Analysis of Magnetic Nozzles for Space Plasma Thrusters

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Analysis of Magnetic Nozzles for Space Plasma Thrusters Universidad Polite´cnica de Madrid Escuela Tecnica´ Superior de Ingenieros Aeronauticos´ Analysis of Magnetic Nozzles For Space Plasma Thrusters An´alisisde Toberas Magn´eticas para Motores Espaciales de Plasma Tesis Doctoral Mario Merino Mart´ınez Ingeniero Aeron´autico Madrid, junio 2013 Departamento de Fundamentos Matem´aticos de la Tecnolog´ıa Aeron´autica Escuela T´ecnica Superior de Ingenieros Aeron´auticos Analysis of Magnetic Nozzles For Space Plasma Thrusters An´alisisde Toberas Magn´eticas para Motores Espaciales de Plasma Tesis Doctoral Autor: Mario Merino Mart´ınez Ingeniero Aeron´autico Director: Eduardo Ahedo Galilea Doctor Ingeniero Aeron´autico Madrid, junio 2013 Tribunal nombrado por el Sr. Rector Magfco. de la Universidad Polit´ecnica de Madrid, el d´ıa ...... de ............... de 2013. Presidente: Francisco Jos´eHiguera Ant´on Vocal: Manuel Mart´ınez S´anchez Vocal: St´ephane Mazouffre Vocal: Davar Feili Secretario: Luis Ra´ul S´anchez Fern´andez Suplente: Juan Arturo Alonso de Pablo Suplente: Claudio Bombardelli Realizado el acto de defensa y lectura de la Tesis el d´ıa 19 de julio de 2013 en la E.T.S.I. Aeron´auticos. Calificaci´on ................................................... EL PRESIDENTE LOS VOCALES EL SECRETARIO Abstract This Thesis presents a theoretical analysis of the operation of magnetic nozzles for plasma space propulsion. The study is based on a two-dimensional, two-fluid model of the supersonic expansion of a hot plasma in a divergent magnetic field. The basic model is extended progressively to include the dominant electron convective terms, the plasma-induced magnetic field, multi-temperature electron populations, and the capability to integrate the plasma flow in the far expansion region. The hyperbolic plasma response is integrated accurately and efficiently with the method of the characteristic lines. The 2D plasma expansion is characterized parametrically in terms of the ion magnetization strength, the magnetic field geometry, and the initial plasma profile. Acceleration mechanisms are investigated, showing that the ambipolar electric field converts the internal electron energy into directed ion energy. The diamagnetic electron Hall current, which can be distributed in the plasma volume or localized in a thin current sheet at the jet edge, is shown to be central for the operation of the magnetic nozzle. The repelling magnetic force on this current is responsible for the radial confinement and axial acceleration of the plasma, and magnetic thrust is the reaction to this force on the magnetic coils of the thruster. The plasma response exhibits a gradual inward separation of the ion streamtubes from the magnetic streamtubes, which focuses the jet about the nozzle axis, gives rise to the formation of longitudinal currents and sets the plasma into rotation. The obtained thrust gain in the magnetic nozzle and radial plasma losses are evaluated as a function of the design parameters. The downstream plasma detachment from the closed magnetic field lines, re- quired for the propulsive application of the magnetic nozzle, is investigated in de- tail. Three prevailing detachment theories for magnetic nozzles, relying on plasma i ii ABSTRACT resistivity, electron inertia, and the plasma-induced magnetic field, are shown to be inadequate for the propulsive magnetic nozzle, as these mechanisms detach the plume outward, increasing its divergence, rather than focusing it as desired. Instead, plasma detachment is shown to occur essentially due to ion inertia and the gradual demagnetization that takes place downstream, which enable the unbounded inward ion separation from the magnetic lines beyond the turning point of the outermost plasma streamline under rather general conditions. The plasma fraction that remains attached to the field and turns around along the magnetic field back to the thruster is evaluated and shown to be marginal. The plasma- induced magnetic field is shown to increase the divergence of the nozzle and the resulting plasma plume in the propulsive case, and to enhance the demagnetization of the central part of the plasma jet, contrary to existing predictions. The increased demagnetization favors the earlier ion inward separation from the magnetic field. The local current ambipolarity assumption, common to many existing magnetic nozzle models, is critically discussed, showing that it is unsuitable for the study of plasma detachment. A grave mathematical inconsistency in a well-accepted model, related to the acceptance of this assumption, is found out and commented on. The formation and 2D shape of electric double layers in the plasma expan- sion is studied with the inclusion of an additional suprathermal electron popu- lation in the model. When a double layer forms, its curvature is shown to in- crease the more peripherally suprathermal electrons are injected, the lower the magnetic field strength, and the more divergent the magnetic nozzle is. The two- electron-temperature plasma is seen to have a greater magnetic-to-total thrust ratio. Notwithstanding, no propulsive advantage of the double layer is found, sup- porting and reinforcing previous critiques to their proposal as a thrust mechanism. Finally, a general framework of self-similar models of a 2D unmagnetized plasma plume expansion into vacuum is presented and discussed. The error as- sociated with the self-similarity assumption is calculated and shown to be small for hypersonic plasma plumes. Three models of the literature are recovered as particularizations from the general framework and compared. iii Resumen (Espa˜nol) Esta tesis presenta un an´alisis te´orico del funcionamiento de toberas magn´eticas para la propulsi´on espacial por plasmas. El estudio est´abasado en un modelo bi- dimensional y bi-fluido de la expansi´on supers´onica de un plasma caliente en un campo magn´etico divergente. El modelo b´asico es ampliado progresivamente con la inclusi´on de t´erminos convectivos dominantes de electrones, el campo magn´etico inducido por el plasma, poblaciones electr´onicas m´ultiples a distintas temperatu- ras, y la capacidad de integrar el flujo en la regi´on de expansi´on lejana. La respuesta hiperb´olica del plasma es integrada con alta precisi´on y eficiencia haciendo uso del m´etodo de las l´ıneas caracter´ısticas. Se realiza una caracterizaci´on param´etrica de la expansi´on 2D del plasma en t´erminos del grado de magnetizaci´on de iones, la geometr´ıa del campo magn´etico, y el perfil inicial del plasma. Se investigan los mecanismos de aceleraci´on, mostran- do que el campo ambipolar convierte la energ´ıa interna de electrones en energ´ıa dirigida de iones. Las corrientes diamagn´eticas de Hall, que pueden hallarse distri- buidas en el volumen del plasma o localizadas en una delgada capa de corriente en el borde del chorro, son esenciales para la operaci´on de la tobera, ya que la fuerza magn´etica repulsiva sobre ellas es la encargada de confinar radialmente y acelerar axialmente el plasma. El empuje magn´etico es la reacci´on a esta fuerza sobre el motor. La respuesta del plasma muestra la separaci´on gradual hacia adentro de los tubos de iones respecto de los magn´eticos, lo cual produce la formaci´on de corrien- tes el´ectricas longitudinales y pone el plasma en rotaci´on. La ganancia de empuje obtenida y las p´erdidas radiales de la pluma de plasma se eval´uan en funci´on de los par´ametros de dise˜no. Se analiza en detalle la separaci´on magn´etica del plasma aguas abajo respecto a las l´ıneas magn´eticas (cerradas sobre s´ımismas), necesaria para la aplicaci´on de la tobera magn´etica a fines propulsivos. Se demuestra que tres teor´ıas existentes sobre separaci´on, que se fundamentan en la resistividad del plasma, la inercia de electrones, y el campo magn´etico que induce el plasma, son inadecuadas para la tobera magn´etica propulsiva, ya que producen separaci´on hacia afuera en lugar de hacia adentro, aumentando la divergencia de la pluma. En su lugar, se muestra que la separaci´on del plasma tiene lugar gracias a la iv ABSTRACT inercia de iones y la desmagnetizaci´on gradual del plasma que tiene lugar aguas abajo, que permiten la separaci´on ilimitada del flujo de iones respecto a las l´ıneas de campo en condiciones muy generales. Se eval´ua la cantidad de plasma que permanece unida al campo magn´etico y retorna hacia el motor a lo largo de las l´ıneas cerradas de campo, mostrando que es marginal. Se muestra c´omo el campo magn´etico inducido por el plasma incrementa la divergencia de la tobera magn´eti- ca y por ende de la pluma de plasma en el caso propulsivo, contrariamente a las predicciones existentes. Se muestra tambi´enc´omo el inducido favorece la desmag- netizaci´on del n´ucleo del chorro, acelerando la separaci´on magn´etica. La hip´otesis de ambipolaridad de corriente local, com´un a varios modelos de tobera magn´etica existentes, es discutida cr´ıticamente, mostrando que es inade- cuada para el estudio de la separaci´on de plasma. Una inconsistencia grave en la derivaci´on matem´atica de uno de los modelos m´as aceptados es se˜nalada y comen- tada. Incluyendo una especie adicional de electrones suprat´ermicos en el modelo, se estudia la formaci´on y geometr´ıa de dobles capas el´ectricas en el interior del plasma. Cuando dicha capa se forma, su curvatura aumenta cuanto m´as perif´ericamente se inyecten los electrones suprat´ermicos, cuanto menor sea el campo magn´etico, y cuanto m´as divergente sea la tobera magn´etica. El plasma con dos temperaturas electr´onicas posee un mayor ratio de empuje magn´etico frente a total. A pesar de ello, no se encuentra ninguna ventaja propulsiva de las dobles capas, reforzando las cr´ıticas existentes frente a las propuestas de estas formaciones como un mecanismo de empuje.
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