lecture11_12

lecture11_12 - 16.522, Space Propulsion Prof. Manuel...

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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 aT T e () T < T e ( ) T > T e T e 6000 7000 K ( ) a 0.8 Si / 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 H 2 and N 2 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 is relevant, and so the proper choice of Φ T ()= kdT k to be used is the averaged value k = 1 T 2 T 1 kdT T 1 T 2 ( 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, c p = 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 , corresponding to < R a x T > T e . This is the only part carrying current. (b) The outer gas, not ionized and with T < T e . 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 1 of 18
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(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 = E x is independent of radius as well, and E r is small. The Ohmic dissipation rate is r j . r E per unit volume, or σ E 2 , since . Here r j = r E varies strongly inside the arc, from zero at r=R a to a maximum c at the centerline; as a rough approximation, we take 1 2 c as a representative average, and so the amount of heat deposited ohmically per unit length is 1 2 π R a 2 c E 2 . This heat must be conducted to 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 2 of 18
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the arc’s periphery, and so it must equal 2 π R a () k T r r = R a . Representing the temperature gradient R a by (roughly) T r R a 2 T c T e R a , we obtain /
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This note was uploaded on 11/07/2011 for the course AERO 16.512 taught by Professor Manuelmartinez-sanchez during the Fall '05 term at MIT.

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lecture11_12 - 16.522, Space Propulsion Prof. Manuel...

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