The most important feature of these equations is the factor = kT -mv which would appear in the denominator. It becomes zero when which is the ion- sonic wave speed, an acoustic wave in which both ions and electrons, undergo compressions and expansions; they are coupled to each other electrostatically and since electrons are hotter, they provide the restoring force kTe, while the ions, more massive, provide the inertia, m Because of this, the gas can accelerate across this speed (of the order of 3000-4000 m/s in Xenon) in one of two modes: (a) Smoothly, if the right-hand sides of all of Equations(29-32)are zero when v,=vs(actually if one of them is zero, the others will also be, at v=v). This imposes an internal condition on the differential equations, to supplement the boundary conditions. The difficulty is that one does not know a-priory where(in x) this condition will occur. It is also difficult o integrate numerical through this point because each derivative is of the -form One needs to use l Hospitals rule to extract the finite ratio (two values normally) (b) Abruptly, if the right-hand sides are nonzero when v, =vs. In this case the derivatives(including Ex) are locally infinite, although one can show that they behave as功/√×-x, and so this infinity is integrable.This can only happen at the open end of the channel, just as with a normal open gas pipe discharging into a vacuum In this case, we impose th end condition V,=Vis (Te) Notice that condition(b) can also occur at the inlet(x=0). Infact, it does occur. This restrict the electron capture to the required I, level this same sheath will then o is because the anode will develop a negative sheath(electron repelling) in order to attract ions, which will therefore enter it at their sonic velocity(a form of Bohms sheath-edge criterion): 6. Boundary Conditions So, in this device we have two sonic points, one(reversed)at inlet, and one (forward )either at the exit plane or somewhere in the channel. This provides either two boundary conditions, or one(Equation (34)) plus one internal condition of smooth sonic passage. Looking at Equations(25-28) we count 6 differential 16.522, Space P pessan Lecture 18 Prof. Manuel martinez Page 10 of 2016.522, Space Propulsion Lecture 18 Prof. Manuel Martinez-Sanchez Page 10 of 20 The most important feature of these equations is the factor 2 e ii 5 kT - m v 2 which would appear in the denominator. It becomes zero when e i is i 5 kT v =v = 3 m (33) which is the ion-sonic wave speed, an acoustic wave in which both, ions and electrons, undergo compressions and expansions; they are coupled to each other electrostatically, and since electrons are hotter, they provide the “restoring force” kTe , while the ions, more massive, provide the inertia, mi . Because of this, the gas can accelerate across this speed (of the order of 3000-4000 m/s in Xenon) in one of two modes: (a) Smoothly, if the right-hand sides of all of Equations (29-32) are zero when v =v i is (actually, if one of them is zero, the others will also be, at v =v i is ). This imposes an internal condition on the differential equations, to supplement the boundary conditions. The difficulty is that one does not know a-priory where (in x) this condition will occur. It is also difficult to integrate numerical through this point, because each derivative is of the 0 0 form. One needs to use L’ Hospital’s rule to extract the finite ratio (two values normally). (b) Abruptly, if the right-hand sides are nonzero when v =v ι is . In this case, the derivatives (including Ex) are locally infinite, although one can show that they behave as 1 x-xs , and so this infinity is integrable. This can only happen at the open end of the channel, just as with a normal open gas pipe discharging into a vacuum. In this case, we impose the end condition v =v T i is e ( ) . Notice that condition (b) can also occur at the inlet (x=0). Infact , it does occur. This is because the anode will develop a negative sheath (electron repelling) in order to restrict the electron capture to the required aI level. This same sheath will then attract ions, which will therefore enter it at their sonic velocity (a form of Bohm’s sheath-edge criterion): ( ) ( ) e i i 3 kT 0 v x=0 =- 5 m (34) 6. Boundary Conditions So, in this device we have two sonic points, one (reversed) at inlet, and one (forward) either at the exit plane or somewhere in the channel. This provides either two boundary conditions, or one (Equation (34)) plus one internal condition of smooth sonic passage. Looking at Equations (25-28) we count 6 differential