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 (<T)(≈6000700 0=a(T-7)(>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 r<R,(x), corresponding to T>T. 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 ( ) 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 < R , corresponding to a (x) T > 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