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 × nullnullnull (1) In our case x=l, x ? ? ? null so y z 0 dB 1 jj=+ dx ≡ μ and calling y B-B≡ , 0 1dB j=- dxμ (2) Ohm’s law (ignoring Hall effect) is () j= E+u × Bσ nullnullnullnullnull (3) 16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 2 of 8 or, using zyx V E E = , B -B , u = u H ≡≡ ()j= E-uBσ (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=jBA=jBwH dx dx × i nullnullnull (6) neglect for now Substitute (2) into (6): 2 00 du 1 dB wH d B m=- B wH=- dx dx dx 2 ?? ?? μμ ?? i (7) Integrate: 22 0 0 00 BB mu+wH =mu +wH 22μμ ii neglect 22 0 0 B-BwH u= 2 m μ i (8) Putting this in Equation (5), () 22 00 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 22 B 0 0 dB x= wH E- B(B -B ) 2m 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 00 BII 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 large as long as this condition prevails, and it must indicate a large value of 0 Lσμ . 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 22 0 B(B -B ), namely, 22 0 B-3B=0 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 2BwH E-uB =E- 33 2mμ i , and, since by assumption, this difference is much less than E, we find 3 0 0 2BwH E 33 2mμ i null (14) or, in terms of voltage and current, 3 2 0 1H I V w 33 m ?? μ ?? ?? i null (15) Returning now to Equation (5), we notice that near both x=0 and x=L, uB<<E, so 0 dB -E dx σμnull , and so the thickness l of the thin current layers (where B varies substantially) can be estimated as follows: 01 1 0e 00 B-B B l ; l EEσμ σμ nullnull (16) where 0 1 B B= 3 . Using (16), () 0e22 00 33-1m 3m l ; l wHB wHBσσ i i nullnull (17) Remembering (from (8)) that the exit velocity is 22 00 e 0 BIwH 1 H u= = 22w mm μ μ ii (18) these results can be written as ()0e0 3 ul 3-1 2 σμ null ; 0e0 3 ul 2 σμ null (19) The non-dimensional group 0 ulσμ is called the Magnetic Reynolds Number (R m ) (based on length l). What we have seen is that this R m , when based on the current 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., l e , l 0 <<L, we can now state that this will occur when () m0e RL uL>>1≡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= wlσ ; e e 4 H 3 R= wlσ (21) (the 4/3 factor accounts for the “triangular” current distribution in the layers) and so the Ohmically dissipated power is 22 00 ee D=IR +IR (22) where () 0 001 00 wBw11 I = B -B = 1- =I 1- 33 ???? ???? μμ ???? (23) and e0 I=I-I=I 3 (24) Substituting, we find () 2 2 22 24 2 00 0 1 1- HI I14 4 H 3 D=I 1- H = 33w w3 3 -1 mw m σμ?? ?? ?? ?? ???? σ μ ii 2 22 24 2 00 e 4 H HI I141H 3 D=I = 3w 39w 3wm m σμ ?? ?? σ ?? μ ii and, in total, 2 24 0 I4H D= w 93 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 3null 16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 6 of 8 times (eV i ) be the effective ionization energy per atom, and e α the degree of ionization at the exit. We then have ei i m ' DeV m α i null (26) or, from (25), 2 2 0i e i ' Im 4H w eV 93 m ?? μ ??α ?? i null (27) This indicates very rapid increase of the ionization fraction as the current increases, or as the flow is reduced. Instability “onset” For a given thruster, as 2 Im i is increased, e α increases rapidly. When it reaches unity, the behavior of the plasma near the exit changes drastically. This is because any extra dissipation cannot be absorbed into ionization anymore, and goes instead directly into heating the plasma (or perhaps the electron component only). This causes conductivity to increase whenever the current concentrates, which leads to further current concentration. We have here the classical prescription for constriction into an arc, and one can expect heavy arcing (with the corresponding damage to electrodes) when e α approaches 1. This behavior has indeed been observed repeatedly, and has been the focus of a lot of attention, because it limits the practically achievable value of 2 Im i . Since (as we will see) efficiency increases with 2 Im i , this is a major hurdle in the path towards efficient MPD operation. It has been dubbed “the onset condition and we are now in a position to see what it implies quantitatively. Setting (27) to unity, we get 2 0 i i ' I eVH93 = w4m m μ i (28) and from the exit velocity expression (18), ii e '' eV eV193 u = = 0.987 24m m (29) The velocity i i ' 2eV m at which the particle’s kinetic energy would be capable of ionizing it is called the “Alfvèn critical speed”. Many years ago Alfvèn used this conversion of kinetic to ionization energy to construct a model of the “condensation” of matter expanding from the proto-Sun to form the existing planets. We see here that the exhaust speed of an MPD thruster (or a PPT, which works on the same principles) is 16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 7 of 8 limited roughly to the Alfvèn critical speed of the gas used (if exit arcs are to be avoided). For various gases, assuming ii ' V=2V, we find Gas Hydrogen Nitrogen Argon Lithium M i (g/mol) 1 14 40 7 V i (volts) 13.6 14.6 15.8 5.4 (I sp ) MAX (s) 5,120 1,420 870 1,220 Expressing i a M m= N (N a =Avogadro’s number), and using kA for I and g/s for m i , Equation (28) can also be rewritten as () 1 2 2 2 i ' kA g/mol IM w 15.4 V g/s H m ?? ?? ?? i null (30) volts For Argon at 6 g/s and w/H=4, this predicts an “onset current” * I18kAnull . Experimental values tend to cluster around * I20-23 kA～ , in reasonable agreement. Much of the difference is simply due to the geometry (coaxial vs. rectangular). The scaling of * I with 1 2 m i and with 1 - 2 Μ is also well documented experimentally. Efficiency Accounting only for the power lost to ohmic dissipation and to near-electrode voltage drops ( cathode anode V= V + V 10-20 Volts?? ? ～ ), we have 2 e 2 e 1 mu 2 = 1 mu +D+I V 2 η ? i i (31) From the equations derived before, 2 2 0 1 = 32 2I V 1+ + 93 I1H m 2w m η ? ?? μ ?? ?? i i 3.05 2 2 0 2 2 224 00 I11H m 22w m = II11H 4H m+ +IV 22w w 93 mm ?? μ ?? ?? η ?? μμ ?? ?? ? ?? ?? ?? i i i ii 16.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 8 of 8 () 2 23 0 1 = 8Vm w 3.05+ HI η ? ?? ?? μ ?? i (32) Even if the voltage drops could be eliminated, this efficiency is no higher than MAX 1 = = 0.328 3.05 η . The best that can be done in the presence of the voltage drop is to approach “onset”, at which point one gets * 1 32 4 i i 0 ' 1 = mwH 3.05+2.88 V eV m η ?? ?? ?? ? ?? ?? μ ?? i (33) For Argon, m=6 g/s i , w =4, V=10 Volts H ? this gives * =0.259η . Values of this order have been reported very often, and it has proven very difficult to exceed =0.3η with Argon at least. Most of the inefficiency is seen to arise from the strong dissipation in the inlet and exit layers. This is intrinsic to the constant area geometry. Although it may not be obvious at this point, a convergent-divergent geometry has the effect of weakening these dissipative layers, and can conceivably be exploited to improve efficiency (and retard “onset”). There is some evidence for this in experiments. The analysis showing the effect of a convergent-divergent geometry (or, in a more limited form, of a purely divergent geometry), can be found in Ref. 1. References: Ref. 1: Martinez-Sanchez, M. “Structure of Self-Field Accelerated Plasma Flows.” J. of Propulsion and Plasma 7, no. 1 (Jan-Feb 1991): 56-64.