16.522, Space Propulsion Prof. manuel martinez-sanchez Lecture 11-12: sIMPlIFiED ANalysIS OF ARcJET OPeraTion 1. Introduction These notes aim at providing order-of-magnitude results and at illuminating the mechanisms involved. Numerical precision will be sacrificed in the interest of physical clarity. We look first at the arc in a cooled constrictor, with no flow, expand the analysi to the case with flow, and then use the results to extract performance parameters for arcjets 2. Basic Physical Assumptions The gas conductivity model will be of the form (r)(a≈08s/m/K) (1) The termal conductivity k of the gas will be modelled as a constant (with possibly a different value outside the arc). This is a fairly drastic simplification, since in H2 and N2 k(T) exhibits very large peaks in the dissociation range(2000-5000K) and in the ionization range(12000-16000K). Because k always multiplies a temperature gradient the combination d(r)= kdT is relevant, and so the proper choice of k to be used is the k= kdT over the range of temperatures intended The arc gas is modelled as ideal, even though its molecular mass shifts strongly and its enthalpy increases rapidly in the dissociation and ionization ranges. In particular has strong peaks, similar to those of k(t), and, once again, we should use temperature-averaged values for it The arc is assumed quasi-cylindrical, with axial symmetry and with gradients which are much stronger in the radial than in the axial direction (similar to boundary layers). The flow region comprises three sub-domains (a) The arc itself, for rT. This is the only part carrying current (b) The outer gas, not ionized and with T<T 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanch Page 1 of 18
16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 11-12: SIMPLIFIED ANALYSIS OF ARCJET OPERATION 1. Introduction These notes aim at providing order-of-magnitude results and at illuminating the mechanisms involved. Numerical precision will be sacrificed in the interest of physical clarity. We look first at the arc in a cooled constrictor, with no flow, expand the analysis to the case with flow, and then use the results to extract performance parameters for arcjets. 2. Basic Physical Assumptions The gas conductivity model will be of the form σ = o a T( ) − Te ⎧ ⎨ ⎩ T Te ( ) (Te ≈ 6000 − 7000K) (a ≈ 0.8Si/m / K) (1) The termal conductivity k of the gas will be modelled as a constant (with possibly a different value outside the arc). This is a fairly drastic simplification, since in H2 and N2 k(T) exhibits very large peaks in the dissociation range (2000-5000K) and in the ionization range (12000-16000K). Because k always multiplies a temperature gradient, the combination dΦ( ) T = kdT is relevant, and so the proper choice of k to be used is the averaged value k = 1 T2 − T1 kdT T1 T2 ∫ (2) over the range of temperatures intended. The arc gas is modelled as ideal, even though its molecular mass shifts strongly and its enthalpy increases rapidly in the dissociation and ionization ranges. In particular, cp = ∂h ∂T ⎛ ⎝ ⎞ ⎠ p has strong peaks, similar to those of k(T), and, once again, we should use temperature-averaged values for it. The arc is assumed quasi-cylindrical, with axial symmetry and with gradients which are much stronger in the radial than in the axial direction (similar to boundary layers). The flow region comprises three sub-domains: (a) The arc itself, for r Te . This is the only part carrying current. (b) The outer gas, not ionized and with T < Te . 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 1 of 18
(c) For the case with coaxial flow, a thin transition layer between(a)and (b) may be necessary for accuracy, but will be ignored in our analysis 3. Constricted Arc With No flow The typical arrangement is a strongly water-cooled cylindrical enclosure, made of mutually insulated copper segments, with the arc burning along its centerline(Fig. 1) Copper Insulators Water Cooling R P Arc Buffer Gas elelelelelelo Constricted Arc Fig. 1. Constricted Arc Except for the near-electrode regions, the arc properties are constant along its length. In a cross-section, the axial electric field E= E is independent of radius as well, and er small. The Ohmic dissipation rate is j E per unit volume, or oE, since j=OE. Here o varies strongly inside the arc, from zero at rRa to a maximum o at the centerline; as a rough approximation, we take -o as a representative average, and so the amount of heat deposited ohmically per unit length is zR E. This heat must be conducted to 16.522, Space Propulsion Lecture 11-12
(c) For the case with coaxial flow, a thin transition layer between (a) and (b) may be necessary for accuracy, but will be ignored in our analysis. 3. Constricted Arc With No Flow The typical arrangement is a strongly water-cooled cylindrical enclosure, made of mutually insulated copper segments, with the arc burning along its centerline (Fig. 1). Fig. 1. Constricted Arc Except for the near-electrode regions, the arc properties are constant along its length. In a cross-section, the axial electric field E = Ex is independent of radius as well, and Er is small. The Ohmic dissipation rate is r j . r E per unit volume, or σE2 , since . Here r j = σ r E σ varies strongly inside the arc, from zero at r=Ra to a maximum σ c at the centerline; as a rough approximation, we take 1 2 σ c as a representative average, and so the amount of heat deposited ohmically per unit length is 1 2 πRa 2 σ cE2 . This heat must be conducted to 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 2 of 18
the arc s periphery, and so it must equal(2rR, & n. Representing the temperature gradient Ra by(roughly)(- we obtain Ra 加2aE2=2tRk2-C R E=2 k(T-7)1 R and since o=aT-T) E=2 k R This important result indicates that the arc field, and hence its voltage, is inversely proportionally radius: the dissipation must increase if the arc is constrained more tightly, which improves its cooling. But note that Ra itself is not yet known, since it is only R, the constrictor diameter that is prescribed The total arc current is 1=[2mm(oEr. Once again, using o=0,we obtain I= TR--E (5) and substituting(4)here, 1=z2(C a Ra =x√2ak(z-7)R (6) Note also that, multiplying(4 )and (6) together we obtain EI=4mk (T-T which is another way to express the heat balance. Its main message is that the arc centerline temperature Tc increases linearly with arc power per unit length, El 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Mar Page 3 of 18
the arc’s periphery, and so it must equal 2πRa ( ) k ∂T ∂r ⎛ ⎝ ⎞ ⎠ r = Ra . Representing the temperature gradient Ra by (roughly) − ∂T ∂r ⎛ ⎝ ⎞ ⎠ Ra ≅ 2 Tc − Te Ra , we obtain π / Ra 2 / 1 2 σ cE2 = 2π /Rakc 2 Tc − Te Ra or E = 2 2kc (Tc − Te ) σ c 1 Ra (3) and since σc = a Tc − Te ( ), E = 2 2 kc a 1 Ra (4) This important result indicates that the arc field, and hence its voltage, is inversely proportionally to its radius: the dissipation must increase if the arc is constrained more tightly, which improves its cooling. But note that Ra itself is not yet known, since it is only R, the constrictor diameter that is prescribed. The total arc current is I = 2πr(σE) . Once again, using o R ∫ dr σ ≅ 1 2 σ c , we obtain I = πRa 2 σ c 2 E (5) and substituting (4) here, I = πRa 2 a T( ) c − Te 2 2 2 ka a 1 Ra I = π 2akc Tc − Te ( )Ra (6) Note also that, multiplying (4) and (6) together we obtain EI = 4πkc Tc − Te ( ) (7) which is another way to express the heat balance. Its main message is that the arc centerline temperature Tc increases linearly with arc power per unit length, EI. 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 3 of 18
Everything covered so far will also apply later to the arc in a flow. The difference is in how the heat is evacuated from the arc periphery. With no flow, this must be accomplished by heat conduction through the buffer gas. If kout denotes its thermal conductivity(probably much lower than ke), and if we ignore cylindrical effects, we must have equality of heat flux(per unit area) on both sides of the arc's edg 2 R R-R where Tw is the temperature of the constrictors wall, controlled externally. Substituting Te-Te from(8)into(6), I=R2ak xR(=) This is a quadratic equation for the arc radius Ra. To simplify algebra, introduce non- dimensional quantities y=nR√2k(-7) 2k (10) R d I=2 (11) Solving for ra, (12 which approaches I from below as I becomes large. We can now obtain other quantities of interest. From( 8), 2 T-T k r-r 16.522 Space Propulsion Lecture 11-12
Everything covered so far will also apply later to the arc in a flow. The difference is in how the heat is evacuated from the arc periphery. With no flow, this must be accomplished by heat conduction through the buffer gas. If kout denotes its thermal conductivity (probably much lower than kc), and if we ignore cylindrical effects, we must have equality of heat flux (per unit area) on both sides of the arc’s edge: kc2 Tc − Te Ra = kout Te − Tw R − Ra (8) where Tw is the temperature of the constrictor’s wall, controlled externally. Substituting Tc-Te from (8) into (6), I = πRa 2ak ( )c 1 2 kout kc Ra R − Ra Te − Tw ( ) This is a quadratic equation for the arc radius Ra. To simplify algebra, introduce nondimensional quantities: I * = I Iref ; Iref = πR 2akc Te − Tw ( ) (9) λ = kout 2kc (10) and ra = Ra R (11) and so I * = λ ra 2 1 − ra (11) Solving for ra, ra = 2 1 + 1+ 4λ I * (12) which approaches 1 from below as I * becomes large. We can now obtain other quantities of interest. From (8), Tc − Te Te − Tw = 1 2 kout kc Ra R− Ra = λ r a 1− ra = 2λ 1 + 4λ I * −1 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 4 of 18
or, rearran 4 1+J1+ 2 which shows how Tc eventually increases linearly with I, but its variation is faster at low current The field itself follows from(4). Define a non-dimensional field E'= E (14) E and then 2 which indicates a decreasing field(and voltage )as the current increases These results are summarized in Fig. 2. calculated with 1=1/4 The negative slope of the line E'=f() is typical of arc discharges, and creates some difficulties in their operation. We note first that the increase ofe as I decreases does not continue indefinitely; below some current level, the thermal power input to the electrodes(particularly the cathode) is insufficient to sustain the electron emission required, and the discharge transitions to a different mode, probably an"anomalous glow discharge” 16.522,S Propulsion Lecture 11-12 Prof. Manuel mar Page 5 of 18
or, rearranging, Tc − Te Te − Tw = 1+ 1 + 4λ I * 2 I * (13) which shows how Tc eventually increases linearly with I * , but its variation is faster ≈ I * ( ) at low current. The field itself follows from (4). Define a non-dimensional field E* = E Eref ; Eref = 2 2kc a 1 R (14) and then E* = 1 ra = 1+ 1 + 4λ I * 2 (15) which indicates a decreasing field (and voltage) as the current increases. These results are summarized in Fig. 2, calculated with λ =1/ 4. The negative slope of the line E* = f I * ( ) is typical of arc discharges, and creates some difficulties in their operation. We note first that the increase of E* as I* decreases does not continue indefinitely; below some current level, the thermal power input to the electrodes (particularly the cathode) is insufficient to sustain the electron emission required, and the discharge transitions to a different mode, probably an “anomalous glow discharge”. 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 5 of 18
Tc-Te 1.6 1.4 0.8 0.6 0.5 15 2.5 Constricted Arc, with Kout/kc=1/2(=1/4) 2 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 6 of 18
Figure 2 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 6 of 18
This happens at current levels in the micro-to milliampere, and the complete V-I curve of an arc then appears as in Fig. 3 Slope Ballast Resistance Battery Line(No Ballast) P Arc Line Battery ballast Line Figure 3. Complete V-I Arc Characteristic Assuming we wish to operate at point P on this line, if we simply connect the arc to a constant-voltage source, such as a battery, we obtain an unstable arrangement. This is because, if current, say, increases slightly above Ip, the arc would now demand less than the equilibirum voltage Vp which the source delivers, and so I would increae even (to the supply limit, or to current fluctuation would cause a rapid snap back to the stable operating point Q(at ver One solution is to insert a series resistance RB(ballast )and increase th open-circuit voltage to, say VB (Fig 3). Since the voltage available to the arc is now VR-RBI, the supply line cuts the arc line form above(Fig 3), and repeating the argument we notice stable operation at P. But, of course, we dissipate in the ballast the L (a-v)which leads to inefficient operation. Notice, however, that AC arcs can be efficiently ballasted by a series inductor, the familiar"choke in fluorescent fixtures Alternatively, one can use a current-regulated power supply, provided its regulating speed and authority are high enough. For space applications, this takes the form of high frequency solid state switching regulators, capable of stabilizing the arc with minimal losses(efficiency>90%). A series inductor can help with the high frequency part of the fluctuation spectrum 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez
This happens at current levels in the micro- to milliampere, and the complete V-I curve of an arc then appears as in Fig. 3: Figure 3. Complete V-I Arc Characteristic Assuming we wish to operate at point P on this line, if we simply connect the arc to a constant-voltage source, such as a battery, we obtain an unstable arrangement. This is because, if current, say, increases slightly above Ip, the arc would now demand less than the equilibirum voltage Vp which the source delivers, and so I would increae even further, and run away (to the supply limit, or to destruction). Conversely, any negative current fluctuation would cause a rapid snap back to the stable operating point Q (at very low current). One solution is to insert a series resistance RB (“ballast”) and increase the source open-circuit voltage to, say VB (Fig. 3). Since the voltage available to the arc is now VB − RBI , the supply line cuts the arc line form above (Fig. 3), and repeating the argument we notice stable operation at P. But, of course, we dissipate in the ballast the power I p(VB − Vp ), which leads to inefficient operation. Notice, however, that AC arcs can be efficiently ballasted by a series inductor, the familiar “choke” in fluorescent fixtures. Alternatively, one can use a current-regulated power supply, provided its regulating speed and authority are high enough. For space applications, this takes the form of high frequency solid state switching regulators, capable of stabilizing the arc with minimal losses (efficiency>90%). A series inductor can help with the high frequency part of the fluctuation spectrum. 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 7 of 18
4. Constricted Arc in a Parallel Flow This is the desired configuration for an arcjet (Fig. 4 ). Here, the heat which arrives through internal conduction at the arc periphery is removed by convection in the outer flow. The gas which comes into contact with the arc is heated as it moves axially and some of it reaches the ionization threshold Te and becomes part of the arc itself. This means the arc grows with distance, just as the thermal boundary layer adjacent to a heated surface does Swirling Propellant Flow Arc Column Anodic Arc Root Cathode Cathodic Arc Root Constricted Arc in Arcjet Fig. 4. Constricted Arc in Arcjet Let cpu), be the axial mass flux at the arc's edge(r=Ra). The amount of" new arc flow in a distance dx is then (ou) 2mR dr dx. The energy absorbed in heating this gas from Tout( the buffer gas temperature)to Te must be supplied by the arc power dissipation. Using(7), we then have (o)2c2(T-Tm)=4(T-7) (16) In order to estimate ou), we now make the further assumption that the quantity puis independent of r at a given x. This is motivated by the observation that in a parallel 16.522, Space Propulsion Lecture 11-12
4. Constricted Arc in a Parallel Flow This is the desired configuration for an arcjet (Fig. 4). Here, the heat which arrives through internal conduction at the arc periphery is removed by convection in the outer flow. The gas which comes into contact with the arc is heated as it moves axially, and some of it reaches the ionization threshold Te and becomes part of the arc itself. This means the arc grows with distance, just as the thermal boundary layer adjacent to a heated surface does. Fig. 4. Constricted Arc in Arcjet Let (ρu)e be the axial mass flux at the arc’s edge (r=Ra). The amount of “new” arc flow in a distance dx is then ( ) ρu e 2πRa dRa dx dx . The energy absorbed in heating this gas from Tout (the buffer gas temperature) to Te must be supplied by the arc power dissipation. Using (7), we then have ( ) ρu e 2πRa dRa dx cp Te − Tout ( )= 4πkc Tc − Te ( ) (16) In order to estimate (ρu)e , we now make the further assumption that the quantity ρu 2 is independent of r at a given x. This is motivated by the observation that in a parallel, 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 8 of 18
inviscid flow, P+pu remains constant along the streamlines, while p is itself independent of r. Thus pu develops gradually at the expense of p, and this should lead to a radius-independent pu. Our flow is not exactly parallel or inviscid, but the approximation(verified from numerical 2-D solutions) is good enough for the present poses We then have (17) RT showing that most of the mass flow must occur in the cool, outside gas, since the numerator in(17)is independent of r We then have (18) and the problem is now to calculate the gas flux cpu) in the buffer gas. The simplest possible approximation is to state that all of the gas flow is carried by this uniform buffer (19) (R2-R2 elds x(2-R)7 and, substituting into(16) -R)78a(-)=2(=Z) The quantity Tc-Te depends on current and arc radius through Eq (6) Substituting, 16.522 Space Propulsion Lecture 11-12
inviscid flow, p + ρu 2 remains constant along the streamlines, while p is itself independent of r. Thus ρu 2 develops gradually at the expense of p, and this should lead to a radius-independent ρu 2 . Our flow is not exactly parallel or inviscid, but the approximation (verified from numerical 2-D solutions) is good enough for the present purposes. We then have, ρu = p ρu 2 ( ) = p ρu2 ( ) RgT (17) showing that most of the mass flow must occur in the cool, outside gas, since the numerator in (17) is independent of r. We then have, ( ) ρu e ( ) ρu out = Tout Te (18) and the problem is now to calculate the gas flux (ρu)out in the buffer gas. The simplest possible approximation is to state that all of the gas flow is carried by this uniform buffer flow: ( ) ρu out ≅ m Ý π R2 − Ra 2 ( ) (19) This yields ( ) ρu e ≅ m Ý π R2 − Ra 2 ( ) Tou t Te (20) and, substituting into (16), m Ý π R2 − Ra 2 ( ) Tout Te Ra dRa dx cp Te − Tout ( ) ≅ 2kc Tc − Te ( ) The quantity Tc-Te depends on current and arc radius through Eq. (6). Substituting, 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 9 of 18
R dR R2) (-T)=k lake r 亟, (21) R2 dx me, (te We again non-dimensionalize using /, r as in (9),(11), plus a non-dimensional distance I nc (22) with the result Since T, is much greater than either T or Tu(which are similar) the temperature function on the right of (23)is close to l, and will be ignored. Integrating(23),from l(o)=ra gi which is an inverse expression for radius vS distance For r,a < I the bracketed quantity in(24)is approximated by Taylor expansion as which indicates rapid arc growth near its upstream(cathodic)end. Therefore it is allowable to use ro =0 in(24), with only minor effect on length The remaining important question is the determination of the pressure(either P the total pressure from upstream, or the pressure Pce at the constrictor exit)required for a flow n, given the current I and geometrical data. For this, recall the assumption that the outside es all the flow, and is undisturbed, because the arc heat has not yet penetrated to it. It then is a subsonic ideal gas flow in a contracting area z(R-R),and will reach sonic conditions at the constrictor exit, provided the initial nozzle divergence 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 10 of 18
m Ý π R2 − Ra 2 ( ) Tou t Te Ra dRa dx cp Te − Tout ( ) ≅ 2kc I π 2akc Ra or Ra 2 R2 − Ra 2 dRa dx = I 2kc a Te Tout m Ý cp (Te − Tout) (21) We again non-dimensionalize using as in (9), (11), plus a non-dimensional distance I * ,r a x* = x xref ; xref = 1 2π m Ýcp kc Tout Te (22) with the result r a 2 1 − r a 2 dra dx* = Te − Tw Te − Tout Tw Tout ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ I * (23) Since Te is much greater than either Tw or Tout (which are similar) the temperature function on the right of (23) is close to 1, and will be ignored. Integrating (23), from r gives a( ) o = rao x * = 1 I * [ln 1 + r a 1 − r a −ra ]rao ra (24) which is an inverse expression for radius vs. distance. For ra << 1 the bracketed quantity in (24) is approximated by Taylor expansion as ra 3 3 which indicates rapid arc growth near its upstream (cathodic) end. Therefore it is allowable to use rao ≅ o in (24), with only minor effect on length. The remaining important question is the determination of the pressure (either P, the total pressure from upstream, or the pressure Pce at the constrictor exit) required for a flow m Ý, given the current I and geometrical data. For this, recall the assumption that the outside layer carries all the flow, and is undisturbed, because the arc heat has not yet penetrated to it. It then is a subsonic ideal gas flow in a contracting area π R2 − Ra 2 ( ), and will reach sonic conditions at the constrictor exit, provided the initial nozzle divergence 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 10 of 18