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lecture22 - 16.522 Space Propulsion Prof Manuel...

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16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 1 of 8 16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 22: A Simple Model For MPD Performance-onset It is well known that rapidly pulsed current tends to concentrate near the surface of copper conductors forming a “skin”. A similar effect occurs when current flows through a highly conductive and rapidly moving plasma: current tends to concentrate near the entrance and exit of the channel. The reason is the appearance of a strong back EMF which tends to block current over most of the channel’s length. This is most easily seen if we “unwrap” the annular chamber of an MPD thruster into a rectangular 1-D channel. Ampère’s law: 0 1 j= B µ × GJ G (1) In our case x =l , x G so y z 0 dB 1 jj = + dx µ and calling y B- B , 0 1dB j=- dx µ (2) Ohm’s law (ignoring Hall effect) is () E+u±×±B σ GG G J G (3)

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16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 2 of 8 or, using zy x V E E = , B -B , u = u H ≡≡ () j= E-uB σ (4) Combining (2) and (4), 0 dB =- dx σµ (5) The flow velocity u evolves along x according to the momentum equation (ignoring pressure forces) x du dP m+ A= j B A = j B w H dx dx × i GJ G (6) neglect for now Substitute (2) into (6): 2 00 du 1 dB wH d B m= - B w H = - dx dx dx 2 ⎛⎞ ⎜⎟ µµ ⎝⎠ i (7) Integrate: 2 2 0 0 B B mu+wH =mu +wH 22 ii neglect 0 0 B-B wH u= 2 m µ i (8) Putting this in Equation (5), 0 dB wH E- B B -B dx 2m ⎡⎤ ⎢⎥ σµ µ ⎣⎦ i (9) If we approximate the conductivity σ as a constant, this can be integrated as 0 B 0 B 0 0 dB x= wH B(B -B ) σµ µ i (10) This integral can actually be calculated analytically, but the resulting expression is not very transparent. It is more useful to examine its behavior qualitatively. The
16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 3 of 8 denominator in the integrand is the driving field (applied field E, minus back emf, uB). The field B 0 at x=0 is a measure of the current I, because integrating (2) between x=0 and x=1 gives L 00 0 0 0 BI I jdx = = B = ww µ µ (11) On the other hand, carrying (10) all the way to x=L, gives 0 B 0 22 0 0 0 dB L= wH E- B(B -B ) 2m σµ µ i (12) where, once I and m i are specified, only E remains as an unknown. This is then the equation for voltage V=EH. Consider conditions where the maximum value of the back emf uB reaches almost the level E. This means that the integrand will be very

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This note was uploaded on 11/07/2011 for the course AERO 16.512 taught by Professor Manuelmartinez-sanchez during the Fall '05 term at MIT.

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lecture22 - 16.522 Space Propulsion Prof Manuel...

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