completely valid. However, for the p₀ = 0Torr cases, the low pressure in the downstream region causes Kn (red line) to rise very quickly, and the fluid numerical method ceases to be valid at a certain distance beyond the discharge chamber exit. While the Kn.0.1limit may be used as a guide, it is by no means a definitive indicator of when the

fluid numerical method breaks down. Hence, the CFD simulation results in the downstream region beyond z >0mm are not used for analysis. Regardless, the CFD simulation results for z ≤ 0mm up to the discharge chamber exit are unaffected by virtue of flow velocity choking.