Experimental Investigation and Numerical Modeling of

Electric Heating Rate in a Generic Electric Propulsion

System

IEPC-2011-063

Presented at the 32nd International ElectricPropulsion Conference,Wiesbaden, Germany

September 11–15, 2011

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 manystudies. In this paper, a computational method for modeling the behavior an arc-jet ispresented. There for the conservation equations of mass, momentum, energy and chargedensity are solved. The energy equation includes an additional source term depending onthe electric charge density of the electric arc. The results are validated on the experimentaldata.

∗

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

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Nomenclature

δ = ionization rate [-]

m = mass [kg]

m = mass flow[kgs

]cp = specific heat capacity

[J

kgK

]n0 = number density of molecules

[1

m3

]t = time [s]

ρ = density[

kgm3

]~u = velocity

[ms

]p = pressure

[kgms2

]T = temperature [K]

φ = electric potential [V ]

Tv = viscous stress tensor[

kgms2

]Ee = total energy density

[m2

s2

]~Ej = electric field intensity

[Vm

]~j = electric current

[Am2

]σ = electric conductivity

[s3A2

m3kg

]q1 = first ionization energy

[J

mol

]σb = Stefan-Boltzman constant 5.67 · 10−8 W

m2K4

k = Boltzman constant 1.38 · 10−23 JK

h = Plank constant 6.62 · 10−34Js

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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 setupis simple. The nozzle consists of an axially symmetric anode and cathode. The anode and the cathode areshaping 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 10−3Pa up to 1Pa (de-pending on the mass-flow through the system). The electrical ignition between the anode and the cathodeis realized by two different power supplies. For the first ignition a high voltage continuous current powersupply with a electrical power of 200W is used. This power supply realized a electrical potential of 2000Vbetween the anode and the cathode. If the propellant is piped through the system the ignition starts and avoltage drop from 2000V to less than 60V is the result. At this point a second high current power supplyassumed the power supply of the experiment. During steady state operation the thruster operates with at aelectric power range of 400W up to 600W , depending on the propellant.

To reestimate the Temperature of the heated experiment a rough approximation was made. It is clearthat the total energy of the power supply is transformed into radiation energy and gas heating energy. So wecombine the ”Stefan-Boltzmann law” Psb = σbAT

4 and the thermodynamic gas heating term Pg = mcp∆T .See Eq. (1).

Ppower = 600W = σbAT4 + mcp∆T (1)

For Argon and Xenon this equation has only one solution that makes sense T ≈ 1860K. The other threesolutions are not real ore negative. By using the ”Saha-Eggers Equation”15 Eq. (2) we are able to reestimatethe ionization rate of the gas.

δ =(2πmkT )

34

h32no

12

e−q1

2kT = 2.2 · 10−14 (2)

The result shows that the ionization of the noble gases is very low. This result is in order with theresearches of Uribari12 and Choueiri13 who are showing, that the influence of electromagnetic field forcesfor thrusters with less than 100kW of power can be neglected. For this reason the influence of the currentinduced 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 modifiedsolver based on the compressible solver rhoCentralFoam. A detailed description of this solver is given byGreenshields9. This solver solves the fundamental fluid dynamic conservation equations of mass, momentumand energy. To model a electric ignition between the anode and the cathode an additional electric heatingtherm is implemented into the source code of the energy Eq. (5). Similar source terms are described byHash3, Papadakis4 and Jones5.

II. The Experiments

During test the electric current is limited to 30A and the voltage depends on the resistance of the ionizedgas and the resistance of the anode and the cathode. The mass-flow through the system is steered by a needle

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valve. For the first tests the noble gas Xenon is used as the propellant, because of the low first ionizationenergy of 1170.4kJ/mol. During the experiments the pressure of the vacuum chamber, the pressure of theignition 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 680Pa inside theignition 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 islimited to 30A. During the steady state operating condition the voltage drop of the thruster is app. 14V .This means a electrical performance of 420W . The inserted electrical performance leads to a pressure of2430Pa inside the ignition chamber.

Xenon - "electric heated"

pres

sure

in P

a

0

1,000

2,000

3,000

4,000

5,000

valu

e

0

10

20

30

40

50

time in s0 5 10 15 20 25 30

"ignition chamber" voltage in V current in A

Xenon - "cold"

pres

sure

in P

a

0

1,000

2,000

3,000

4,000

5,000

time in s0 10 20 30 40

"ignition chamber"

680 Pa

2500 Pa

Figure 2. left hand side - measurement without electrical ignition of Xenon / right hand side - measurementwith electrical ignition of the Xenon

Argon - "electric heated"

pres

sure

in P

a

0

1,000

2,000

3,000

4,000

5,000

valu

e

0

10

20

30

40

50

time in s60 80 100 120 140

"ignition chamber"voltage in Vcurrent in A

ArXe

2300 Pa

2300 Pa

Argon - "cold"

pres

sure

in P

a

0

1,000

2,000

3,000

4,000

5,000

time in s10 15 20 25

"ignition chanber"

660 Pa

Figure 3. left hand side ” not ignited measurement” without electrical ignition / right hand side - measurementwithout electrical ignition of the Xenon - Argon mixing experiment with electrical ignition . The thruster wasstarted with Xenon and changed over to Argon. The change of the propellant is represented by the hatchedarea

In a second test the noble gas Argon was used as propellant. Because of the higher ionization energyof 1520.6kJ/mol the flash over needs more power to establish a continuous electrical ignition between theanode and the cathode. To reduce the required electrical energy a gas mixing gadget was used. The intentionof the mixing gadget is to heat up the thruster with Xenon and to take over with Argon. In this way thehigher temperature of the ignition chamber relieved the establishing of the contentious Argon ignition. Theresult of this experiment is presented in the plot Fig. 3.

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The black line represents the voltage drop between the anode and the cathode, the green line the currentflow 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 solversolves the conservation of mass Eq. (3), the Navier-Stokes-Equation Eq. (4) and the conservation of energyEq. (5). The detailed equations are presented below. The general derivations of these three equation aregiven in Oertel14.

∂ρ

∂t+∇ · (ρ~u) = 0 (3)

ρ

[∂~u

∂t+ ~u · ∇~u

]= −∇p−∇ · Tv (4)

∂ρEe

∂t+∇[~u(ρEe)] = −∇ · [p~u]−∇ · [Tv · ~u]− ~Ej ·~j (5)

The continuity of mass Eq. (3) defines that the change of density ∂ρ/∂t inside the system is equal to the flux ofmass ∇·(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 viscousstress tensor Tv. The conservation of energy Eq. (5) include the total energy density Ee = 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 Lin6

that the electric conductivity of a gas is proportional to the absolute temperature T , the pressure of the gasp, the first ionization potential of the gas q1 and the Boltzman constant k.

σ ∝ T34

p12

· e−q1

2kT (6)

~j = σ ~Ej (7)

~Ej = −∇φ (8)

The ”Ohm’s Law” Eq. (7) describes the relation for the electric current fluxdensity ~j, the electric

field intensity ~Ej and the electric conductivity σ. The electric field ~Ej is calculated by the voltage dropbetween the anode and the cathode Eq. (8). The solver implements a ”Sutherland Transport Model”7−8 fortemperature depending viscosity modeling.

Standard pressure correction methods often return bad agreements of computational and measurementdata with compressible hypersonic flows. Therefore a different method is used implementing own numericalprocedures. The solver includes an explicit predictor equation and an implicit corrector equation for thediffusion of primitive variables instead of a pressure correction. The procedure is well-known as ”Kurganov’sMethod”10−11.

B. Numerical Boundary Conditions

For the numerical simulations a hexagonal mesh with nearly 100, 000 cells was used. Fig. 4 shows the ignitionchamber of the thruster, the most interesting part of the mesh. The inlet is colored yellow. At this patch themass-flow through this patch is steered by the solver. The temperature is fixed at 300K. The green patchrepresents the cathode. At this patch the velocity is zero. The same boundary condition is used for the blueanode. For the modulation of the electric ignition a voltage drop of 0.1V between the anode and the cathodeis the additional boundary condition for the heat production term. The outlet is not shown in this pictureand the pressure at the outlet is defined as wave trans massive at 1Pa. This boundary condition allowspressure variabilities at the outlet patch. This is necessarily because of the flow induced pressure variationsbehind the nozzle (shock and expansion waves).

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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 = 1.1 · 106 A2s2

m3√

kg·m.

σ = C · T34

p12

· e−q1

2kT (9)

∇2φ = 0 (10)

For the numerical simulation the potential of the anode is zero and the potential of cathode φc =

−0.11kgm2

s3A was defined as a boundary condition. The potential drop between the anode and the cathodewas calculated by the solver using the Eq. (10). 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 consistof 100,000 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 10 are showing the final result of the numerical simulation of an Argon gas flow of16mg/s. The plots comparing the final solution of the not ignited and the ignited operation condition of thethruster.

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Fig 5 show the isolines of the electric potential. The potential of the anode is zero and the potential ofthe cathode is −0.11V . The potential drop between the anode and the cathode is caused by Eq. (10).

Fig. 6 shows the vector field of the electric currant. All vectors are pointing from the anode to thecathode. The maximum magnitude of these vectors is in the area where the distance between anode andcathode 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 ofthe ignited operation condition. At the not ignited operation condition the temperature is 300K at the inletpatch and the temperature is falling inside the nozzle, because of the gas expansion. At the ignited operationcondition Fig. 8, the highest temperatures are at the surface of the cathode. Close to the point where thelocal currant density is high and the local velocity is low. The absolute value of the local temperaturedepends on the local mass and the velocity. Inside the nozzle the temperature falls, because of the expansionof the gas.

Figure 9. velocity field before ignition Figure 10. velocity field and mach number after ignition

Fig. 9 and 10 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 than500m/s. At the ignited operation condition the velocity maximum is higher than 1200m/s and the machnumber is 2.2. The highest local mach number and local velocity is behind the nozzle where the propallantexpand into the vacuum chamber.

The benefit of the electrical ignition is the higher local mach number. Because of this the local velocitiesof the ignited gas flow is higher, than the not ignited. In this way the local impulse is higher.

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The comparison between the experiments and the numerical solutions for the same mass flows of Argonand Xenon leads to Fig. 11. This plot shows the pressure inside the ignition chamber, because this wasmeasured at the experiment and was analyzed in the numerical simulation. The horizontal lines in Fig. 11represent the lowered steady state pressure inside the ignition chamber for the experiment. The pressurerise during the start up of the experiment has not been measured because of a to small measuring frequencyof the measuring technique. The dots representing the numerical solution. At the numerical solution wehave the ability to dissolve the time depending pressure rise inside the ignition chamber. So it is shown thatthe start-up time for continuous flow conditions is in range of 4ms up to 6ms, depending on heating andpropellant.

time depending pressure rise

pres

sure

in P

a

0

250

500

1,500

2,000

2,500

time in s0 0.001 0.002 0.003 0.004 0.005 0.006

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

experiment - Argon 2300Pa

experiment - Xenon 2500Pa

experiment - Argon 660Pa

experiment - Xenon 680Pa

Figure 11. pressure rise inside ”ignition chamber” with time - top OpenFOAM simulation result with electricignition - 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 experimentwe can detect, that for an Argon gas flow of 16mg/s the results of the experiment is app. 660Pa and theresult of the numerical solver is 659Pa. For a Xenon gas flow of 33mg/s the result of the experiment app.680Pa and the result of the numerical solver is 677Pa.

For the ignited operation condition the result of the same Argon gas flow is app. 2300Pa for theexperiment and 2324Pa for the numerical model. The result of the Xenon gas flow is app. 2500Pa for theexperiment and 2567Pa for the numerical model.

If we compare this results of the experiment (lines in Fig. 11) and the final numerical solution (dotsin Fig. 11) it is quite evident that the solutions of the experiment and the numerical solver are in good

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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 wouldlike to thank Torben Schadowski and Fabian Fastabend for their support during the test period.

References

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13 Choueir, E. Y., Okuda, H., ”Anomalous Ionization in the MPD Thruster” Air Force Office of ScientificResearch, No. F49620-93-1-0222, pp. 1, 8.

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