NASA-CR-191716 // /// _o ...._) DEPARTMENT OF MATHEMATICAL SCIENCES COI.IFGE OF SCIENCES OLD DOMINION UNIVERSITY NORFOLK, VIRGINIA 23529 :c) r- j THEORETICAL STUDIES OF A MOLECULAR BEAM GENERATOR © By John H. Heinbockel, Principal Investigator Progress Report o> For the period May 16, 1992 to November 15, 1992 Prepared for National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-2199 © Under Research Grant NAG-l-1424 Dr. Sang H. Choi, Technical Monitor SSD-High Energy Science Branch (NASA-CR-191716) THEORETICAL N93-169_5 STUDIES OF A MOLECULAR BEAM GENERATOR Progress Report, 16 May - 15 Nov. 1992 (Old Oominion Univ.] Unclas 52 p G3/72 0139659 D January 1993 DEPARTMENT OF MATHEMATICAL SCIENCES COLLEGE OF SCIENCES OLD DOMINION UNIVERSITY NORFOLK, VIRGINIA 23529 THEORETICAL STUDIES OF A MOLECULAR BEAM GENERATOR By John H. Heinbockel, Principal Investigator Progress Report For the period May 16, 1992 to November 15, 1992 Prepared for National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-2199 Under Research Grant NAG-l-1424 Dr. Sang H. Choi, Technical Monitor SSD-High Energy Science Branch Submitted by the Old Dominion University Research Foundation P.O. Box 6369 Norfolk, Virginia 23508-0369 January 1993 PROGRESS REPORT RESEARCH GRANT NAG-l-1424 ODU RESEARCH FOUNDATION GRANT NUMBER 126611 THEORETICAL STUDIES OF A MOLECULAR BEAM GENERATOR MOLECULAR BEAM GENERATOR MODEL The following is a proposed baseline model that is being develope for the simulation of hydrodynamic generator, which can be converted at a later date to a magnetohydrodynamic MHD thruster by adding the necessary electric and magnetic fields. The following development will include the electric and magnetic terms, however, the initial computer program will not include these terms. The analysis that follows is for a one species, single temperature model constructed over the domain D defined by the region enclosed by ABCDEF illustrated in the figure 1. Figure 1. Geometry of thruster MHD model. CONTINUITY EQUATION The continuity equation expresses conservation of mass and is given by cOp o-7+ v .(pff) = o (1) 1 where p = p(r, O,z, t) is the density of the gas, and I_ = Vr _r + VO _o + Vz _z is the velocity. In cylindrical coordinates the equation (1) has the form Op io(,-pV,)iO(pvo) O(pv ) -=+ + + -o. (2) Ot r Or r (9t9 Oz CONSERVATION OF MOMENTUM The equation for conservation of linear momentum is given by n p---_ + p(V. V)V = ___ ffi (3) i=1 where _n=l ffi represents a summation of body forces per unit volume acting upon a control volume within the domain D. We consider initially the pressure force F1 = -VP (4) where pressure and density are related by the equation of state gas law P = p*RT, Where p* is the density in mole/m 3. i.e. p*W = p where W is the molecular weight in kg/mole. The force due to viscosity is (5) if2 = rl {v2_ + V(V. ¢)} - _V(71V. I7)+ 2(Vr/- V)V + Vr/× (V × V) where r1=1"2510-19 5 ( Mm-k_ BT)8niA1/2 (2kB_ T)2=C1T 5/2 (6) is the plasma viscosity, with o_2 A = 2(log(1 + a2) I+o12 ) a constant which depends upon the ionization factor a, (a = 1 for a fully ionized gas). The additional constants are M the ion mass (Kg), m the electron mass, e the electron charge, kB Boltzmann's constant, T is the absolute temperature, and ni = 1 for a singly ionized plasma. For oL= 0, we employ an empirical curve fit for the viscosity as a function of temperature. The magnetic force is given by g3 =fxB (7) 2 2 where f is the current density. The gravitational force is given by if4 = P_'. The electric force is given by All additional forces are represented by 12 i'6 and are neglected for the present. CONSERVATION OF ENERGY Representing the internal energy by u = CpT where Cp is the specific heat at constant pressure and T is the absolute temperature, the energy equation can be written in the form of an energy b alatlce a_s O(CpT) " P 0t + p(f" V)(CpT) = V(KTVT) + _ ¢i (8) i=1 where both the specific heat Cp and thermal conductivity KT are treated as functions of temperature T. The thermal conductivity KT of the medium is given by the Spitzer-Harris relation 4.4 10-l° T5/2 KT = (9) 23-1og[ l'22103nl/2]Ta/2 and n is the plasma number density in particles/m 3. In addition to the heat loss term the right hand side of equation (8) contains the terms ¢1=(_+ f x B). Y-0f" (I0) which represents joule heating, -87/] (0v0 v0) + + \-o-7 + 0r ) + k, Or r 2} (11) Iov + A\-6V + -7 + -_z ] 3 which represents viscous dissipation. Here r/and A are viscosity coefficients satisfying A+ _q = 0. In addition there is the radiation loss term. Various forms of the radiation term exists in the literature. As a first approximatrion we take the radiation loss term from reference 11 which can be expressed ,4 ¢3=wR v(_T 4) where )_R is the Rosseland mean free path (,kR = 1/a R where _R is the Rosseland absorption coefficient (cm-1)), and a is the Stephan Boltzman constant. The remaining terms )--_n=4 ¢i represents additional energy considerations which are initially neglected. MAXWELL'S EQUATIONS Maxwell's equations in the MKS Rational system of units can be expressed v.g=o (12) Gauss's law for magnetism v.b=p_ (13) Gauss's law for electricty (Coulomb's law) Ampere's law VxH=f+ 0/_- (14) Ot Vxg= og Faraday's law at (15) CONSTITUTIVE EQUATIONS Assuming an isotropic, homogeneous medium we adopt the constitutive equations /)=e/_ and /_=#H. OHM'S LAW Ohm's law is written in the form f_ -. ..f= o(_i+ 17xg)- _.(J x .g) (16) where f is the current density, /_ is the electric field, /_ is the induced magnetic field and a is the electric conductivity with milts of mho/m and f/is the Hall parameter given by (reference 1) £l = 9.6(1016)(Ta/2B/Zn logA) 4 4 with Coulomblogarithm givenby logA _ 23- log(1.22x 103nU2/T312) with n the plasma number density. ELECTROMAGNETIC FIELD EQUATIONS Neglecting the displacement current modifies Ampere's law to v×g=#Z (17) Assumming that the charge density is constant implies the equation of continuity of charge is V. f = 0. (Note that the divergence of equation (17) also gives this result.) From Jackson, reference 10, along with neglecting the displacement current, it is appropriate to ignor Coulomb's law as its effects are negligible. We thus obtain the electromagnetic field equations v×g=j (is) Vx._- ag at Using equation (17) in Ohm's law we solve for/_ and write = _v x g- 17x g + _'(v x fi) x g (19) where/3 = _t/tzBa and a -- 1/ap. We substitute the results from equation (19) into Faraday's law and write o.d -.-_-.-= v x (av x/3)- v x (17x/3) + v x ,8(v x/_) x B (20) and since/3 is a function of T we find ag -Vx(aVx/3)-Vx(17xB)+V/_x(Vx/3) x/3+/3Vx(Vx/3) x/3 (21) Ot 5 SUMMARY OF BASIC EQUATIONS USED FOR MODELING Continuity Op o-_+ v(p¢) = o Moment um P-k- + p(p v)_ = _._ i=1 Energy n + p('V. V)(CpT) = V(KTVT ) + _ ¢i i=1 Electromagnetic field equations og --- = v x(_ x_)- _ x(_x_)+ v xZ(vx_)x Ot This produces a system of eight simultaneous partial differential equations in the eight unknowns Br, BO, Bz ,p, Vr, Vo, Vz ,T. Throughout the calculations the following quantities can be generated in terms of the above variables. Y=ivx_ # (22) ._= o_,f- ff x g + #B(Zx g) SCALAR FORM OF FIELD EQUATIONS Assuming symmetry with respect to the 0 variable, all derivatives with respect to 0 are neglected. The following set of scalar equations then results Continuity op ia(_py_) o(pyz) --_-+ + -o r Or Oz 6 Momentum or, (v,°v" or, v_ =_(_,), P---_- + O t --'_'V -l- Vz Cgz r i=n1 n p---_- + p Vp + Vz -]- =i=1 n P---_--}-P Vr-_r +Vz'-_-z J i=1 Energy p (gp + Ta_6____ _T agp'_ a_T _-T= \ ux l I'i + _,_i aT \\-bT) + \_--T) ]+KTI_2+-/a---/+-_ 2] i=1 7 Electromagnetic field equations OBr OBz OVr 02 Br 02Bz OBr Br OVz _-gy + _O-_z + V,-N-z + -_-z- y, & Oz Oz - _ B"-0O2B_o2z + OOB_zOBoOz + B" (k0_2oB+-o_7z o--7-+-g7 \ o, + +82_[ z 02 +B,.\ Or + OB 0 02Bo 02Bo _ OB o ol OB z OVo + -_Bo- Vo_ Bz O---;- Ot c_ Oz 2 oL 0r 2 r Or OBr OVo + v____ + Bo-O-V_z_- + v,-O-gBoV + Bo-OgV-_,.- vo O--g--B, O--;- OBz "_ 2Be OBe ) +5 Bz Oz 2 OrOz + _ Oz Or r Oz +B, \ o-7_z o_2. + _ \ Oz 0_ ]J+ O_ [L,.,_OO-B_0z+ B, (OBOr z O_Bz_]] + -_z Bz Oz Or - Bo \ Or OBz 02Br 02Bz _ OBr _ OBz OBr OVz v_ 0---7--B, 0--7- Ot - _ OrOz -_ + r Oz r Or V, OBz OV, 1 + --_-r +Bz-_r +r(VrBz-VzBr) + fl Bzo-0--2_BzO + OOBrz OOBz0 + Br\ (02BOor 2 +-_ o_ +--g# k-g-, + + r _j +_ Bz--0T+B" \ 0_ + These equations are subject to certain boundary and initial conditions which are now discussed. BOUNDARY AND INITIAL CONDITIONS With reference to the figure 1, the line AF has the input conditions P = P0 = constant T = To = constant V,. = Vo =0 Vz = Vo = constant OB,. OBo OBz Oz Oz Oz 8 8