layer thickness, is of order unity. Since we started out by assuming conditions when these layers are thin i.e. le, lo< <L, we can now state that this will occur when Rm(L)=cHou L>>1 (20) This is indeed the condition for operation in the pure mPd regime. Effects of Dissipation The high-current inlet and exit layers are very dissipative. Their resistances can be estimated as (21) owo (the 4/3 factor accounts for the"triangular"current distribution in the layers)and so the ohmically dissipated power is D=IRo+IR。 (22 ere Io B1) d I=I-5=/3 Substituting, we find 1 Dn=2|1- 1 H aHui √3(H)aI W3|√3-1mw =33wm D.=n13帅=41邮 and, in total D √3 (25) Part of this dissipation goes to heating the gas, but the major portion is used in ionizing and exciting(followed by radiation) the gaseous atoms. Let ev=2 to 3 16.522, Space P pessan Lecture 22 Prof. Manuel martinez Page 5 of 816.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 5 of 8 layer thickness, is of order unity. Since we started out by assuming conditions when these layers are thin, i.e., le, l0<<L, we can now state that this will occur when R L u L >> 1 m 0e ( ) ≡ σµ (20) This is indeed the condition for operation in the pure MPD regime. Effects of Dissipation The high-current inlet and exit layers are very dissipative. Their resistances can be estimated as 0 0 4 H 3 R = σwl ; e e 4 H 3 R = σwl (21) (the 4/3 factor accounts for the “triangular” current distribution in the layers) and so the Ohmically dissipated power is 2 2 D=IR +IR 00 ee (22) where ( ) 0 0 01 0 0 w 11 wB I = B -B = 1- =I 1- 3 3 ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ µ µ ⎝ ⎠⎝ ⎠ (23) and e 0 I =I-I =I 3 (24) Substituting, we find ( ) 2 22 24 2 2 0 0 0 1 1 - 14 4 H HI I 3 D =I 1- H = 3 33 3 3w w3 3 - 1 m w m ⎛ ⎞ σ µ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ σ µ i i 22 24 2 2 0 0 e 4 H 1 41 H 3 HI I D =I = 3 w 39 w 3wm m σ µ ⎛ ⎞ ⎜ ⎟ σ ⎝ ⎠ µ i i and, in total, 2 2 4 0 4 H I D = 9 3 w m ⎛ ⎞ µ ⎜ ⎟ ⎝ ⎠ i (25) Part of this dissipation goes to heating the gas, but the major portion is used in ionizing and exciting (followed by radiation) the gaseous atoms. Let ' i eV 2 to 3