A coaxial Teflon pulsed plasma thruster, UIUC PPT-7, was tested and results were reported in a previous paper. More thrust data is taken while varying the stored energy for different geometries. A thermal model is used to determine energy lost as heat from the thruster to be 14% of the available energy. A similar thermal model is used to estimate the portion of heat lost due to conduction into the Teflon fuel. The plasma current is curve-fit to reveal a linearly increasing plasma resistance. The plasma temperature is estimated to vary from 9000 K to 21000 K. Introduction The pulsed plasma thruster is a robust, solid state device that has flown on a number of missions. In a previous paper 2 the performance of UIUC PPT-7, an electrothermal, coaxial pulsed plasma thruster was reported. The Two-Stream model was used to describe the thruster efficiency. Since then, more tests have been performed, resulting in a greater understanding of the performance versus energy of the thruster. Also, temperature data is examined to help discuss the heat loss of the thruster. Experiments Apparatus Experiments were performed to help determine the effects of geometry and energy on the performance of an electrothermal thruster. The thruster Tests with stored energy less than 50 J use 14 parallel mica capacitor sections, 3 with a total capacitance of 9.2 µF. In order to test at 70 J, 7 more sections are added, increasing the capacitance to 14.1 µF. The 50 J baseline current, shown in where ∫ = Ψ dt I 2 and E pl is the energy in the plasma. However, when this resistance is used to curve-fit to the experimental data, it becomes apparent that the resistance is not constant throughout the pulse. The plasma current is best fit with a linearly increasing resistance, as shown in A number of diagnostics were used during these tests. The UIUC compact thrust stand 4 was used in single pulse mode to measure the thrust. At least 10 thrust measurements were taken for each geometry tested. The current was measured using a Rogowski coil on the central electrode. A 1000:1 high voltage probe (Tektronix P6015) was used to measure the capacitor discharge voltage at the vacuum tank feed-through. Temperature measurements of the propellant and one capacitor pack were taken using type K thermocouples. The location of the thermocouples is shown on The thruster and capacitor surface temperature measurements were used to determine the heat loss from the thruster and the transfer loss. Starting from room temperature with a constant pulse rate, the capacitor temperature was observed to rise linearly at the outer surface. Assuming that the temperature is uniform throughout the capacitor the power lost to heating is calculated from the measured constant temperature rise rate and the thermal mass (mC p ), according to The thermal mass of the capacitors, listed in Because the heat from the Teflon cavity is flowing radially through the thruster body, the temperature is not uniform throughout. For this reason, the above equation cannot be used to determine the power lost as heat in the thruster. A heat flow model described later is employed to match the temperature rise rate at the outer surface of the fuel to the power lost to heating. Tests Each geometry tested is characterized by the cavity diameter near the rear electrode, the cavity length, and the cavity exit diameter. The baseline constant diameter geometry for these experiments is 14/35/14, denoting a 14 mm rear cavity diameter, 35 mm length, and a 14 mm front cavity diameter. From this baseline case, three other diameters and two other lengths were tested. Tests were also performed without a nozzle in order to determine its effect on performance. Two lengths of tapered cavities were also tested: 3.5/20/14 and 3.5/35/14 with 15° and 10° half angle respectively. These test parameters are illustrated in Test procedure Each test consisted of 1000 shots repetitive firing at 1 Hz to warm up the thruster and burn-in the fuel tube. After burn-in, 10 thrust measurements were taken in single pulse mode, followed by another 1000 shots to reduce error in the mass loss measurement. Each pulse produces a thrust stand position transducer (LVDT) output which is a slowly decaying sinusoid. 3 The initial (t = 0) slope is determined by curve-fitting an analytical damped sinusoid to the LVDT waveform, which gives the post-pulse thrust stand platform velocity u p . The impulse bit is u p multiplied by the platform mass, which is weighed for each test setup. Thermocouple data, including the capacitor and fuel temperature, were recorded during all repetitive pulsing phases of the test. The second set of data, measuring the variation of thrust versus energy at 8 mm diameter, were performed with a slightly different procedure. After the pumps reached operating pressure, the thruster was fired 1000 times at 50 J, 1 Hz to warm up the fuel and thruster. Then, approximately 10 single shot thrust measurements were taken at energies ranging from 50 J to 10 J. If the thruster fired repeatably at lower energies, thrust was also measured at energies below 10 J. Results The following plots represent the data taken. The specific mass loss (ML) vs energy is then curvefit using a cubic polynomial fit. The specific impulse vs energy curve fit is then calculated using the specific thrust and specific mass loss fits acording to the equation I sp = T sp /(g o ML). Specific thrust and specific mass loss vs diameter and length is also curve-fit using the above polynomial fit. The corresponding specific impulse curve fit is calculated as above. Some of the polynomial curve fits were determined without consideration of possible data outside the range of the data points. For this reason, extrapolation using these curve fits may be erronious. Analysis The thruster efficiency η t can be expressed as the product of component sub-efficiencies. 8 These efficiencies include the pulse energy transfer η tr , thruster heat loss η h , frozen flow η f , exhaust beam divergence η div , and exhaust velocity distribution efficiency η dist . The transfer efficiency takes into account the energy lost to the equivalent series resistance (ESR) of the capacitor. This energy is estimated from the temperature increase during a 50 W test, and the thermal mass of the capacitor. A small portion of the heating power is due to heat flow from the thruster, however that amount is insignificant compared to the thruster heating. For the baseline case, Q = 2.5 W This is 5.0% of the 50 W making the η tr = 0.95. Another way to estimate the transfer efficiency is from the relation capacitor heating/pulse = Ψ × ESR . A separately-measured capacitor ESR 3 of 2.8 mΩ for the baseline case gives 4.5 W, making η tr = 0.91. We adopt a value of η tr = 0.93. The energy in the Plasma E pl is therefore the stored capacitor energy reduced by capacitor and wall/electrode heat losses, or E pl = η tr η h E o . Mass loss. The pulse specific mass loss (µg/J) is relatively constant with energy for the baseline (14/35/14) geometry. For constant energy (50 J), specific mass loss is seen to increase for decreasing 4 diameters The specific thrust of the baseline geometry is relatively constant with energy at 35 -38 µN-s/J from 20 -70 J, and then decreases significantly (Eq. 3) at lower energies. For the 8 mm constant diameter cases, specific thrust increases slightly with energy from 20 -50 J. Below 20 J, the specific thrust decreases similarly to the baseline case. This behavior is consistent with a heat loss model of ∆Q loss = B + AE o . Assuming that the specific thrust scales as T sp = C E pl /E o , it can be shown that: which is mathematically of the same form as the curve fit of Eq. 3. By comparing Eq. 5 to the curve fit Eq. 3, values of the constant can be extracted. For the baseline geometry, the constant C is 46 µN-s/J, A = 0.06, and B = 2.8 J, so that T sp = 46(0.87 -2.8/E o ) and the heat loss is ∆Q loss = 2.8 + 0.06E o . At E o = 50 J, ∆Q loss = 5.8 J, approximately 50% from the constant term B and 50% from the term proportional to discharge energy. We have speculated as to the origin of the constant term of 2.8 J. One possibility is that the plasma pulse quickly raises the Teflon surface temperature to the vaporization point, where it is capped by the sublimation process, freezing the temperature profile and therefore the heat conducted into the solid. This picture suggests that shorter pulse lengths would reduce the constant term in the ∆Q loss equation. Applying this model to the specific thrust curve fit of Specific Impulse. Specific impulse generally increases with energy, from 330 s at 10 J to 490 s at 70 J (baseline geometry). With diameter, I sp reaches a maximum of 450 s at 14 mm (baseline). I sp increases to above 500 s for lengths < 25 mm, and some benefits (610 s) is shown for a short tapered cavity. Heat loss In this paper, we are mostly concerned with the heat loss efficiency, the other efficiencies are described elsewhere. where T is the temperature, r is the radius, r i is the inner radius, r o is the outer radius, and k 1 , k 2 , k 3 , kp 1 , kp 2 , kp 3 , are all constants. The model begins with temperature of 0° everywhere. Since all the calculations are based on temperature differences, starting from room temperature does not affect the outcome. The boundary conditions required involve a constant influx of energy at the inner radius, and zero efflux of energy at the outer radius. Due to a numerical problem, it is not possible to make both the inner and outer radius boundary conditions contain only derivative terms with k 3 or kp 3 equal to zero, so a small temperature dependence is added to the efflux of energy at the outer radius. The empirically determined constants for the baseline case are listed in where k is the solid Teflon thermal conductivity, A is the surface area of the inside of the tube, and ∂T ∂r = k 3 . Surface Heat Conduction From the above model, it was determined that 6.4 Joules of energy is lost to thruster heating for every 50 J pulse. A portion of that heat is deposited into the electrode through sheath losses. The rest is deposited into the Teflon. As energy is deposited into the Teflon fuel, three processes are occurring. Heat is conducted into the fuel, heat is radiated out of the fuel, and heat is lost when fuel evaporates. This paper considers the convection of heat into the solid Teflon from the high temperature plasma. The timedependent temperature distribution inside the Teflon immediately after one pulse is obtained by using a model similar to the one used in the heat transfer analysis, with constant initial temperature throughout the Teflon except at the surface, according to the following equation. . the constants a and b are the inner and outer radius of the Teflon fuel, v 1 and v 2 are the surface temperatures at the inner and outer surface respectively. α n is the nth root of where J 0 and Y 0 are Bessel functions. U 0 is the relation and κ is the diffusivity which is K/ρCp or 7.8x10 -8 for Teflon. 6 The model requires knowing the surface temperature of the Teflon. It is assumed that the pressure in the cavity is the same as the vapor pressure of the Teflon. The vapor pressure of Teflon follows the equation: where p c = 1.84 x 10 15 and T c = 20800 K. Eq. 11 is a curve fit of data from reference 10, using the ClausiusClapeyron equation to fit the data. 1 The fit and data are shown in To estimate the length of time that the Teflon surface temperature remains at 675°C the sonic velocity of the plasma during the current pulse is determined by estimating plasma temperature. The plasma resistance, determined by subtracting the ESR from the circuit resistance used to curve-fit the current data lnΛ is determined according to: ln Λ = 23.4 -1.15 Log(n e ) + 3.45 Log(T ev ) The electron density n e (in cm -3 ) needed for lnΛ is determined using the Saha equation for a gas mixture, assuming only single ionization 13 : The partition function ratios are found from spectral data 14 and are 0.53 for carbon and 1.68 for fluorine. The following equations are then used to solve for the electron density n e p = (n e + n C + n C + + n F + n F + )kT (15) A rough estimate of conductive and radiative heating rate of the Teflon surface exposed to this temperature plasma at several atm predicts that the temperature rises to 675°C in a short time compared to the pulse length. Using the sonic velocity determined above, the length of time it would take an expansion wave to travel from the cavity entrance to the rear electrode is approximately 8 µs. For this model, since parts of the Teflon surface would be cooled before the expansion wave would reach the end of the cavity, it is assumed that the Teflon surface no longer absorbs heat from the plasma 4 µs after the current pulse. The measured heat loss of 6.4 J/pulse can be accounted for from two sources: a) heat stored in a thin surface layer of Teflon, and b) heat transferred to the electrodes by the voltage sheaths during the pulse. By using the thermal conduction model, a temperature distribution extending approximately 5 µm inside the Teflon is developed at the end of the pulse. From Ref 10 the specific heat of Teflon includes a phase change from crystalline to amorphous, corresponding to 59 J/g at 600 K. Using a constant C p of 1.4 J/kg-K the temperature distribution corresponds to 4.2 Joules of energy stored in the Teflon in the form of heat. This value is increased by 10% to account for the phase change to a final value of 4.6 Joules. This accounts for 72% of the energy lost due to thruster heating. The remaining heat (1.8 J) can be accounted for by the sheath loss at the electrodes, assuming ~20 V total drop. Conclusions The specific thrust vs energy for geometries with a smaller diameter cavity than the baseline show a specific Thrust variation of T sp =a-b'/E o , similar to the baseline case. A thermal model was used to show the 7 transfer efficiency η tr =0.93 and the heat loss efficiency η h = 0.86, corresponding to 6.4 J lost to the wall out of 50 J stored. The plasma current was curve-fit to reveal a linearly increasing plasma resistance from which the plasma temperature is estimated to vary from 9000 K to 21000 K. A thermal model is used to estimate that 72% of heat lost is by convection into the Teflon fuel and the remainder is due to the sheath drop
