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Experimental and numerical study of an electrothermal pulsed plasma thruster for small satellites
ARTICLE IN PRESS
Vacuum 80 (2006) 1223–1228 www.elsevier.com/locate/vacuum
Experimental and numerical study of an electrothermal pulsed plasma thruster for small satellites Toshiaki Edamitsu, Hirokazu Tahara Division of Mechanical Engineering, Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Abstract A pulsed plasma thruster (PPT) with a propellant feeding mechanism was designed using two poly tetrafluoroethylene (PTFE) bars as propellants. The PPT was mounted on a thrust stand with a 1-m-long perpendicular pendulum which was developed for precise measurements of impulse bits. Initial thrust performance showed thrust-to-power ratio of 43–48 mN/W, specific impulse of 470–500 s and thrust efficiency of 10–12% with energy of 4.5–14.6 J. Ten thousand shots achieved a total impulse of approximately 3.6 Ns, and the PTFE bars were consumed approximately 2 mm in length. However, uneven receding of the PTFE surface was observed. In order to investigate physical phenomena in a whole system, an unsteady numerical simulation of initial discharge, generation of plasma, heat transfer to the PTFE, heat conduction inside the PTFE, ablation from the PTFE surface and acceleration of plasma was performed. The calculated results were used to explain the physical phenomena in the cavity, especially ablated mass of the PTFE. r 2006 Elsevier Ltd. All rights reserved. Keywords: Electric propulsion; Electrothermal pulsed plasma thruster; Solid propellant; Thrust performance; Unsteady numerical simulation
1. Introduction A pulsed plasma thruster (PPT) is expected to be used as a thruster for a small satellite. The PPT has some features superior to other kinds of electric propulsion [1]. It has no sealing part, simple structure and high reliability. These are the benefits of using a solid propellant, mainly poly tetrafluoroethylene (PTFE). Especially, an electrothermal PPT has some advantages of high thrust-to-power ratio and large thrust efficiency compared with an electromagnetic PPT. However, performances of PPTs are generally low compared with other electric thrusters. We have studied the reasons for the low performance using the preliminary PPT shown in Fig. 1. It has a discharge cavity inside a PTFE tube between the anode and cathode. Energy of discharge is initially stored in a capacitor, and discharge is started by an ignition. The PTFE is ablated, heated and ionized. Then, the plasma with high pressure is Corresponding author. Tel./fax: +81 6 6850 6179.
E-mail address: [email protected] (T. Edamitsu). 0042-207X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2006.01.055
accelerated downstream through the nozzle. The following results are mainly obtained with the preliminary study [2,3]:
A small cross-sectional area of a discharge cavity improves the performance, because it decreases transmission energy loss in the discharge circuit due to high plasma resistance and increases pressure in the cavity. Due to interactions between the discharge circuit and the plasma, there is an optimum cavity length (distance between electrodes) for thrust efficiency, because a short cavity increases transmission energy loss in the discharge circuit and a long cavity increases acceleration energy loss in the PPT.
Based on these results, we designed and tested a PPT with a propellant feeding mechanism. Furthermore, an unsteady numerical simulation was performed in order to investigate physical phenomena in the whole system. It mainly consists of models of discharge, generation of plasma, heat transfer to the PTFE, heat conduction inside
ARTICLE IN PRESS 1224
T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228
Fig. 1. Configuration of electrothermal pulsed plasma thruster for preliminary study.
the PTFE, ablation from the PTFE surface and acceleration of plasma. 2. Experimental apparatus 2.1. Electrothermal pulsed plasma thruster Fig. 2 shows a PPT with a propellant feeding mechanism [4]. A cavity is formed between two PTFE bars with 6 mm in thickness and two PYREXs glass bars. The crosssectional area of the cavity (6.5 mm2) is approximately the same as the preliminary model. The cavity length is 12 mm. The PTFE bars are provided by spring loads of approximately 3 N. The glass bars are fixed to the body. An igniter is mounted in the cathode nozzle. The nozzle has a length of 28 mm and a half angle of 301.
Fig. 2. Configuration of PPT with propellant feeding mechanism. (a) Accelerator, (b) cross section of cavity.
2.2. Thrust stand and vacuum chamber In order to measure impulse bit precisely, a vacuum chamber and a thrust stand were developed, as shown in Fig. 3. The thrust stand has a 1-m-long perpendicular pendulum. The PPT and the energy storage capacitors are mounted on the pendulum, which rotates around fulcrums of two knife edges without friction. The displacement of the pendulum is detected by an eddy-current-type gap sensor (non-contacting micro-displacement meter) near the PPT, which resolution is about 70.5 mm. The electromagnetic damper is used to suppress mechanical noises and to decrease quickly the amplitude for the next measurement after firing the PPT. The damper consists of a permanent magnet fixed to the pendulum and two coils fixed to the supporting stand. A control circuit differentiates the output voltage of the displacement sensor and supplies the current proportional to the differentiated voltage to the coil. As a result, the damper works as a viscosity resistor. The damper is turned off just before firing the PPT for a measurement without damping. The calibration of impulse is carried out by a collision of a ball to the pendulum. The sensitivity of the thrust stand is variable by changing the weight mounted on the top of the pendulum as shown Fig. 4. The vacuum chamber pressure was kept at about 4 103 Pa under operations.
Fig. 3. Configuration of thrust stand and vacuum chamber.
2.3. Discharge circuit Six capacitors of mica-paper type with a total capacitance of 9.0 mF are used to store energy. Charge voltage is varied from 1000 to 1800 V, corresponding to stored energy of 4.5–14.6 J. A discharge current waveform is measured by a Rogowski coil.
ARTICLE IN PRESS T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228
qðAruÞ q qA þ ½Aðru2 þ pÞ ¼ p fLcir , qt qx qx qðAeÞ q þ ½Auðe þ pÞ qt qx ¼ AðQj Qrad Qconv þ Qab FÞ,
Amplitude per impulse bit, m / Ns
10
1
0.1
0.01 2
1225
3
4 Additional weight, kg
5
6
Fig. 4. Sensitivity of thrust stand dependent on top weight.
3. Calculation model 3.1. Assumptions Basic assumptions for the plasma flow are: (1) local thermodynamic equilibrium (LTE) is established: T e ¼ T i ¼ T n T, (2) ionization equilibrium (ionization degree is estimated by Saha’s equation), (3) one-fluid plasma, (4) quasi-neutral plasma, (5) effects of magnetic field are negligible. The additional assumptions for the plasma near the PTFE surface are: (6) the following relationship is satisfied between the energy-transfer mean free paths of electron-heavy particle leh,E and heavy particle-heavy particle lhh,E [4]: Cavity length ð12 mmÞ4leh;E 4typical cavity radius ð0:8 mmÞblhh;E , (7) heat conduction by heavy particles inside the plasma is negligible. From assumption (6), the temperature jump of heavy particle on the surface is negligible. Therefore, the temperature of heavy particle near the surface is lower than that on the axis. On the other hand, electrons moving towards the surface do not lose their energy, and therefore the temperature of electron near the surface is approximately the same as that on the axis. Then T e bT i ¼ T n T s (Ts: temperature of PTFE surface) is assumed near the surface. In the axial direction, LTE is approximately established. 3.2. Equations and method of calculation Axial components of mass, momentum and energy conservations are: qðArÞ qðAruÞ þ ¼ Lcir G, qt qx
ð1Þ
where A is cross sectional area, r mass density, Lcir circumference of the cross section, G ablation mass flux, u axial velocity, p pressure, f momentum loss on the wall surface, e total energy: p=ðg 1Þ þ ru2 =2, and F energy loss due to viscosity. The specific heat ratio g is assumed to be a constant, 1.1 [5]. Qj is energy of Joule heating, Qrad emission energy of bremsstrahlung (braking radiation), Qconv convection energy loss, and Qab thermal energy of ablated PTFE, which are written as follows: Qj ¼ Zj 2 ;
Qrad ¼ 1:57 1040 n2e T 1=2 ,
Qconv ¼ ðqe þ qi þ qn ÞLcir =A, Qab ¼ G=mi 2kT s Lcir =A,
ð2Þ
where j is discharge current density, Z resistivity of plasma in the cavity, ne plasma density, qe, qi and qn convection energy flux to the propellant surface due to electrons, ions and neutral particles, respectively, considering effect of sheath on the PTFE surface, and Ts PTFE surface temperature. Considering assumption (6), Qconv is approximately written as [4]: 1 Lcir . (3) Qconv ji 2kT þ ef 2 A Ion flux ( ¼ electron flux) to the surface ji and potential drop in the sheath f satisfy 1 8kT s 1=2 ; ji ne 4 pmi
f
kT Tmi ln . 2e T s me
(4)
The discharge circuit is modeled as a LCR series circuit including the plasma resistance: Z € L0 Q þ R0 þ ðZ=AÞ dx Q_ þ Q=C ¼ 0, _ J ¼ Q, Z¼
ln L me 3kT 1=2 þ s n , en n me 1:53 102 T 3=2 ne e2
ð5Þ
where J is discharge current, sen cross-sectional area of electron neutral, and ln L the Coulomb logarithm. The resistance R0 and inductance L0 of the circuit are measured with frequency response method [2]. Assuming heat-layer thickness is much smaller than the radius of the PTFE bar’s curve, heat transfer equation inside the PTFE is as follows: 2 qY l q Y q2 Y ¼ þ 2 , qt rC p qx2 qr qY A DH G, ð6Þ l ¼ ðQconv þ Qrad Qab Þ qr r¼0 Lcir
ARTICLE IN PRESS T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228
8.8J (1400V) Calculation
4000
Experiment 3000
2000
1000
0
-1000
-2000
0
5
10
15
Time, s
Fig. 5. Calculated and experimental discharge current waveforms.
1000
300
4. Results and discussion
Exp. Impulse bit Cal. Impulse bit
4.1. Initial performances and physical phenomena Impulse bit, µN s
200 600 150 400 100 200
0
50
5
(a)
15
500 15 400 300 10 Specific impulse Thrust efficiency
200 100 (b)
0
20
4.2. 10 000-shot operation Ten thousand shot operation is conducted with a frequency of 0.5 Hz. Figs. 9 and 10 show the change in performance during the test and the shape of PTFE bars before and after the test, respectively. Both the impulse bit
10 Stored energy E0, J
600
Specific impulse, s
The calculated current waveform is agreed well with the experimental results as shown in Fig. 5. Initial thrust performance showed impulse bit of 192–686 mN s corresponding thrust-to-power ratio of 43–47 mN/W, specific impulse of 470–500 s and thrust efficiency of 10–12% when the stored energy was 4.5–14.6 J shown in Fig. 6. It is remarkable that thrust performance is kept high even at a low energy of 4.5 J. The calculated mass loss per shot agrees well with the experimental results. It is considered that phenomena in the cavity (condition of plasma, heat transfer to/inside the PTFE, ablation) are simulated well with this model. However, the calculated impulse bit is larger than the experimental result. The peaks of plasma temperature, discharge current, ablation rate, pressure and thrust at the center of cavity length appear in this order as shown in Fig. 7. After the ablation is completed, gas starts to expand, and thrust is generated until over 25 ms. Fig. 8 shows the temporal and spatial variation of Mach number. The Mach number increases in the cavity and becomes unity at the exit of the cavity ðx ¼ 12 mmÞ. The plasma is efficiently accelerated in the divergent nozzle.
250
Exp. ∆m Cal. ∆m
800
Mass loss per shot ∆m, µg
where pc ¼ 1.84 1015 Pa and Tc ¼ 20 815 K are the characteristic pressure and temperature, respectively. Momentum and energy loss due to viscosity f and F are expediently estimated with radial distribution of axial velocity which is assumed to be a distribution of ideal laminar or turbulent flow considering local Reynolds number. The calculation domain for plasma flow is between the anode and the nozzle exit. The plasma flow and the LCR series circuit are calculated with the TVD MacCormack scheme and the Runge–Kutta scheme, respectively. Initial energy is arranged in capacitors. In order to start the discharge, the plasma with 1.7 103 J and 1.1 mg, which are much less than the initial energy in capacitors and mass loss per shot, respectively, are arranged in the calculation domain as initial conditions.
5000
5
10 Stored energy E0, J
15
Thrust efficiency, %
where Y is temperature inside the PTFE, and DH the unzipping energy of PTFE: 1.5 106 J/kg [1]. The following ablation flux G is calculated with Langmuir’s law [6] using equilibrium vaporizing pressure pvap of PTFE [1,7]: 1=2 mi G¼ pvap ; pvap ¼ pc expðT c =T s Þ, (7) 2pkT s
Discharge current, A
1226
0
Fig. 6. Initial performance vs. stored energy. (a) Impulse bit and mass loss per shot, (b) Specific impulse and thrust efficiency.
ARTICLE IN PRESS T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228 800
10
20
E0=8.8J, x=6m m
700
Plasma temperature 8
Pressure, atm Ablation rate, g/s Thrust, N
Ablation rate Pressure
60
6
Thrust 40
4
20
2
600 Impulse bit, Ns Mass loss per shot, g Specific impulse, s
Current
Current, kA Plasma temperature, eV
80
Impulse bit Mass loss per shot Specific impulse Thrust efficiency
500 10
400 300
5
200 0
0 0
5
10
15
Time, µs
Fig. 7. Calculated temporal variations of plasma temperature, discharge current, ablation rate, pressure and thrust.
15 Thrust efficiency, %
100
1227
100 0
0
2000
4000 6000 Shot number
8000
10000
0
Fig. 9. Changes in performances during 10 000-shot operation.
Fig. 8. Calculated temporal and spatial variations of Mach number.
and the thrust efficiency gradually decrease with shot number, because gaps are gradually generated between the PTFE bars. These gaps do not derive a severe leakage of gas because there are glass bars on the two ends of the cavity. However, increase in cavity volume lead to decrease in pressure in the cavity. Heat loss to the glass should be also considered. Actually, mass loss of glass bars is observed, which is added to the mass loss and considered in estimating the specific impulse and thrust efficiency. Fig. 11 shows the calculated distribution of ablated mass per area versus time. This distribution qualitatively agrees with the observed uneven receding of the surface. The reason for this phenomenon is that the plasma density near the cavity exit is small because of the high velocity. Then, heat convection to the PTFE is small near the cavity exit. The mass loss per shot gradually decreases with shot
Fig. 10. Feature of PTFE bars before/after 10 000-shot operation. (a) Schematic, (b) Photograph after operation (top view).
number because heat loss to the glass surface increases. In spite of the uneven receding, the PTFE is also ablated near the cavity exit, and each bar was moved in the distance of approximately 2 mm toward the cavity during the 10000 shot operation. As a result, a total impulse of approximately 3.6 Ns is obtained. 5. Conclusions The following results were obtained in this study. 1. A PPT with a propellant feeding mechanism showed initial performances of thrust-to-power ratio of
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T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228
performances, especially, discharge current waveform and mass loss per shot. 3. Ten thousand-shot operation was conducted, and a total impulse of 3.6 Ns was achieved. The propellant feeding mechanism worked well. However, both the impulse bit and the thrust efficiency gradually decreased because of uneven receding of the PTFE surface. The shape of the surface was qualitatively explained with the calculation. It was suggested that the axial distribution of ablation flux mainly depends on that of plasma density.
References
Fig. 11. Calculated distribution of ablated mass of PTFE per area.
43–47 mN/W, specific impulse of 470–500 s and thrust efficiency of 11–12% with stored energy of 4.5–14.6 J. 2. An unsteady calculation, which simultaneously simulates discharge, heat transfer to/inside the PTFE, ablation and plasma flow, was carried out. The calculated results roughly agreed with measured initial
[1] Burton L, Turchi J. J Propul Power 1998;14(5):716–35. [2] Edamitsu T, Tahara H, Yoshikawa T. In: Proceedings of Asian joint conference on propulsion and power, Seoul, Korea, 2004. p. 324–34. [3] Edamitsu T, Tahara H, Yoshikawa T. In: Proceedings of 24th international symposium on space technology and science, Miyazaki, Japan, ISTS-2004-b-6, 2004. p. 112–8. [4] Edamitsu T, Tahara H, Yoshikawa T. In: Proceedings of Asian joint conference on propulsion and power, Kitakyusyu, Japan, AJCPP200522083, 2005. [5] Kovitya P. IEEE Trans Plasma Sci 1984;PS-12(1):38–42. [6] Keider M, Boyd D, Beilis I. IEEE Trans Plasma Sci 2000;28(2):376–85. [7] Keider M, Boyd D, Beilis I. J Propul and Power 2003;19(3):424–30.
Vacuum 80 (2006) 1223–1228 www.elsevier.com/locate/vacuum
Experimental and numerical study of an electrothermal pulsed plasma thruster for small satellites Toshiaki Edamitsu, Hirokazu Tahara Division of Mechanical Engineering, Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Abstract A pulsed plasma thruster (PPT) with a propellant feeding mechanism was designed using two poly tetrafluoroethylene (PTFE) bars as propellants. The PPT was mounted on a thrust stand with a 1-m-long perpendicular pendulum which was developed for precise measurements of impulse bits. Initial thrust performance showed thrust-to-power ratio of 43–48 mN/W, specific impulse of 470–500 s and thrust efficiency of 10–12% with energy of 4.5–14.6 J. Ten thousand shots achieved a total impulse of approximately 3.6 Ns, and the PTFE bars were consumed approximately 2 mm in length. However, uneven receding of the PTFE surface was observed. In order to investigate physical phenomena in a whole system, an unsteady numerical simulation of initial discharge, generation of plasma, heat transfer to the PTFE, heat conduction inside the PTFE, ablation from the PTFE surface and acceleration of plasma was performed. The calculated results were used to explain the physical phenomena in the cavity, especially ablated mass of the PTFE. r 2006 Elsevier Ltd. All rights reserved. Keywords: Electric propulsion; Electrothermal pulsed plasma thruster; Solid propellant; Thrust performance; Unsteady numerical simulation
1. Introduction A pulsed plasma thruster (PPT) is expected to be used as a thruster for a small satellite. The PPT has some features superior to other kinds of electric propulsion [1]. It has no sealing part, simple structure and high reliability. These are the benefits of using a solid propellant, mainly poly tetrafluoroethylene (PTFE). Especially, an electrothermal PPT has some advantages of high thrust-to-power ratio and large thrust efficiency compared with an electromagnetic PPT. However, performances of PPTs are generally low compared with other electric thrusters. We have studied the reasons for the low performance using the preliminary PPT shown in Fig. 1. It has a discharge cavity inside a PTFE tube between the anode and cathode. Energy of discharge is initially stored in a capacitor, and discharge is started by an ignition. The PTFE is ablated, heated and ionized. Then, the plasma with high pressure is Corresponding author. Tel./fax: +81 6 6850 6179.
E-mail address: [email protected] (T. Edamitsu). 0042-207X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2006.01.055
accelerated downstream through the nozzle. The following results are mainly obtained with the preliminary study [2,3]:
A small cross-sectional area of a discharge cavity improves the performance, because it decreases transmission energy loss in the discharge circuit due to high plasma resistance and increases pressure in the cavity. Due to interactions between the discharge circuit and the plasma, there is an optimum cavity length (distance between electrodes) for thrust efficiency, because a short cavity increases transmission energy loss in the discharge circuit and a long cavity increases acceleration energy loss in the PPT.
Based on these results, we designed and tested a PPT with a propellant feeding mechanism. Furthermore, an unsteady numerical simulation was performed in order to investigate physical phenomena in the whole system. It mainly consists of models of discharge, generation of plasma, heat transfer to the PTFE, heat conduction inside
ARTICLE IN PRESS 1224
T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228
Fig. 1. Configuration of electrothermal pulsed plasma thruster for preliminary study.
the PTFE, ablation from the PTFE surface and acceleration of plasma. 2. Experimental apparatus 2.1. Electrothermal pulsed plasma thruster Fig. 2 shows a PPT with a propellant feeding mechanism [4]. A cavity is formed between two PTFE bars with 6 mm in thickness and two PYREXs glass bars. The crosssectional area of the cavity (6.5 mm2) is approximately the same as the preliminary model. The cavity length is 12 mm. The PTFE bars are provided by spring loads of approximately 3 N. The glass bars are fixed to the body. An igniter is mounted in the cathode nozzle. The nozzle has a length of 28 mm and a half angle of 301.
Fig. 2. Configuration of PPT with propellant feeding mechanism. (a) Accelerator, (b) cross section of cavity.
2.2. Thrust stand and vacuum chamber In order to measure impulse bit precisely, a vacuum chamber and a thrust stand were developed, as shown in Fig. 3. The thrust stand has a 1-m-long perpendicular pendulum. The PPT and the energy storage capacitors are mounted on the pendulum, which rotates around fulcrums of two knife edges without friction. The displacement of the pendulum is detected by an eddy-current-type gap sensor (non-contacting micro-displacement meter) near the PPT, which resolution is about 70.5 mm. The electromagnetic damper is used to suppress mechanical noises and to decrease quickly the amplitude for the next measurement after firing the PPT. The damper consists of a permanent magnet fixed to the pendulum and two coils fixed to the supporting stand. A control circuit differentiates the output voltage of the displacement sensor and supplies the current proportional to the differentiated voltage to the coil. As a result, the damper works as a viscosity resistor. The damper is turned off just before firing the PPT for a measurement without damping. The calibration of impulse is carried out by a collision of a ball to the pendulum. The sensitivity of the thrust stand is variable by changing the weight mounted on the top of the pendulum as shown Fig. 4. The vacuum chamber pressure was kept at about 4 103 Pa under operations.
Fig. 3. Configuration of thrust stand and vacuum chamber.
2.3. Discharge circuit Six capacitors of mica-paper type with a total capacitance of 9.0 mF are used to store energy. Charge voltage is varied from 1000 to 1800 V, corresponding to stored energy of 4.5–14.6 J. A discharge current waveform is measured by a Rogowski coil.
ARTICLE IN PRESS T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228
qðAruÞ q qA þ ½Aðru2 þ pÞ ¼ p fLcir , qt qx qx qðAeÞ q þ ½Auðe þ pÞ qt qx ¼ AðQj Qrad Qconv þ Qab FÞ,
Amplitude per impulse bit, m / Ns
10
1
0.1
0.01 2
1225
3
4 Additional weight, kg
5
6
Fig. 4. Sensitivity of thrust stand dependent on top weight.
3. Calculation model 3.1. Assumptions Basic assumptions for the plasma flow are: (1) local thermodynamic equilibrium (LTE) is established: T e ¼ T i ¼ T n T, (2) ionization equilibrium (ionization degree is estimated by Saha’s equation), (3) one-fluid plasma, (4) quasi-neutral plasma, (5) effects of magnetic field are negligible. The additional assumptions for the plasma near the PTFE surface are: (6) the following relationship is satisfied between the energy-transfer mean free paths of electron-heavy particle leh,E and heavy particle-heavy particle lhh,E [4]: Cavity length ð12 mmÞ4leh;E 4typical cavity radius ð0:8 mmÞblhh;E , (7) heat conduction by heavy particles inside the plasma is negligible. From assumption (6), the temperature jump of heavy particle on the surface is negligible. Therefore, the temperature of heavy particle near the surface is lower than that on the axis. On the other hand, electrons moving towards the surface do not lose their energy, and therefore the temperature of electron near the surface is approximately the same as that on the axis. Then T e bT i ¼ T n T s (Ts: temperature of PTFE surface) is assumed near the surface. In the axial direction, LTE is approximately established. 3.2. Equations and method of calculation Axial components of mass, momentum and energy conservations are: qðArÞ qðAruÞ þ ¼ Lcir G, qt qx
ð1Þ
where A is cross sectional area, r mass density, Lcir circumference of the cross section, G ablation mass flux, u axial velocity, p pressure, f momentum loss on the wall surface, e total energy: p=ðg 1Þ þ ru2 =2, and F energy loss due to viscosity. The specific heat ratio g is assumed to be a constant, 1.1 [5]. Qj is energy of Joule heating, Qrad emission energy of bremsstrahlung (braking radiation), Qconv convection energy loss, and Qab thermal energy of ablated PTFE, which are written as follows: Qj ¼ Zj 2 ;
Qrad ¼ 1:57 1040 n2e T 1=2 ,
Qconv ¼ ðqe þ qi þ qn ÞLcir =A, Qab ¼ G=mi 2kT s Lcir =A,
ð2Þ
where j is discharge current density, Z resistivity of plasma in the cavity, ne plasma density, qe, qi and qn convection energy flux to the propellant surface due to electrons, ions and neutral particles, respectively, considering effect of sheath on the PTFE surface, and Ts PTFE surface temperature. Considering assumption (6), Qconv is approximately written as [4]: 1 Lcir . (3) Qconv ji 2kT þ ef 2 A Ion flux ( ¼ electron flux) to the surface ji and potential drop in the sheath f satisfy 1 8kT s 1=2 ; ji ne 4 pmi
f
kT Tmi ln . 2e T s me
(4)
The discharge circuit is modeled as a LCR series circuit including the plasma resistance: Z € L0 Q þ R0 þ ðZ=AÞ dx Q_ þ Q=C ¼ 0, _ J ¼ Q, Z¼
ln L me 3kT 1=2 þ s n , en n me 1:53 102 T 3=2 ne e2
ð5Þ
where J is discharge current, sen cross-sectional area of electron neutral, and ln L the Coulomb logarithm. The resistance R0 and inductance L0 of the circuit are measured with frequency response method [2]. Assuming heat-layer thickness is much smaller than the radius of the PTFE bar’s curve, heat transfer equation inside the PTFE is as follows: 2 qY l q Y q2 Y ¼ þ 2 , qt rC p qx2 qr qY A DH G, ð6Þ l ¼ ðQconv þ Qrad Qab Þ qr r¼0 Lcir
ARTICLE IN PRESS T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228
8.8J (1400V) Calculation
4000
Experiment 3000
2000
1000
0
-1000
-2000
0
5
10
15
Time, s
Fig. 5. Calculated and experimental discharge current waveforms.
1000
300
4. Results and discussion
Exp. Impulse bit Cal. Impulse bit
4.1. Initial performances and physical phenomena Impulse bit, µN s
200 600 150 400 100 200
0
50
5
(a)
15
500 15 400 300 10 Specific impulse Thrust efficiency
200 100 (b)
0
20
4.2. 10 000-shot operation Ten thousand shot operation is conducted with a frequency of 0.5 Hz. Figs. 9 and 10 show the change in performance during the test and the shape of PTFE bars before and after the test, respectively. Both the impulse bit
10 Stored energy E0, J
600
Specific impulse, s
The calculated current waveform is agreed well with the experimental results as shown in Fig. 5. Initial thrust performance showed impulse bit of 192–686 mN s corresponding thrust-to-power ratio of 43–47 mN/W, specific impulse of 470–500 s and thrust efficiency of 10–12% when the stored energy was 4.5–14.6 J shown in Fig. 6. It is remarkable that thrust performance is kept high even at a low energy of 4.5 J. The calculated mass loss per shot agrees well with the experimental results. It is considered that phenomena in the cavity (condition of plasma, heat transfer to/inside the PTFE, ablation) are simulated well with this model. However, the calculated impulse bit is larger than the experimental result. The peaks of plasma temperature, discharge current, ablation rate, pressure and thrust at the center of cavity length appear in this order as shown in Fig. 7. After the ablation is completed, gas starts to expand, and thrust is generated until over 25 ms. Fig. 8 shows the temporal and spatial variation of Mach number. The Mach number increases in the cavity and becomes unity at the exit of the cavity ðx ¼ 12 mmÞ. The plasma is efficiently accelerated in the divergent nozzle.
250
Exp. ∆m Cal. ∆m
800
Mass loss per shot ∆m, µg
where pc ¼ 1.84 1015 Pa and Tc ¼ 20 815 K are the characteristic pressure and temperature, respectively. Momentum and energy loss due to viscosity f and F are expediently estimated with radial distribution of axial velocity which is assumed to be a distribution of ideal laminar or turbulent flow considering local Reynolds number. The calculation domain for plasma flow is between the anode and the nozzle exit. The plasma flow and the LCR series circuit are calculated with the TVD MacCormack scheme and the Runge–Kutta scheme, respectively. Initial energy is arranged in capacitors. In order to start the discharge, the plasma with 1.7 103 J and 1.1 mg, which are much less than the initial energy in capacitors and mass loss per shot, respectively, are arranged in the calculation domain as initial conditions.
5000
5
10 Stored energy E0, J
15
Thrust efficiency, %
where Y is temperature inside the PTFE, and DH the unzipping energy of PTFE: 1.5 106 J/kg [1]. The following ablation flux G is calculated with Langmuir’s law [6] using equilibrium vaporizing pressure pvap of PTFE [1,7]: 1=2 mi G¼ pvap ; pvap ¼ pc expðT c =T s Þ, (7) 2pkT s
Discharge current, A
1226
0
Fig. 6. Initial performance vs. stored energy. (a) Impulse bit and mass loss per shot, (b) Specific impulse and thrust efficiency.
ARTICLE IN PRESS T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228 800
10
20
E0=8.8J, x=6m m
700
Plasma temperature 8
Pressure, atm Ablation rate, g/s Thrust, N
Ablation rate Pressure
60
6
Thrust 40
4
20
2
600 Impulse bit, Ns Mass loss per shot, g Specific impulse, s
Current
Current, kA Plasma temperature, eV
80
Impulse bit Mass loss per shot Specific impulse Thrust efficiency
500 10
400 300
5
200 0
0 0
5
10
15
Time, µs
Fig. 7. Calculated temporal variations of plasma temperature, discharge current, ablation rate, pressure and thrust.
15 Thrust efficiency, %
100
1227
100 0
0
2000
4000 6000 Shot number
8000
10000
0
Fig. 9. Changes in performances during 10 000-shot operation.
Fig. 8. Calculated temporal and spatial variations of Mach number.
and the thrust efficiency gradually decrease with shot number, because gaps are gradually generated between the PTFE bars. These gaps do not derive a severe leakage of gas because there are glass bars on the two ends of the cavity. However, increase in cavity volume lead to decrease in pressure in the cavity. Heat loss to the glass should be also considered. Actually, mass loss of glass bars is observed, which is added to the mass loss and considered in estimating the specific impulse and thrust efficiency. Fig. 11 shows the calculated distribution of ablated mass per area versus time. This distribution qualitatively agrees with the observed uneven receding of the surface. The reason for this phenomenon is that the plasma density near the cavity exit is small because of the high velocity. Then, heat convection to the PTFE is small near the cavity exit. The mass loss per shot gradually decreases with shot
Fig. 10. Feature of PTFE bars before/after 10 000-shot operation. (a) Schematic, (b) Photograph after operation (top view).
number because heat loss to the glass surface increases. In spite of the uneven receding, the PTFE is also ablated near the cavity exit, and each bar was moved in the distance of approximately 2 mm toward the cavity during the 10000 shot operation. As a result, a total impulse of approximately 3.6 Ns is obtained. 5. Conclusions The following results were obtained in this study. 1. A PPT with a propellant feeding mechanism showed initial performances of thrust-to-power ratio of
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T. Edamitsu, H. Tahara / Vacuum 80 (2006) 1223–1228
performances, especially, discharge current waveform and mass loss per shot. 3. Ten thousand-shot operation was conducted, and a total impulse of 3.6 Ns was achieved. The propellant feeding mechanism worked well. However, both the impulse bit and the thrust efficiency gradually decreased because of uneven receding of the PTFE surface. The shape of the surface was qualitatively explained with the calculation. It was suggested that the axial distribution of ablation flux mainly depends on that of plasma density.
References
Fig. 11. Calculated distribution of ablated mass of PTFE per area.
43–47 mN/W, specific impulse of 470–500 s and thrust efficiency of 11–12% with stored energy of 4.5–14.6 J. 2. An unsteady calculation, which simultaneously simulates discharge, heat transfer to/inside the PTFE, ablation and plasma flow, was carried out. The calculated results roughly agreed with measured initial
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