16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 1 of 11 16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 15: Thrust Calculation (Single Grid, Single Potential) 2 x d1 E dx 2 ?? ?? ?? d dd 2 0xxx ch x 0 x 00 0 EFdE = E dx = E dx = Ad ε ρε ∫∫ 43 a x =-V d ?? φ ?? ?? 13 a x V 4x E= d3 d ?? ?? ?? x 0 E=0 a x d V4 E= 3d 22 0a ax 0 VVF 16 8 == A29 9dd ? ?ε ε ? ? ? ? Alternative: 33 22 aa axi i 00 ii V2eV VFm m42 e 8 =jc= = Ae e9 m m 9dd εε m i j from Child-Langmuir 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 2 of 11 Bohm velocity: Why? collisional “drag” ii iii ix in dv dP nmv + =enE -F dx dx ee eee ex en dv dP nmv + =-enE -F dx dx small small not near wall, ei n=n. Add: constant () ei i eii in dP+Pdv nmv + - F dx dx null and ei nv=const= i Γ () 2 eii e in i d nmv +P +P - F dx null ( ) ei ee i ee P+P=nkT+T nkTnull i iii e in i d mv+kT =-F dx v ??Γ Γ ?? ?? min. at e i i kT v= m 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 3 of 11 The lines i e i kT v Γ and iii mvΓ must cross at e i i kT v m = , where their sum is minimum. So, no solution at e iB i kT v> =v m Ions accelerate to iB v=v in the quasineutral plasma. Beyond that, eix n<<n , E becomes very strong, and ions just free-fall to wall (in the sheath) so, entering sheath, iB v=v. How big is the sheath? In sheath, say e n0null Child-Langmuir: 3 2 s iii 0 2 i V42 e j=env= 9m ε δ But also sh e ie i kT jen m null sh 3 2 se 0e2 ii VkT42 e en 9m m ε δ null 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 4 of 11 sh sh 3 3 2 2 2 0s 0 e s 2 eee e VkTV42 1 42 = 9n9 kTen ekT ??εε δ ?? ?? null sh ee -1 n=nexp 2 ∞ ?? ?? ?? 1 3 2 4 0e s -1 2 2 ee kT eV42 kTen 9e ∞ ??ε δ ?? ?? null 1.018 d Debye () () e -5 Debye 17 -3 e TK 3×11600 d 69 69 =2.4×10 m=24 m 3×10nm μ= ～ If wall not biased (insulator), 34i se DD e m eV kT ln 5 d 3d m ?? ～→δ～?? ?? null For sheath in front of extractor grid, s V 1000V～ D 78dδ null e kT 3V e ～ 1.9 mmδ null This approximately sizes the extractor holes. 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 5 of 11 If holes much bigger than ,δ If much smaller, ions lost plasma would escape to grid Space charge effects in the accel-decel gap For d 0x<< , 2 i 2 00i 2 00 i endjj =- =- =- vdx 2e v- m φ εε φ ε 2 0 0 2 0 i 1d j d =- +c 2dx 2e v- m φ φφ?? ?? ε φ?? ∫ Note: Slope here not necessarily zero. Change integration variable : 2 0 i 2e v= v - m φ () 22i 0 m =v-v 2e φ i m d=- vdv e φ () 0 i 2 v 2ii 000 000v m - vdv mm1d j j j 2e e =c- =c- v-v=c- v- v- 2dx v e e m ?? φφ?? ?? ?? εεε ?? ?? ∫ 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 6 of 11 where 2 T 0 i 2eV v= m () 2 T iT 0i i 2e V -m2eV 1d 2j =2c- - 2dx e m m ?? φ φ?? ?? ?? ε ?? ?? () TiT 0i i d dx = 2e V - 2jm 2eV 2c - - em m φ ?? φ ?? ε ?? () TN =V -V d 0 TiT 0i v d d= c, 2e V - 2jm 2eV 2c - - em m φ φ ? ?? φ ?? ε ?? ∫ then all profiles follows. In the limit when the second gap becomes choked as well, (as in the case with no decel grid, in which case d d is the downstream sheath thickness) d iT N x=d 0i i jm 2eV 2eVd =0 c= - dx e m m ?? φ?? ?? ? ?? ε ?? ?? () TN =V -Vφ Then () TN V-V d 0 TiN 0ii d d= 2e V -2jm 2eV - em m φ ?? φ ?? ε ?? ∫ Change again variable: i m d=- vdv e φ () T i 2e V - =v m φ 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 7 of 11 N i T NT i 2eV i m v 0i d v2eV N iN m 0i m -vdv m v dv e d= = 2je v-v 2jm 2ev v- em ε ?? ?? ε ?? ∫∫ N 2d v - v () T T N N v v 0i 0i dNNTNTTN v v mm4 d = 2v v-v - 2 v-v dv = v -v 2v - v -v 2je 2je 3 ?? εε ???? ?? ?? ?? ?? ?? ?? ∫ () T N v 3 2 N v 4 v-v 3 T N 24 v+ v 33 () 0i dTNTN m2 d= v-v v+2v 3ej ε () 1 1 4 4 111 1 0 222 2 dTNTN i 2×2 e d= V -V V +2V 3jm ??ε ?? ?? Now 3 2 T 0 2 i a V42 e j= 9md ε , so 1 3 4 4 11 T 42 a01 2 i V2e d= ×2 3m j ?? ε ?? ?? define N T V R= V ()() 1 12 12 12 122 TN T T d 3 4 a T V-V V+2V d = d V ()() 1 112 d 22 a d =1-R 1+2R d 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 8 of 11 Appendix B ELECTRON DIFFUSION IN A MAGNETIC FIELD 1) No B nullnull Field In electron momentum balance, main forces are pressure gradient and collisional retardation (no inertia): e eee P-nmv?ν nullnull null (1) Also eee P=nkT, eee PkTn.??null Solve for flux: e e ee ee kT nv =- n m ? ν nullnull (2) This is Fick’s law of diffusion e eee nv =-D n? nullnull , with a diffusivity e e ee kT D= m ν (3) ( j eje e =ncQν ∑ , collision frequency) 2) With () e Bto P⊥? nullnull Add magnetic force: ee eeeee P=-nmv -env×B?ν nullnullnullnullnullnull (4) To solve for e e nv nullnull , form () ee eee e P ×B=-m n v ×B-en v ×B ×B?ν nullnullnullnullnullnullnullnullnullnullnullnull , and use 0 () () 2 eeev×B×B=Bv . B-Bv. nullnullnullnullnullnullnullnullnullnullnullnullnullnull Eliminate () ev×B nullnullnullnull between these two equations, simplify: 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 9 of 11 ee ee22 ee ee e e 2 ee kT ekT -n+n×B m m nv = eB 1+ m ?? ν ν ?? ?? ν ?? nullnull nullnull (5) NOTE: This leaves the E null field out. To include it, just replace e n? by e e e en n+ E kT ? null Define the nondimensional factor c ee e eB β m ω ≡≡ νν (Hall parameter) (6) where c e eB = m ω is the cyclotron frequency for electrons. Then () e eeee2 1 nv = -D n - ×D n 1+ ?β? β nullnullnull (7) Of these two terms, the second is perpendicular to both, B nullnull and e n? , and is called the “ e P×B? nullnull drift”. The main interest is on the first term, which is along e -n? , as a regular diffusion. We see that this “cross-field diffusion” is governed by e ee nv =-D n ⊥ ? nullnull , with e 2 D D= 1+ ⊥ β (8) So, a high Hall parameter βcan greatly reduce diffusion, compared to that in the absence of a magnetic field. High β means both, high B and/or low collision frequency. In an ion engine, with e T = 4eV = 46400 K, the e-n and e-i cross-section are roughly -19 2 en Q10 m,null -18 2 ei Q4×10 m,null and 6e e e kT8 c= 1.34×10 m/s. mπ null If also 17 -3 e n2.8×10 m,null 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 10 of 11 18 -3 n n7.4×10 m,null then 18 6 -19 5 -1 en 6-1 e 17 6 -18 6 -1 ei = 7.4×10 ×1.34×10 ×10 = 9.9×10 s =2.49×10 s = 2.8×10 ×1.34×10 ×4×10 =1.50×10 s ?ν ? ν ? ν ? ? At a point in the engine where B = 100 gauss = 0.01 Tesla, -19 -2 9-1 c c -30 e 1.6×10 ×10 = =1.76×10 s = = 706 >>1 0.91×10 ω ω?β ν Under these conditions, (8) reduces to e 2 D D ⊥ β null (9) or ee ee 22 2 ec c kT kT D mm+ ⊥ ??νν ?? νω ω ?? = null (10) This last form shows that collisions favor diffusion. In contrast, recall Equation (3), (no magnetic field, or <<1β ), which shows that in that case, collisions impede diffusion, Equation (10) also shows that D ⊥ scales as 2 1 B : ee eee 22 2 2 e 2 e kT m kT D = m e B e B m ⊥ νν null and so, increasing B should provide very strong confinement of electrons. With the given numbers, we find -23 52e e -30 6 ee kT 1.38×10 ×46400 D = = = 2.83×10 m /s m 0.91×10 ×2.49×10ν and 5 2 2 2.83×10 D= =0.57 m/s 706 ⊥ A diffusing substance spreads (in 1-D) roughly as x2Dt～ . So, to spread by 1 cm, electrons would require a time 16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 11 of 11 2-42 -5 -4 2 x10 m t= =4.5×10 s 4D 4×0.57×10 m /s ～ It turns out, however, that electrons can diffuse faster than this in most cases. The physical reasons are apparently related to the “equivalent collisionality” produced by scattering of the electrons by small-scale plasma density fluctuations which are almost always present. This is the same situation that has kept tokamaks from delivering fusion power (only in that case it is the H + ions that “leak through” the confining B nullnull field). Bohm obtained an empirical expression (with some theoretical guidance) for this so- called “anomalous diffusion” e Bohm kT D= 16eB (11) and experiments in ion engines and Hall thrusters appear to confirm the 1/B dependence, but also seem to indicate a somewhat smaller diffusivity magnitude. An often used expression is () e anomalous B B kT D = c 16 -100 ceB ～ (12) It is of some interest to see what collision frequency would produce the same diffusivity as these fluctuations: e e e anomalous c anomalous22 BB kT m kT = ceB ceB νω ≡ ? ν (13) and so B c can be thought of as the “Anomalous Hall Parameter”. For modeling purposes, one often adds together collision anomalous +ν ν in calculating diffusivity D ⊥ by Equation (10).