lecture15

lecture15 - 16.522, Space Propulsion Prof. Manuel...

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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 0x xx ch x 0 x 00 0 E Fd E = E dx = E dx = Ad x 2 ε ρε ∫∫ 43 a x =-V d φ 13 a x V4x E= d3 d x 0 0 a x d V 4 3d 22 0a a x 0 VV F 16 8 == A2 9 9 ε ε Alternative: 33 aa a xi i ii V2 e V V Fm m 42 e 8 =j c = = Ae e 9 m m 9 εε m i j from Child-Langmuir
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16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 2 of 11 Bohm velocity : Why? collisional “drag” ii i ix i n dv dP nmv + =enE -F dx dx ee eee ex e n dv dP + =-enE -F dx dx small small not near wall, ei n=n . Add: constant () i i i n dP+P dv + - F dx dx ± and nv=const= i Γ 2 i e i n i d nmv +P +P - F dx ± ( ) ee i e e P+P=nkT+T nkT ± i iii e i n i d mv+ k T = - F dx v ⎡⎤ Γ Γ ⎢⎥ ⎣⎦ min. at e i i kT v= m
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16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 3 of 11 The lines i e i kT v Γ and iii mv Γ 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 v=v in the quasineutral plasma. Beyond that, ei x n<<n , E becomes very strong, and ions just free-fall to wall (in the sheath ) so, entering sheath, v=v . How big is the sheath ? In sheath, say e n0 ± Child-Langmuir: 3 2 s ii i 0 2 i V 42 e j=env= 9m ε δ But also sh e ie i kT je n m ± sh 3 2 se 0e 2 Vk T e en m ε δ ±
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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 ee e e Vk T e V 42 1 = 9n 9 k T en ekT ⎛⎞ εε δ ⎜⎟ ⎝⎠ ± sh -1 n= ne x p 2 1 3 2 4 0e s -1 2 2 e e kT eV kT 9e ε δ ± 1.018 d Debye () e -5 Debye 17 -3 e TK 3×11600 d 69 =2.4×10 m=24 m 3×10 nm µ = If wall not biased (insulator), 34 i se DD e m eV kT ln 5 d 3d m ∼→ δ ± For sheath in front of extractor grid, s V 1000V D 78d δ ± e kT 3V e 1.9 mm δ ± This approximately sizes the extractor holes.
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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
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lecture15 - 16.522, Space Propulsion Prof. Manuel...

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