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 near the entrance and exit of the channel. The reason is the appearance of a strong through a highly conductive and rapidly moving plasma: current tends to concentrat back EMF which tends to block current over most of the channels length this is most easily seen if we" unwrap"the annular chamber of an MPd thruster into a ectangular 1-D channel B=0 了 H Amperes law j=-V×B (1) In our case and calling dB Ohms law(ignoring Hall effect)is j=G(E+uxB Pro f a spa m artie ssn Lecture 22 1 of 8
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 × G JG (1) In our case = l,x x ∂ ∇ ∂ G so y z 0 1 dB j j=+ dx ≡ µ and calling B -By ≡ , 0 1 dB j=- µ dx (2) Ohm’s law (ignoring Hall effect) is j = E + u × B σ ( ) G GGJG (3)
or, using j=σ(E-uB) 4 Combining(2)and (4) d (E The flow velocity u evolves along x according to the momentum equation(ignoring pressure forces neglect for now Substitute(2)into(6) u wh d B2 μodx Integrate mu+wH B WH B2-B2 Putting this in Equation(5), B(B:-B2 If we approximate the conductivity o as a constant this can be integrated as B(B2-B2) 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 2 of 8
16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 2 of 8 or, using z yx V E E = , B -B , u = u H ≡ ≡ j = E - uB σ ( ) (4) Combining (2) and (4), 0 ( ) dB = - E - uB 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 = jBwH dx dx × i G JG (6) neglect for now Substitute (2) into (6): 2 0 0 du 1 dB wH d B m = - B wH = - dx dx dx 2 ⎛ ⎞ ⎜ ⎟ µ µ ⎝ ⎠ i (7) Integrate: 2 2 0 0 0 0 B B mu + wH = mu + wH 2 2 µ µ i i neglect 2 2 0 0 wH B -B u = 2 m µ i (8) Putting this in Equation (5), ( ) 2 2 0 0 0 dB wH =- E- B B -B dx 2 m ⎡ ⎤ σµ ⎢ ⎥ ⎣ ⎦ µ i (9) If we approximate the conductivity σ as a constant, this can be integrated as B0 0 2 2 B 0 0 dB x = wH E - B(B - B ) 2 m 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
denominator in the integrand is the driving field(applied field E, minus back emf, uB). The field Bo at x=0 is a measure of the current I, because integrating(2) between x=0 and x=l gives I Bo B dI (11) On the other hand, carrying(10) all the way to x=L, gives d B (12) B where, once i and m 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 large as long as this condition prevails, and it must indicate a large value of ooL. By the same token, Equation(5)says that B will remain flat when E-uB<<E, and from (2), there will be little current in this region. Schematically TTTTTTTTTTTT We see here that two strong current concentrations develop near x=0 and x=L Let us investigate when this situation will arise. From (12), the denominator is minimum at a B value that maximizes B(B:-B2), namely, B2-3B2=0 16.522, Space P pessan Lecture 22 Prof. Manuel martinez Page 3 of 8
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 B0 at x=0 is a measure of the current I, because integrating (2) between x=0 and x=1 gives L 0 0 0 0 0 I B I jdx = = B = w w µ ⇒ µ ∫ (11) On the other hand, carrying (10) all the way to x=L, gives B0 0 2 2 0 0 0 dB L = wH E - B(B - B ) 2 m 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 large as long as this condition prevails, and it must indicate a large value of σµ0L . By the same token, Equation (5) says that B will remain flat when E-uB<<E, and from (2), there will be little current in this region. Schematically: We see here that two strong current concentrations develop, near x=0 and x=L. Let us investigate when this situation will arise. From (12), the denominator is minimum at a B value that maximizes 2 2 B(B - B ) 0 , namely, 2 2 B - 3B = 0 0
nd th (E-UBMIN=E WH 2B 33 and since by assumption, this difference is much less than E, we find H B 3√ or, in terms of voltage and current, 1H),I3 33 Returning now to Equation (5), we notice that near both x=0 and x=L, uB<<E,so d B dx - OHoE, and so the thickness i of the thin current layers(where B varies substantially) can be estimated as follows HDE where B Using(16) 3(3-1) aWHB2 (17) NHB Remembering(from( 8)) that the exit velocity is Bo 1 HH. these results can be written as cHou.lo The non-dimensional group foul is called the Magnetic Reynolds Number(rm) (based on length I). What we have seen is that this rm, when based on the current 16.522, Space Propulsion Lecture 22 Prof. Manuel martinez-s of 8
16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 4 of 8 MAX 0 1 (uB) B B Β = 3 ≡ (13) and then ( ) 3 0 MIN 0 wH 2B E - uB = E - 3 3 2 m µ i , and, since by assumption, this difference is much less than E, we find 3 0 0 wH 2B E 3 3 2 m µ i (14) or, in terms of voltage and current, 3 2 0 1H I V 3 3 w m ⎛ ⎞ µ ⎜ ⎟ ⎝ ⎠ i (15) Returning now to Equation (5), we notice that near both x=0 and x=L, uB<<E, so 0 dB - E dx σµ , and so the thickness l of the thin current layers (where B varies substantially) can be estimated as follows: 0 1 1 0 e 0 0 B -B B l ; l σµ σµ E E (16) where 0 1 B B = 3 . Using (16), ( ) 0 e 2 2 0 0 3 3 -1 m 3m l ; l σ σ wHB wHB i i (17) Remembering (from (8)) that the exit velocity is 2 2 0 0 e 0 wH 1 H B I u= = 2 2w m m µ µ i i (18) these results can be written as 0 e0 ( ) 3 u l 3 -1 2 σµ ; 0 e0 3 u l 2 σµ (19) The non-dimensional group 0 σµ ul is called the Magnetic Reynolds Number (Rm) (based on length l). What we have seen is that this Rm, when based on the current
layer thickness, is of order unity. Since we started out by assuming conditions when these layers are thin i.e. le, lo< <L, we can now state that this will occur when Rm(L)=cHou L>>1 (20) This is indeed the condition for operation in the pure mPd regime. Effects of Dissipation The high-current inlet and exit layers are very dissipative. Their resistances can be estimated as (21) owo (the 4/3 factor accounts for the"triangular"current distribution in the layers)and so the ohmically dissipated power is D=IRo+IR。 (22 ere Io B1) d I=I-5=/3 Substituting, we find 1 Dn=2|1- 1 H aHui √3(H)aI W3|√3-1mw =33wm D.=n13帅=41邮 and, in total D √3 (25) Part of this dissipation goes to heating the gas, but the major portion is used in ionizing and exciting(followed by radiation) the gaseous atoms. Let ev=2 to 3 16.522, Space P pessan Lecture 22 Prof. Manuel martinez Page 5 of 8
16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 5 of 8 layer thickness, is of order unity. Since we started out by assuming conditions when these layers are thin, i.e., le, l0<<L, we can now state that this will occur when R L u L >> 1 m 0e ( ) ≡ σµ (20) This is indeed the condition for operation in the pure MPD regime. Effects of Dissipation The high-current inlet and exit layers are very dissipative. Their resistances can be estimated as 0 0 4 H 3 R = σwl ; e e 4 H 3 R = σwl (21) (the 4/3 factor accounts for the “triangular” current distribution in the layers) and so the Ohmically dissipated power is 2 2 D=IR +IR 00 ee (22) where ( ) 0 0 01 0 0 w 11 wB I = B -B = 1- =I 1- 3 3 ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ µ µ ⎝ ⎠⎝ ⎠ (23) and e 0 I =I-I =I 3 (24) Substituting, we find ( ) 2 22 24 2 2 0 0 0 1 1 - 14 4 H HI I 3 D =I 1- H = 3 33 3 3w w3 3 - 1 m w m ⎛ ⎞ σ µ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ σ µ i i 22 24 2 2 0 0 e 4 H 1 41 H 3 HI I D =I = 3 w 39 w 3wm m σ µ ⎛ ⎞ ⎜ ⎟ σ ⎝ ⎠ µ i i and, in total, 2 2 4 0 4 H I D = 9 3 w m ⎛ ⎞ µ ⎜ ⎟ ⎝ ⎠ i (25) Part of this dissipation goes to heating the gas, but the major portion is used in ionizing and exciting (followed by radiation) the gaseous atoms. Let ' i eV 2 to 3