16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 10: Electric Propulsion - Some Generalities on Plasma (and Arcjet Engines) Ionization and Conduction in a High-pressure Plasma A normal gas at T <3000K is a good electrical insulator, because there are almost no ～ free electrons in it. For pressure > ～ 0.1 atm, collision among molecule and other particles are frequent enough that we can assume local Thermodynamic Equilibrium, and in particular, ionization-recombination reactions are governed by the Law of Mass Action. Consider neutral atoms (n) which ionize singly to ions (i) and electrons (e): n R e +i (1) One form of the Law of Mass Action (in terms of number densities n= P j kT , where T j is the same for all species) is nn ei () (2)=S T n n Where the “Saha function” S is given (according to Statistical Mechanics) as 3 ()ST i q n = 2 q ? π e ? h 2 ? 2m kT ? ? ? eV i e kT - (3) q i = Ground state degeneracy of ion (= 1 for H + ) q n = Ground state degeneracy of neutral (= 2 for H) m e = mass of electron = 0.91 × 10 -30 Kg k = Boltzmann constant = 1.38 ×10 -23 J/K (Note: k = R/Avogadro’s number) h = Plank’s constant = 6.62x10 -34 J.s. V i = Ionization potential of the atom (volts) (V i = 13.6 V for H) Except for very narrow “sheaths” near walls, plasmas are quasi-neutral: e n= n i (4) So that e n n n ()=S T (3’) can be used. 2 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 1 of 12 2 Given T, this relates n e to n n . A second relation is needed and very often it is a specification of the overall pressure P = (n + n i + n )kT = (2n + n )kT (5) e n e n Combining (3’) and (5), 2 () ? ? P -2n e ? = ST n = ST ? kT ? ()(n-2n e ) e ? P Where n= is the total member density of all particles. kT We then have 2 n+ 2Sn -Sn = 0 e e S 2 (6) n= -S + + Sn = n e 1+ 1 + n ST() Since S increases very rapidly with T, the limits of (6) are → Sn ( → 0 ) (Weak ionization) n ????? e T0 T0→ → ????? n n e T→∞ → 2 (Full ionization) G Once an electron population exists, an electric field Ewill drive a current density G j through the plasma. To understand this quantitatively, consider the momentum balance of a “typical” electron. It sees an electrostatic force G G F= -eE (7)E It also sees a “frictional” force due to transfer of momentum each time it collides with some other particle (neutral or ion). Collisions with other electrons are not counted, because the momentum transfer is in that case internal to the electron species. The ions and neutrals are almost at rest compared to the fast-moving electrons, and we define an effective collision as one in which the electron’s directed momentum is fully given up. Suppose there are ν e of these collisions per second ( ν e =collision frequency per electron). The electron loses momentum at a rate JG JJG -m Ve ν , where V e =mean directed velocity of electrons, and so e e G JG FFriction = -m Ve ν (8) e e On average, 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 2 of 12 G G F+ Friction =0, E or JG G mVeν =-e E e e JG ? e ? G V=- ? ? E (9)e ν ? m ee ? The group μ e = e is the electron “mobility” ((m/s)/ (volt/m)). The current density m ν ee is the flux of charge due to motion of all charges. If only the electron motion is counted (it dominates in this case) G JG j-en V ≡ e e (10) and from (9), 2 G ? en ? G e j= ? ? m ν ? ? E (11) ? ee ? The group 2 e ν e σ = (12) m ν ee is the conductivity of the plasma (Si/m). Let us consider the collision frequency. Suppose a neutral is regarded as a sphere with a cross-section area Q en . Electrons moving at random with (thermal) velocity c e intercept the area Q en at a rate equal to their flux nc Q en . Since a whole range of ee speeds ce exists, we use the average value ce for all electrons. But this is for all electrons colliding with one neutral. We are interested in the reverse (all neutrals, one electron), so the part of ν e due to neutrals should be nc Q en . Adding the part due to ions, e n ν = n cQ en + n cQ ei (13) e e n e i 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 3 of 12 NOTE: c e is very different (usually much larger) than V e . Most of the thermal motion is fast, but in random directions, so that on average it nearly cancels out. The non- cancelling remainder is V e . Think of a swarm of bees moving furiously to and fro, but moving (as a swarm) slowly. G The number of electrons per unit volume that have a velocity vector c e ending in a “box” 3 dc dc dc ≡ d c in velocity space is defined as e e z ee x y G 3 fc ) d c ( e e e ( G JG Where fc , x ) is the Distribution function of the e e JG electrons which depends (for a given location x and time G t) on the three components of c e . In an equilibrium JJG situation all directions are equally likely, so f= f e ( c c ) = f ()only, and one can e e e e show that the form is Maxwellian. 3 2 2 mc 2 2 2 2 - ee f= n ? m ? ? e 2kT e ; ( c = c + c ey + c ez ) (14) e e ? e e ex 2kT e ?? π With the normalization 3 fd c = n . e e∫∫∫ e The mean velocity is then 3 ce ≡ ∫∫∫ c f d c ee e and direct calculation gives 8 kT e (15)c=e π m e For Hydrogen atoms, ce = 6210 T (m/s, with T e in K) (16) e NOTE: If there is current, the distribution cannot be strictly Maxwellian (or even isotropic). But since V e << c e , the mean thermal velocity is very close to Equation (15) anyway. 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 4 of 12 Regarding the cross sections Q en , Q ei , they depend on the collision velocityc, e especially Q ei . This is because the e-i coulombic interaction is “soft”, so a very fast electron can pass nearly undeflected near an ion, whereas a slow one will be strongly deflected. The complete theory yields an expression Q ei  2.95 ×10 -10 ln Λ (m 2 ) (17) T (T in Kelvin) where 1 ln Λ  -11.35 + 2ln T(K)- ln P(atm) (18) 2 so that ln Λ is usually around 6-12, and can even be taken as a constant (~8) in rough calculations. For the neutral hydrogen atoms, the collisions are fairly “hard”, and one can use the approximation 2 Q  2 ×10 -19 m (19) e H Numerical Example Consider Hydrogen at P=1 atm. For T ≥ 4000K , diatomic H 2 is not present anymore (H 2 → 2H). So ionization is from atomic hydrogen, H, for which V = 13.6 volts, i q= 1, i q= 2, n so that 3 2 157,800 - e T S= 2.42 ×10 21 T (m -3 ). We also find 7.34 ×10 27 (m -3 ),n= T c = e 6210 T (m/s), σ = 2.821×10 -8 n ν e e (ν = ν ei + ν ). e en 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 5 of 12 The results are shown below. T(K) 5,000 6,000 7,000 8,000 10,000 12,000 14,000 16,000 -3 ) 1.47x10 24 1.22x10 24 1.048x10 24 9.18x10 23 7.34x10 23 6.12x10 23 5.24x10 23 4.59x10 23 nm S(m -3 ) ( 1.68x10 13 4.26x10 15 2.30x10 17 4.70x10 18 3.393x10 20 6.189x10 21 5.104x10 22 2.55x10 23 -3 ) 4.97x10 18 7.216x10 19 4.906x10 20 2.072x10 21 1.545x10 22 5.565x10 22 1.203x10 23 1.716x10 23 nm e ( α 3.4x10 -6 5.9x10 -5 4.7x10 -4 0.0023 0.0215 0.1000 0.298 0.597 ce (m/s) 4.39x10 5 4.81x10 5 5.20x10 5 5.56x10 5 6.21x10 5 6.80x10 5 7.35x10 5 7.86x10 5 ln Λ 5.68 6.05 6.36 6.62 7.07 7.44 7.74 ν 8.01 ei (s -1 ) 1.46x10 8 1.72x10 9 9.76x10 9 3.51x10 10 2.00x10 11 5.77x10 11 1.030x10 12 1.244x10 12 ν (s -1 ) 1.29x10 11 1.15x10 11 1.09x10 11 1.02x10 11 8.73x10 10 6.81x10 10 4.16x10 10 1.82x10 10 en ν (s -1 ) 1.29x10 11 1.17x10 11 1.19x10 11 1.37x10 11 2.87x10 11 6.45x10 11 1.072x10 12 1.262x10 12 e σ (Si/m) 1.09 17.4 116.3 426.7 1519 2434 3166 3836 Notice (a) Coulomb-dominated ( ν >> ν en ) for T > 8000 K ei ～ (b) Rapid rise of ionization fraction α for 7000 < T < 10, 000 K ～ ～ 100 Si/m for T > 7000 K ～ (c) Conductivity above 1, 000 Si/m T > 9000 K ～ Ohmic Dissipation - Stability, constriction The conductivity σ increases rapidly with T in the fully Coulomb-dominated range, 0.0153 T 3 2 e σ = (T e in K, σ in Si/m). ln Λ Notice also how, in this limit (which occurs at high temperature, as α approaches 1), the conductivity becomes independent of the kind of gas in question, except for small influences hidden in ln Λ . 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 6 of 12 V One important consequence of σ = σ ( Τ ) is the tendency for current to concentrate into “filaments”, or “arcs”. To understand this, consider the amount of work done by electric forces to overcome the “friction” on the electrons due to collisions. The force G on the n e electrons in a unit volume is-en E, and these electrons reach a terminal JG e velocity V e as they slide against friction. Hence the power dissipated per unit volume is GJG JG G D OH =-en E.V =- ( en V ) . E e e e e GG = j . E (Ohmic dissipation) (20)D OH G G Since Ohm’s law gives j= σ E , we can put j 2 = σ Ε 2 or D OH = (21)D OH σ The simplest situation is one with an initially uniform plasma subject G to a constant applied field E, such as would occur between the plates of a plane capacitor: Regardless of the path taken by the current, if the plates are large and the gap is small, the field E= remains unchanged. If we d now look at = σ Ε 2 ,D OH we see that the dissipation becomes large wherever the conductivity (hence the temperature) is large. Starting from uniform temperature, if a small non-uniformity arises such that T is higher along a certain path, that path becomes more conductive, heats up due to extra Ohmic dissipation, and this reinforces the initial nonuniformity. The result is a constriction of the current into a filament or “arc”. In principle, the constriction process would continue indefinitely and lead to arcs of zero radius and infinite current density. But as the temperature profile steepens, heat will increasingly diffuse away from the hot core to the cooler surroundings, and, provided it can be removed efficiently from there, an equilibrium is eventually reached at some finite arc radius and arc core temperature. Clearly, the detailed end result will depend on the details of the thermal management of the gas: the more efficient the cooling of the background, the more the constriction can progress, and the hotter the eventual arc. This counter-intuitive result (more cooling leads to hotter arcs) is one of several paradoxical properties of arcs, all of them related to their being the result of a statically unstable situation. We analyze this behavior next. 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 7 of 12 Sample Physical Properties of High-temperature Gases (Near Equilibrium) Figure 1: Equilibrium Composition of Nitrogen at P=1 atm. Note: N 2 replaced by N at ~ 7,000K, then by N + at ~ 14,000K and then by N ++ at ~ 29,000K. Electron density satisfies n e = n + + 2n ++ For Hydrogen, similar, but all transitions at lower temperature (and, of course, there is no H ++ ) 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 8 of 12 Figure 2, 3: Electrical Conductivity of Nitrogen and Oxygen gas type Note: Weak dependence on Pressure Units are mho/cm ≡ Si/cm. 1 Si/cm = 100 Si/m 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 9 of 12 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 10 of 12 Figure 4, 5: Thermal Conductivity of Nitrogen Note: The molecular contribution Km is due to the transport of energy by the random thermal motion of N 2 molecules. It increases weakly with T when there are molecules, but, of course it disappears when they dissociate (T ≥ 7000K). Similar physics applies to the atoms contribution K a , the ion contribution K i and the electron contribution, K e (these rise rapidly at first, when these species first appear, then drop rapidly out when they in turn dissociate). The most striking new feature are K D (the “dissociation contribution”), and K I (the “ionization contribution”). These are akin to heat-pipe effects, and they appear in temperature ranges where dissociation (or ionization) are very sensitive to T (around 7000K for K D , around 15,000K for K I ). The mechanism is as follows: When there is a temperature gradient, molecules are dissociating at a higher rate 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 11 of 12 than they are forming by recombination in high-T sections, while the reverse happens in low-T parts. The dissociation products are continuously diffusing from hot to cold regions. When they are created (high T) they absorb the heat of dissociation (very large) and they deliver it when they recombine (at low T). This creates a net (strong) heat transport from high to low T, and hence a thermal conductivity K D . The same description applies to K I , except the heat transported is now the ionization energy. Note the very strong K D impact in the natural scale Figure (5). Figure 6: Thermal Conductivity of Hydrogen Qualitatively similar to Nitrogen, the K I component is here even more evident (around 16,000 K). 16.522, Space Propulsion Lecture 10 Prof. Manuel Martinez-Sanchez Page 12 of 12