Experimental Investigation and Numerical Modeling of Electric Heating Rate in a Generic Electric Propulsion System

Transcription

1 Experimental Investigation and Numerical Modeling of Electric Heating Rate in a Generic Electric Propulsion System IEPC Presented at the 32 nd International Electric Propulsion Conference, Wiesbaden, Germany September 11 15, 211 S. Reichel, R.Groll and H.J. Rath ZARM - University of Bremen, Bremen, 28359, Germany Electric propulsion systems and the behavior of dilute gases have been subjects in many studies. In this paper, a computational method for modeling the behavior an arc-jet is presented. There for the conservation equations of mass, momentum, energy and charge density are solved. The energy equation includes an additional source term depending on the electric charge density of the electric arc. The results are validated on the experimental data. Dipl.-Ing., University of Bremen, reichel@zarm.uni-bremen.de Dr.Ing., University of Bremen, groll@zarm.uni-bremen.de Prof. Dr., University of Bremen, rath@zarm.uni-bremen.de 1 of 9

2 δ = ionization rate [-] m = mass [kg] [ ] ṁ = mass flow kg s [ = specific heat capacity c p J kgk n = number density of molecules [ 1 t = time [s] ρ u p [ ] kg m 3 = density = velocity [ ] m = pressure [ s ] kg ms 2 T = temperature [K] φ = electric potential [V ] T v E e j E j σ = viscous stress tensor = total energy density [ ] kg ms 2 [ ] m 2 s 2 ] = electric field intensity [ V m = electric current [ A m 2 ] = electric conductivity [ s 3 A 2 m 3 kg Nomenclature ] ] m 3 ] q 1 = first ionization energy [ J mol σ b = Stefan-Boltzman constant W m 2 K 4 k = Boltzman constant J K h = Plank constant Js ] 2 of 9

3 I. Introduction In this paper a numerical simulation and experiments of a dilute gas flow through an arc-jet is presented. The experimental setup is placed inside a low pressure chamber. The geometry of the experimental setup is simple. The nozzle consists of an axially symmetric anode and cathode. The anode and the cathode are shaping a ring shaped gap. Fig. 1 shows the nozzle geometry of the experimental setup. Figure 1. CAD - cut of the thruster geometry The environmental pressure of the experiment during the test is in a range of 1 3 P a up to 1P a (depending on the mass-flow through the system). The electrical ignition between the anode and the cathode is realized by two different power supplies. For the first ignition a high voltage continuous current power supply with a electrical power of 2W is used. This power supply realized a electrical potential of 2V between the anode and the cathode. If the propellant is piped through the system the ignition starts and a voltage drop from 2V to less than 6V is the result. At this point a second high current power supply assumed the power supply of the experiment. During steady state operation the thruster operates with at a electric power range of 4W up to 6W, depending on the propellant. To reestimate the Temperature of the heated experiment a rough approximation was made. It is clear that the total energy of the power supply is transformed into radiation energy and gas heating energy. So we combine the Stefan-Boltzmann law P sb = σ b AT 4 and the thermodynamic gas heating term P g = ṁc p T. See Eq. (1). P power = 6W = σ b AT 4 + ṁc p T (1) For Argon and Xenon this equation has only one solution that makes sense T 186K. The other three solutions are not real ore negative. By using the Saha-Eggers Equation 15 Eq. (2) we are able to reestimate the ionization rate of the gas. δ = (2πmkT ) 3 4 h n o 2 e q 1 2kT = The result shows that the ionization of the noble gases is very low. This result is in order with the researches of Uribari 12 and Choueiri 13 who are showing, that the influence of electromagnetic field forces for thrusters with less than 1kW of power can be neglected. For this reason the influence of the current induced electro magnetic field forces on the charged ions are neglected in our following considerations. The simulations are conducted with the OpenFOAM (Version 1.7.1) simulation software. The modified solver based on the compressible solver rhocentralfoam. A detailed description of this solver is given by Greenshields 9. This solver solves the fundamental fluid dynamic conservation equations of mass, momentum and energy. To model a electric ignition between the anode and the cathode an additional electric heating therm is implemented into the source code of the energy Eq. (5). Similar source terms are described by Hash 3, Papadakis 4 and Jones 5. (2) II. The Experiments During test the electric current is limited to 3A and the voltage depends on the resistance of the ionized gas and the resistance of the anode and the cathode. The mass-flow through the system is steered by a needle 3 of 9

4 valve. For the first tests the noble gas Xenon is used as the propellant, because of the low first ionization energy of 117.4kJ/mol. During the experiments the pressure of the vacuum chamber, the pressure of the ignition chamber, the current and the voltage are measured. The results are presented below Fig 2. The plot on the left hand side shows the pressure inside the ignition chamber at the not ignited operating condition (without electric ignition). The measurement shows a pressure of 68P a inside the ignition chamber. The plot on the right hand side shows the pressure of the ignition chamber for the ignited operating condition. In this case the same mass flow is piped through the system. The current is limited to 3A. During the steady state operating condition the voltage drop of the thruster is app. 14V. This means a electrical performance of 42W. The inserted electrical performance leads to a pressure of 243P a inside the ignition chamber. Xenon - "cold" Xenon - "electric heated" 5, 4, "ignition chamber" 5, 4, "ignition chamber" voltage in V current in A 5 4 3, 2, 3, 2, 25 Pa 3 2 value 1, 68 Pa 1, Figure 2. left hand side - measurement without electrical ignition of Xenon / right hand side - measurement with electrical ignition of the Xenon Argon - "cold" Argon - "electric heated" 5, 4, "ignition chanber" 5, 4, Xe Ar "ignition chamber" voltage in V current in A 5 4 3, 2, 3, 2, 23 Pa 23 Pa 3 2 value 1, 66 Pa 1, Figure 3. left hand side not ignited measurement without electrical ignition / right hand side - measurement without electrical ignition of the Xenon - Argon mixing experiment with electrical ignition. The thruster was started with Xenon and changed over to Argon. The change of the propellant is represented by the hatched area In a second test the noble gas Argon was used as propellant. Because of the higher ionization energy of 152.6kJ/mol the flash over needs more power to establish a continuous electrical ignition between the anode and the cathode. To reduce the required electrical energy a gas mixing gadget was used. The intention of the mixing gadget is to heat up the thruster with Xenon and to take over with Argon. In this way the higher temperature of the ignition chamber relieved the establishing of the contentious Argon ignition. The result of this experiment is presented in the plot Fig of 9

5 The black line represents the voltage drop between the anode and the cathode, the green line the current flow and the red line the pressure inside the ignition chamber of the thruster. III. The Numerical Simulations A. The Numerical Solver The simulations where conducted with a modified OpenFOAM solver rhocentralfoam. This basic solver solves the conservation of mass Eq. (3), the Navier-Stokes-Equation Eq. (4) and the conservation of energy Eq. (5). The detailed equations are presented below. The general derivations of these three equation are given in Oertel 14. ρ + (ρ u) = (3) t ] [ u ρ + u u t = p T v (4) ρe e + [ u(ρe e )] = [p u] [T v u] E t j j (5) The continuity of mass Eq. (3) defines that the change of density ρ/ t inside the system is equal to the flux of mass (ρ u) over the system faces. The Navier-Stokes-Equation Eq. (4) consist of the unsteady acceleration u/ t, of the convective acceleration ( u u), of the pressure gradient ( p), and the divergence of the viscous stress tensor T v. The conservation ( of energy Eq. (5) include the total energy density E e = e + u 2 /2 and includes an additional source term Ej j). This term describes the heating of the electric arc between the anode and the cathode. For modeling of the electric arc the conductivity of the gas is calculated with Eq. (6). It is shown by Lin 6 that the electric conductivity of a gas is proportional to the absolute temperature T, the pressure of the gas p, the first ionization potential of the gas q 1 and the Boltzman constant k. σ T 3 4 p 1 2 e q 1 2kT (6) j = σ E j (7) E j = φ (8) The Ohm s Law Eq. (7) describes the relation for the electric current fluxdensity j, the electric field intensity E j and the electric conductivity σ. The electric field E j is calculated by the voltage drop between the anode and the cathode Eq. (8). The solver implements a Sutherland Transport Model 7 8 for temperature depending viscosity modeling. Standard pressure correction methods often return bad agreements of computational and measurement data with compressible hypersonic flows. Therefore a different method is used implementing own numerical procedures. The solver includes an explicit predictor equation and an implicit corrector equation for the diffusion of primitive variables instead of a pressure correction. The procedure is well-known as Kurganov s Method B. Numerical Boundary Conditions For the numerical simulations a hexagonal mesh with nearly 1, cells was used. Fig. 4 shows the ignition chamber of the thruster, the most interesting part of the mesh. The inlet is colored yellow. At this patch the mass-flow through this patch is steered by the solver. The temperature is fixed at 3K. The green patch represents the cathode. At this patch the velocity is zero. The same boundary condition is used for the blue anode. For the modulation of the electric ignition a voltage drop of.1v between the anode and the cathode is the additional boundary condition for the heat production term. The outlet is not shown in this picture and the pressure at the outlet is defined as wave trans massive at 1P a. This boundary condition allows pressure variabilities at the outlet patch. This is necessarily because of the flow induced pressure variations behind the nozzle (shock and expansion waves). 5 of 9

6 Figure 4. used mesh for the numerical calculation / cathode green / anode blue / inlet yellow colored For the ignited operation condition the electric conductivity of Eq. (6) was connected to a constant C see Eq. (9). For the present results in this paper the constant is defined as C = A2 s 2 m 3 kg m. σ = C T 3 4 p 1 2 e q 1 2kT (9) 2 φ = (1) For the numerical simulation the potential of the anode is zero and the potential of cathode φ c =.11 kgm2 s 3 A was defined as a boundary condition. The potential drop between the anode and the cathode was calculated by the solver using ( the ) Eq. (1). Eq. (8) delivers the electrical field and Eq. (7) the electrical current. Now the heating term Ej j of energy Eq. (5) generates energy that depends on the local electric field and the local currant. IV. Results The simulation was conducted on the basic generic geometry which is shown in Fig. 4. This mesh consist of 1, hexagonal cells. The mass flow through the system is controlled by the numerical solver. Figure 5. isoline of the potential field Figure 6. vector field of the electric currant The plots Fig. 5 to 1 are showing the final result of the numerical simulation of an Argon gas flow of 16mg/s. The plots comparing the final solution of the not ignited and the ignited operation condition of the thruster. 6 of 9

7 Fig 5 show the isolines of the electric potential. The potential of the anode is zero and the potential of the cathode is.11v. The potential drop between the anode and the cathode is caused by Eq. (1). Fig. 6 shows the vector field of the electric currant. All vectors are pointing from the anode to the cathode. The maximum magnitude of these vectors is in the area where the distance between anode and cathode is low. Figure 7. temperature field before ignition Figure 8. temperature field after ignition Fig. 7 and 8 are showing isoshapes of the local temperature behind the nozzle of the not ignited and of the ignited operation condition. At the not ignited operation condition the temperature is 3K at the inlet patch and the temperature is falling inside the nozzle, because of the gas expansion. At the ignited operation condition Fig. 8, the highest temperatures are at the surface of the cathode. Close to the point where the local currant density is high and the local velocity is low. The absolute value of the local temperature depends on the local mass and the velocity. Inside the nozzle the temperature falls, because of the expansion of the gas. Figure 9. velocity field before ignition Figure 1. velocity field and mach number after ignition Fig. 9 and 1 are showing the velocity field (colored in red, green and blue) and the local mach number (contour plot in black and white). At the not ignited operation condition the velocity maximum is less than 5m/s. At the ignited operation condition the velocity maximum is higher than 12m/s and the mach number is 2.2. The highest local mach number and local velocity is behind the nozzle where the propallant expand into the vacuum chamber. The benefit of the electrical ignition is the higher local mach number. Because of this the local velocities of the ignited gas flow is higher, than the not ignited. In this way the local impulse is higher. 7 of 9

8 The comparison between the experiments and the numerical solutions for the same mass flows of Argon and Xenon leads to Fig. 11. This plot shows the pressure inside the ignition chamber, because this was measured at the experiment and was analyzed in the numerical simulation. The horizontal lines in Fig. 11 represent the lowered steady state pressure inside the ignition chamber for the experiment. The pressure rise during the start up of the experiment has not been measured because of a to small measuring frequency of the measuring technique. The dots representing the numerical solution. At the numerical solution we have the ability to dissolve the time depending pressure rise inside the ignition chamber. So it is shown that the start-up time for continuous flow conditions is in range of 4ms up to 6ms, depending on heating and propellant. time depending pressure rise 2,5 experiment - Xenon 25Pa experiment - Argon 23Pa 2, 1, experiment - Xenon 68Pa experiment - Argon 66Pa Argon mass flow 16 mg/s final pressure 659 Pa Xenon mass flow 33 mg/s final pressure 677 Pa Argon mass flow 16 mg/s final pressure 2324 Pa Xenon mass flow 33 mg/s final pressure 2567 Pa Figure 11. pressure rise inside ignition chamber with time - top OpenFOAM simulation result with electric ignition - bottom OpenFOAM simulation result without electric ignition V. Conclusion If we compare the results of the not ignited operation condition of the numerical solver and the experiment we can detect, that for an Argon gas flow of 16mg/s the results of the experiment is app. 66P a and the result of the numerical solver is 659P a. For a Xenon gas flow of 33mg/s the result of the experiment app. 68P a and the result of the numerical solver is 677P a. For the ignited operation condition the result of the same Argon gas flow is app. 23P a for the experiment and 2324P a for the numerical model. The result of the Xenon gas flow is app. 25P a for the experiment and 2567P a for the numerical model. If we compare this results of the experiment (lines in Fig. 11) and the final numerical solution (dots in Fig. 11) it is quite evident that the solutions of the experiment and the numerical solver are in good 8 of 9

9 agreement with the steady state solution of the numerical model. Acknowledgments Special thanks go to Dipl.-Ing. Ronald Mairose for his advice and the technical support. I also would like to thank Torben Schadowski and Fabian Fastabend for their support during the test period. References Periodicals 1 Fox J. N., Hobson R. M., Temperature dependence of Dissociative Recombination Coefficients in Argon, Physical Review Letters, Vol. 17, No. 4, 1966, pp. 161, Revel I.,Pitchford L. C., Boeuuf J. P., Calculated gas temperature profiles in argon glow discharges, Journal of applied Physics, Vol. 89, No. 5, 2, pp. 2234, Hash D. B., Bose D.,Rao M. V. V. S., Cruden B. A., Mayyappan M., Sharma S. P., Impact of gas heating in inductively coupled plasmas, Journal of applied Physics, Vol. 9, No. 5, 21, pp. 2148, Papadakis A. P., Georghiou G. E., Metaxas A. C., Simulation for the transition from non-thermal to thermal discharges, Plasma Sources Science and Technology, Vol. 14, No. 5, 25, pp. 25, Jones J. E., On corona-induced gas motion and heating I: Field equations, modeling and vortex formation, Journal of Electrostatics, Vol. 66, No. 4, 28, pp. 84, Lin S. -C., Resler E. L., and Kantrowitz A., Electrical Conductivity of Highly Ionized Argon Produced by Shock Waves, Journal of applied Physics, Vol. 26, No. 1, 1955, pp. 95, Stier L. G., The Coefficients of Thermal Diffusion of Neon and Argon and Their Verification with Temperature, Physical Review, Vol. 62, 1942, pp. 548, Lysenko V. I., Effect of the specific heat ratio on the stability and laminar-turbulent transition of a supersonic boundary layer Fluid Dynamics, Vol. 24, 1989, pp. 317, Greenshields, C. J., Weller H. G., Gasparini, L. and Reese, J. M., Implementation of semi-discrete, non-straggered central schemes in a collocated, polyhedral, finite volume framework, for high-speed viscous flows International Journal for Numerical Methods in Fluids, 29 1 Kurganov, A., Tadmor, E., New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations Journal of Computational Physics, Vol. 16, 2, pp. 241, Kurganov, A., Noelle, S. and Petrova, G., Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations Society for Industrial and Applied Mathematics, Vol. 23, No. 3, 21, pp. 77, Uribarri, L. and Choueiri, E. Y., The Onset of Voltage Hash and its Relationship to Anode Spots in Magnetoplasmadynamic Thrusters IEPC-paper, Vol. 29, No. 84, Choueir, E. Y., Okuda, H., Anomalous Ionization in the MPD Thruster Air Force Office of Scientific Research, No. F , pp. 1, 8. Books 14 Oertel, H., Böhle, M. and Dohrmann, U., Strömungsmechanik, Vieweg+Teubner Verlag; 4th ed., Wiesbaden, 26, Chaps. 3, Meschede, D., Gerthsen Physik, Springer Verlag; 23rd ed., Berlin Heidelberg, 26, Chaps. 8, of 9