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 18m 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