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Gas Flow Unit Conversions and Cathode Pressure Estimates 465 gas constant, and T is the temperature of the gas. The measured gas flow rate, or the gas throughput, is given by the ideal gas law: Q = PmVm = Nm RoTm , (B-7) where Pm is the pressure and Vm is the volume where the flow is measured for gas at a temperature Tm , and Nm is the mole flow rate. The mole flow rate is then Nm = PmVm / RoTm . Defining Tr = T / Tm and substituting the mole flow rate into Eq. B-1 gives the measured flow as a4 P12 –P2 a4 P12–P2 Q=16 l T Tm=16 l T . (B-8) r Putting this in useful units and writing it in terms of a conductance of the tube, which is defined as the gas flow divided by the pressure drop, gives Q = 1.28 d 4 (P12 P2 ), (B-9) Tr l where Q is the flow in sccm, is the viscosity in poises, d is the orifice diameter and l the orifice length in centimeters, and the pressures are in torr. The pressure upstream of the cathode orifice is then 2 0.78Q Trl 1/2 P1= P2+ d4 . (B-10) While Eq. (B-10) requires knowledge of the downstream pressure, for this rough estimate it is acceptable to assume P2 << P1 and neglect this term. For xenon, the viscosity in poises is = 2.3 10 4T(0.71+0.29/Tr ) for T >1, (B-11) rr where Tr = T ( ̊K)/289.7. The viscosity in Eq. (B-11) is different than Eq. (6.5-9) because 1 Ns/m2 = 10 poise. It should be noted that the temperature of the gas in the hollow cathode can exceed the temperature of the cathode by factors of 2 to 4 due to charge-exchange heating with the ions, which then affects the viscosity. As an example, take the NASA Solar Electric Propulsion Technology Applications Readiness (NSTAR) discharge cathode operating at a nominalPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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