#
Comprehensive Data Reduction for N_{2}O/HDPE Hybrid Rocket Motor Performance Evaluation

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## Abstract

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_{2}O) as the oxidizer and high-density polyethylene (HPDE) as the fuel are analyzed using a novel approach to data reduction that allows histories for fuel mass consumption, nozzle throat erosion, characteristic exhaust velocity (c

^{∗}) efficiency, and nozzle throat wall temperature to be determined experimentally. This is done by firing a motor under the same conditions six times, varying only the burn time. Results show that fuel mass consumption was nearly perfectly repeatable, whereas the magnitude and timing of nozzle throat erosion was not. Correlations of the fuel regression rate result in oxidizer port mass flux exponents of 0.62 and 0.76. There is a transient time in the c

^{∗}efficiency histories of around 2.5 s, after which c

^{∗}efficiency remains relatively constant, even in the case of excessive nozzle throat erosion. Although nozzle erosion was not repeatable, the erosion onset factors were similar between tests, and greater than values in previous research in which oxygen was used as the oxidizer. Lastly, nozzle erosion rates exceed 0.15 mm/s for chamber pressures of 4 to 5 MPa.

## 1. Introduction

_{2}O) for its low toxicity and self-pressurization ability [8]. To clarify the self-pressurizing ability of N

_{2}O, values for density, ρ, in kg/m

^{3}and vapor pressure, P

_{v}, in Pa are plotted as functions of temperature in Figure 1 based on data from the National Institute of Standards and Technology (NIST) online database [9]. The main conclusion that can be drawn from this figure is that N

_{2}O can be self-pressurized as a liquid with densities ranging from 700 kg/m

^{3}to 900 kg/m

^{3}at pressures ranging from 4 MPa to 6 MPa for a range of temperatures typically permissible to satellite operators—roughly 280 K to 300 K.

_{2}O. These include the aforementioned SpaceShipTwo, as well as its predecessor vehicle, SpaceShipOne [10], the highly successful Stuttgart University student-based hybrid sounding rocket, HEROS 3, reported by Kobald et al. [11], the rocket test sled trials of Muroran Institute of Technology reported by Nakata et al. [12], the small launch vehicles of TiSPACE Inc. reported by Chen and Wu [13], as well as the sounding rockets of Space Forest Ltd. reported by Gamal et al. [14]. However, it is also important to point out that N

_{2}O can be pressurized above its vapor pressure to increase the flow rate capability of, and prevent two-phase flow within, the feed system. For example, Whitmore et al. used helium to pressurize a liquid N

_{2}O feed system for closed-loop throttling and thrust control [15]. Similarly, the Peregrine sounding rocket project of NASA, Stanford University, and SPG Inc. reported by Zilliac et al. used helium to pressurize an N

_{2}O reservoir to slightly above the vapor pressure [16]. The current research uses a similar feed system, which will be described in detail in the Materials and Methods section.

_{2}O hybrid rocket projects to achieve high thrust-to-weight ratios. One strategy is to use non-liquifying fuels with multiple ports and/or chambers to increase the burning surface area and/or create a vortex-like flow field. This is true for SpaceShipOne/Two, and the hybrid rockets reported by Chen and Wu, and Whitmore et al. in [1,13,15]. The other strategy is to use liquefying fuels with a single port, in which case the entrainment of liquid droplets from the surface of the melting fuel leads to high burning rates. This is true for the hybrid rockets reported by Kobald et al., Gamal et al., and Zilliac et al. in [11,14,16]. In a 5 kN thrust-class sounding rocket launch reported by Nagata et al., the former strategy was employed with great success using a high-density polyethylene (HDPE) fuel of the Cascaded Multistage Impinging-jet (CAMUI) design and liquid oxygen as the oxidizer [17]. With a characteristic exhaust velocity, c

^{∗}, efficiency, η

^{∗}, of 99% and a sustained acceleration during launch of 4 G to 5 G, the major concern of this development project shifted to nozzle thermochemical erosion. The tests in follow-on research to investigate nozzle erosion consistently showed values of η

^{∗}greater than 95%. Furthermore, the mass “equivalent” regression rates, based on the fuel grain outer dimensions and mass consumption rates, reached values upwards of 3 mm/s, which is comparable to or exceeding that of liquefying propellants [18]. The main drawback of using a CAMUI-type fuel, or something similar in nature, is the rigor involved in the fuel design itself. It can be said that the main benefit of using liquefying fuels in place of geometrically complex fuels, like CAMUI, is the simplicity of the single port fuel design that can be used. Mazzetti et al. make a strong argument on behalf of hybrid rockets in general, but specifically for liquefying fuel-based hybrid rockets because of the combination of the high fuel regression rate and fuel design simplicity [19].

_{2}O) was selected to be the oxidizer for its long-term storability, non-toxicity, non-corrosiveness, low cost, widespread commercial availability, self-pressurizing ability, and successful role in the numerous projects mentioned in the previous paragraphs. High-density polyethylene (HDPE) was also selected for its non-toxicity, low cost, and widespread commercial availability, as well as for its mechanical strength and heritage in the development of the CAMUI-type hybrid rocket by the authors and their predecessors.

_{2}O/HDPE, has mostly gone overlooked in previous research due to the heavy focus on the development of launch vehicles with high thrust-to-weight ratios. The latest version of the SpaceShipTwo engine may be an exception to this claim; however, the details of the current propellant combination have not been made public yet. The two previous publications that were closely related to the proposed work were an experimental study conducted by Doran et al. on a laboratory-scale N

_{2}O/HDPE hybrid rocket [20], and a numerical study of nozzle thermochemical erosion conducted by Bianchi and Nasuti [21]. Doran et al. reported η

^{∗}values ranging from 90% to 95%, and a weak correlation between fuel regression rates—ranging from 0.4 mm/s to 0.6 mm/s—and oxidizer port mass flux. Bianchi and Nasuti predicted that the erosion rate of graphite is 0.06 mm/s when the equivalence ratio is close to 1 and the pressure is 1 MPa. Moreover, they demonstrated that as a general trend among all propellant combinations, the erosion rate decreases with an increasing equivalence ratio and is linearly dependent on pressure.

^{2}-s. Also, Doran et al. supplied N

_{2}O to the motor as a gas, whereas it is crucial for the purposes of this study to supply the oxidizer as a liquid. Bianchi and Nasuti’s results will prove to be consistent with this study, but they are only valid when the nozzle throat temperature has reached a steady state.

_{2}O/HDPE, the authors found it necessary to conduct basic research to form empirical correlations for the fuel regression rate, c

^{∗}, efficiency, η

^{∗}, and nozzle erosion for the first time. Of the multitude of measurement techniques introduced in previous research, the latest versions of the data reduction methods referred to as ballistic reconstruction techniques appeared to offer the most effective means of pursuing this research in a cost-effective and expedient way. In general, ballistic reconstruction techniques only require some combination of the following commonly measured experimental values: (1) Oxidizer mass flow rate, ${\dot{m}}_{ox}$; (2) chamber pressure, P

_{c}; (3) thrust, F; (4) overall fuel mass consumed, ΔM

_{fu}; and (5) final nozzle throat diameter, d

_{t,f}. This means that the same measurement equipment can be used regardless of fuel design, configuration, or scale. This is done by using either the c

^{∗}equation, thrust equation, or both, in an iterative algorithm to determine instantaneous values of the oxidizer-to-fuel mass ratio, ξ; c

^{∗}efficiency, η

^{∗}; thrust correction factor, λ; and, recently, nozzle throat area, A

_{t}. Wernimont and Heister, and Nagata et al. introduced ballistic reconstruction techniques, which used the c

^{∗}equation to determine fuel mass consumption under the assumption that η

^{∗}is constant and nozzle throat erosion is negligible [22,23]. Carmicino and Sorge, and Nagata et al. alleviated the need to treat η

^{∗}as a constant by measuring thrust and incorporating the thrust equation [24,25]. Most recently, Kamps et al. combined these works, incorporating both the c

^{∗}and thrust equations, and treating η

^{∗}as a constant in order to determine the history of the nozzle throat area in an effort to investigate nozzle throat erosion [26]. This ballistic reconstruction technique was titled “Nozzle Throat Reconstruction Technique” or “NTRT,” and was later used in a follow-on study by Kamps et al., which demonstrated how to use the results of the NTRT with thermocouple measurements from within the nozzle to determine the wall temperature history at the nozzle throat [27]. This follow-on technique was titled the “Throat Temperature Reconstruction Technique” or “TTRT.”

^{∗}can be used to overcome this problem in the data reduction of hybrid rocket firing tests using oxygen. However, as will be evident in the following section, these approximations are not suitable when using N

_{2}O as the oxidizer because the region of multiple solutions that exists is too large. The objective of this research is to further develop the experimental methodology introduced by Kamps et al. so that the analysis of tests conducted under the propellant combination of N

_{2}O/HDPE yields results for the histories of fuel consumption, nozzle erosion, and characteristic exhaust velocity efficiency regardless of the oxidizer-to-fuel-mass ratio. The purpose of this research is to lay the groundwork for effectively conducting basic combustion research in the development of a hybrid rocket apogee kick motor using N

_{2}O as the oxidizer and non-liquefying fuels, such as HDPE, as the fuel.

## 2. Materials and Methods

_{t}and ξ histories through the data reduction of commonly measured experimental values with the aim of enabling hybrid rocket researchers to accurately and cost-effectively investigate nozzle erosion experimentally [26]. The two governing equations of this method are the thrust equation shown by (1), and the characteristic exhaust velocity equation shown by (2):

_{e}is the nozzle exit velocity in m/s, P

_{e}and P

_{a}are the nozzle exit pressure and atmospheric pressure in Pa, and A

_{t}and A

_{e}are the nozzle throat area and exit area in m

^{2}. The thrust correction factor, λ, accounts for the momentum losses due to non-one-dimensionality and non-isentropicity of flow, as well as the momentum losses in the axial direction due to the divergence angle of the nozzle exit. The propellant mass flow rate can be shown in terms of the oxidizer-to-fuel-mass ratio according to Equation (3):

_{e}is calculated implicitly from Equation (4):

_{e}is calculated explicitly from Equation (5):

_{c}and ξ [29]. Although it is not a problem in the tests conducted in this study, it is worth noting that Equations (4) and (5) are not valid if shocks occur in the nozzle, which is possible in ground tests where the chamber pressure is relatively low. Figure 2 plots F as a function of ξ for the case where d

_{t}is 4 mm, P

_{c}is 4.7 MPa, and ${\dot{m}}_{ox}$ is 36 g/s—which is representative of the tests conducted in this study. Here, the efficiency terms, η

^{∗}and λ, were assumed to be 1 (i.e., 100%). The range of ξ from 4.0 to 7.1 results in roughly the same value for thrust. Specific impulse, I

_{sp}, has also been plotted in Figure 2 to show that optimal performance is achieved when ξ is roughly equal to 7. This is important because in a typical hybrid rocket, the value of ξ increases in time. Thus, for the goal of maximizing I

_{sp}in long duration firing tests, it may be advantageous to begin operations in the multiple solutions region of ξ. Furthermore, from the vantage point of nozzle erosion prevention, it is advantageous to operate with the smallest value of ξ that is possible [21].

_{fu}, which is a constant value input to the original NTRT, will be replaced by a fuel mass consumption history, m

_{fu}. Thus, η

^{∗}can be solved at every time, rather than be treated as a constant. Since the outputs of the data reduction method in this research are in effect the same as the NTRT, the method in this paper will be referred to as “NTRT plus” or “NTRT+.”

^{∗}history. Convergence of loop A depends on reducing the final throat diameter residual, ψ

_{d}, to zero, according to Equation (6):

_{t}(t

_{b}) is the calculated value for the nozzle throat diameter at the end of the firing duration. The calculated nozzle throat diameter history is backed out of the value for the nozzle throat area, A

_{t}, which is determined by rearranging the terms in Equation (2), as shown by Equation (7):

_{F}, to zero, according to Equation (8):

^{∗}is iterated until the thrust calculation matches the measured value at that time.

_{ox}is in kg/m-s

^{2}, and r

_{fu}is the fuel port radius in m. Two methods of fuel regression rate correlations are compared in this study. In both cases, the concept behind the correlations is to run tests under the same oxidizer mass flow rate, varying only the combustion time. As the fuel burns, the port diameter will increase and the oxidizer port mass flux will decrease accordingly. This is different than the traditional approach of limiting the combustion time and varying the oxidizer mass flow rate to vary the oxidizer port mass flux. With that in mind, the correlation procedure remains essentially the same. The more conventional correlation procedure is to use only the endpoint data (i.e., changes in value before and after firing) for fuel mass consumption and the measurement of the oxidizer mass flow rate. This method is referred to as the “endpoint” method. By separating the variables of the port radius, r

_{fu}and time, t, Equation (9) can be integrated as follows in Equation (10):

_{b}is the burn time in s. Burn time is defined as the duration of time from the moment the (gauge) chamber pressure reaches 10% of the maximum value to the time at the aft-bisector of the pressure drop at the end of burning. This designation is outlined in [31] (p. 459), and was used in [26,27]. The fuel port radius in Equation (9) is calculated based on the fuel mass, M

_{fu}, fuel density, ρ

_{fu}, and the outer diameter and length of the fuel, D

_{fu}and L

_{fu¬}, according to Equation (11):

_{fu}, D

_{fu}, and L

_{fu}are in m, M

_{fu}is in kg, and ρ

_{fu}is in kg/m

^{3}. The constant, a, is determined by using the least-squares method (for a given value of n) on Equation (10), and the exponent, n, is determined by finding the value that maximizes the correlation of determination (i.e., R

^{2}) of the results of a. The second fuel regression correlation method attempted in this paper simply considers the solution to Equation (11) for all times, and performs the least-squares method directly on Equation (9). This is possible because we can integrate the fuel mass consumption history trendline from the tier I operations to determine the history of M

_{fu}.

_{w}. Convergence of the TTRT depends on reducing the thermocouple temperature residual, ψ

_{T}, to zero:

_{n1}is the temperature at the position of the thermocouple placed closest to the nozzle throat in K. In general, the governing differential equation for 1D conductive heat flux in cylindrical coordinates reduces to Equation (13) by assuming a negligibly small axial temperature gradient and no internal heat generation:

^{2}/s, T is the local instantaneous temperature in K, and r is the radial position from the centerline of the nozzle in m. Approximating the partial derivatives in Equation (13) by the first term of the Taylor series expansions near the points of interest, consolidating terms, and rearranging yields the following finite difference equation:

_{w}, and the other set as the nozzle thermocouple measurement temperature, T

_{n2}, we can solve for the temperature distribution history within the nozzle according to Equation (15):

^{−5}m and a time step of Δt = 0.5 s was used for the calculations in this study.

_{2}O reservoir; a purge gas line with an N

_{2}gas tank; and an ignition assist oxidizer line with an O

_{2}gas tank. The flow in each line was controlled using a Swagelok pneumatic ball valve operated by an SMC solenoid valve. Solenoid valve operations were controlled in LABVIEW7. The N

_{2}O in the reservoir, which was stored at outdoor atmospheric temperature (≈290 K), was pressurized to 7.6 MPa using Ar gas to ensure that it remained in a liquid phase until injection into the combustion chamber. The oxidizer mass flow rate was determined by measuring the pressure drop across an orifice plate with an orifice diameter of 1 mm located between the N

_{2}O reservoir and the motor. The equation used to calculate oxidizer mass flow rate is (16):

_{or}is a dimensionless orifice flow coefficient determined experimentally to be 0.92, d

_{or}is the orifice hole diameter in m (i.e., 0.001 m), ρ

_{ox}is the oxidizer density in kg/m

^{3}, and P

_{up}and P

_{dw}are the pressure upstream and downstream of the orifice in Pa, respectively. Note that the density of N

_{2}O is a function of temperature, and was calculated based on the following equation determined from the NIST database [9]:

_{2}O in °C.

_{2}H

_{4}) with a density of 955 kg/m

^{3}as the fuel. Fuel grains were assembled from multiple short cylindrical fuel blocks, which allowed for a pseudo-one-dimensional evaluation of the fuel mass consumption. As is shown in Figure 5b,c, the initial fuel port diameter of the first four blocks (blocks ①–④) was 20 mm and that of the last four blocks (blocks ⑤–⑧) was 30 mm. The step-increase in port diameter between blocks ④ and ⑤ was used to force a disturbance in the boundary layer and improve mixing. After assembly, fuel grains were loaded into glass fiber-reinforced plastic (GFRP) insulating tubes and sealed in a steel motor case. An impinging-type injector with 4 × 0.8 mm holes at a convergence angle of 45° was used to promote atomization. The nozzles used in all tests were manufactured using the same grade of isotropic graphite, Tokyo Tokai Carbon Ltd. G347. The density and thermal conductivity at atmospheric conditions are listed by the manufacturer to be ρ

_{n}= 1850 kg/m

^{3}and k = 116 W/m-K, respectively [32]. The temperature dependency of these and other properties of G347 graphite is not specified by the manufacturer, and so values were referenced from previous research on similar graphite. An empirical correlation of data for thermal conductivity, k, based on Figure 1 in [33] yields Equation (18):

_{p}, based on Figure 1 in [34] yields Equation (19):

_{p}is in J/kg-K, and T is in K. The initial nozzle throat diameter, d

_{t,o}, was 4 mm in all tests, and the nozzle exit diameter, d

_{e}, was 8 mm in all, but one test. The nozzle exit diameter in a single test was lowered to 6 mm to examine the effect of the nozzle expansion ratio, ε, on the nozzle throat erosion history and thrust correction factor. Thus, in all tests, with the exception of one, ε is 4, and in one test, ε is 2.25. There are three thermocouple measurement ports at the nozzle throat plane. The initial radial distances from the thermocouple measurement points to the throat are 5 mm, 8 mm, and 10 mm, as shown in Figure 5c.

_{2}) to ensure heating and gasification of fuel. After two seconds of O

_{2}ignition, the O

_{2}line was closed, and the main oxidizer (N

_{2}O) line was opened. Firing tests were shut down by closing the N

_{2}O line and opening the N

_{2}purge line. The chamber was purged with N

_{2}for a minimum of 30 s, both to extinguish the combustion of fuel and to cool the nozzle to less than 500 K for handling.

_{t,o}and d

_{t,f}, were taken by the image analysis of digital photographs of the nozzle before and after firing. A length scale was established by placing a plaque of 1 mm-spacing grid paper next to the nozzle being photographed. The nozzle throat diameter was computed based on the area of the lighted region at the throat using ImageJ software [35].

_{y}, in some operational output, y, as a function of the inputs, x

_{i}:

_{xi}terms on the right-hand side represent the uncertainties in the x

_{i}measurements. The partial derivative terms in Equation (20) represent the sensitivity of the reconstructed solution to each input. Since the algorithms in most of the data reduction operations are coupled non-linear problems, the partial derivative terms in Equation (20) are approximated as (21):

_{i}, has been perturbed by the amount of 1% of the nominal value.

## 3. Results

_{t}, are listed in mm, the values of overall fuel mass consumption, ΔM

_{fu}, are listed in g, the time-averaged thrusts, $\overline{F}$, are listed in N, the time-averaged oxidizer mass flow rates, ${\overline{\dot{m}}}_{ox}$, are listed in g/s, the time-averaged chamber pressures, ${\overline{P}}_{c}$, are listed in MPa, and the time-averaged nozzle temperatures, ${\overline{T}}_{10mm}$, ${\overline{T}}_{8mm}$, and ${\overline{T}}_{5mm}$, are listed in K. An inspection of the overall fuel mass consumption and final nozzle throat diameters of Tests-10s-I through to III reveals that nozzle throat erosion was not the same even though the initial test conditions were identical, whereas the values of fuel mass consumption were within 1% of one another. Test-3s did not have a measurable amount of nozzle throat erosion, whereas the remaining tests did. The primary reason for this is that the nozzle throat wall temperature did not become high enough to allow for chemical reactions with the combustion gas. This is further discussed in Section 3.4. The greater than symbol “>” means that the thermocouple overheated during the firing, and thus the time-averaged temperature shown is an underestimate of the true value.

#### 3.1. Tier I Results: Fuel Mass Consumption History Trendline

#### 3.2. Tier II Results

#### 3.2.1. Fuel Regression Correlations

_{ox}> 50 kg/m

^{2}-s. The most important discrepancy is the large difference in the solution of the fuel regression rate exponent, n. The history correlation results in an exponent of 0.76, which is very close to the theoretical value of 0.8 that is expected by Marxman et al.’s diffusion-limited boundary layer combustion theory [30,41], whereas the endpoint method results in an exponent of 0.62, and Doran et al. reports a value of 0.331 [20]. The departure of the fuel regression correlation exponent, n, from a value of 0.331 in Doran et al.’s study [20] to either 0.62 or 0.76 in this study, depending on whether the endpoint method or history method is used, warrants a discussion.

_{ox}tested in this study is too narrow. The time-averaged values of G

_{ox}varied from 50 kg/m

^{2}-s to over 300 kg/m

^{2}-s in Doran et al.’s test series, but only from 50 kg/m

^{2}-s to 100 kg/m

^{2}-s in this study. With that stated, it is also clear from this study that the largest changes in G

_{ox}take place in the first three seconds of firing, which constitute half of the burn time in Doran et al.’s tests. Given that even the time-averaged endpoint values from this study overestimate the prediction of Doran et al. by nearly 50%, it seems possible that the low exponent observed by Doran et al. may simply be due to the uncertainty introduced by using time-averaged values in the fuel regression rate correlation of Equation (9). If so, the consequences of this are grave, because the increase or decrease in time of the value of ξ depends on whether the exponent, n, is greater than or less than 0.5. In other words, the results of this study will predict an increase in ξ in time, whereas the results of Doran et al. will predict a decrease of ξ in time.

#### 3.2.2. Results of the NTRT+

^{∗}efficiency, ${\overline{\eta}}^{\ast}$, are listed in Table 2. The results of the NTRT+ for oxidizer-to-fuel-mass ratio, ξ, history are shown in Figure 10a, while the results for nozzle throat diameter history, d

_{t}, and c

^{∗}efficiency history, η

^{∗}, are shown in Figure 10b. The histories of the oxidizer mass flow rate, ${\dot{m}}_{ox}$, were added to Figure 10a to show that increases in ${\dot{m}}_{ox}$ correspond to increases in ξ. The close agreement of all throat diameter histories with the known value for d

_{t,o}in the beginning of the tests bolsters the reliability of the NTRT+ results, since d

_{t,o}is a known value that is not used in the NTRT+ algorithm. It can be seen from these results that the timing of the onset of nozzle throat erosion varies by 2 s between tests, from around 3 s in Tests-4s and 10s-I to around 5 s in Tests-10s-II and 15s. Also, the erosion rate of Test-10s-I is noticeably smaller than that of the other tests. Test-10s-I is the one test which employed a smaller nozzle expansion ratio than the other tests—2.25 versus 4.0. Thus, it is likely that the nozzle expansion ratio effected some aspect of the nozzle throat erosion mechanisms. Similar to the chamber pressure and thrust histories, the histories of η

^{∗}were the same during the first three seconds of burn time, during which time they sharply rose to a steady-state value. The time averages of the steady-state values of η

^{∗}are listed in Table 2. The values differ between tests in the range of 0.85 < η

^{∗}< 0.95, which is consistent with the results of Doran et al. [20].

#### 3.3. Tier III Results: Results of the TTRT

_{5mm}, but in two occasions T

_{check}

_{,}failed due to overheating. This typically happened around 5 s into the burn time. The results of the TTRT for nozzle throat wall temperature histories, T

_{w}, in K are shown in red in Figure 11a. The two thermocouple measurement histories used as input data to the TTRT, at depths of 10 mm (T

_{n2}) and 5 mm (T

_{n1}) from the nozzle throat, are also shown in in Figure 11a in black and blue, respectively. The procedure for confirming the results of T

_{w}is depicted in Figure 11b, where the thermocouple located at a depth of 8 mm from the nozzle throat (T

_{check}) was used to verify the calculated value at that position within the nozzle. The solid black lines in Figure 11b show the calculated temperature profiles in the nozzle at 1 s intervals. The close agreement between calculated and measured values in Test-3s as shown in Figure 11b was also observed in all tests for the range of times where the thermocouples did not fail due to overheating.

#### 3.4. Erosion Onset Factor

_{sto}(ξ

_{sto}= 9.4 for N

_{2}O/HDPE), and the measured value of oxidizer-to-fuel mass ratio, ξ. Since the timing of the onset of erosion can be identified from Figure 10b, a brief analysis of Π will be conducted in this section. The equation for Π is (22):

_{Π}is a dimensionless empirical constant, and E

_{Π}is an empirical constant with units of K. Based on the results of [27], the following values of empirical constants will be used: n

_{Π}= 1.03 and E

_{Π}= 1408 K. Here, it is necessary to identify the pressure at the nozzle throat and the temperature of the nozzle throat wall position at the onset of nozzle throat erosion—listed as P

_{Π}in Pa and T

_{Π}in K in Table 2—so that values for the erosion onset factor may be calculated. The subscript, Π, is used to indicate the data taken at the time of the onset of erosion. Here, the value of P

_{Π}was taken to be the pressure at the nozzle throat, P

_{t}, which was approximated by the equation for the isentropic expansion of an ideal gas:

_{Π}, were chosen to be the time when the d

_{t}histories in Figure 10b departed from the value of d

_{t,o}and had an erosion rate of more than 0.02 mm/s for longer than 1 s. The values for nozzle throat wall temperature at the onset of erosion were obtained from the T

_{w}histories plotted in Figure 11a. The range of equivalence ratios, Φ, was not wide enough to carry out a correlation of Π, such as that in [27]. Thus, the data from [27] was used as a reference to evaluate the relative erosion characteristics observed in this study. Fortunately, there is a group of data and a trendline from [27] that lines up with the values of Φ observed in this study. This is shown in Figure 12. We can draw two useful conclusions from this comparison. The first conclusion is that even though nozzle erosion histories were not repeatable between tests in this study, the values of Π were. This suggests that the onset of erosion was essentially chemical-controlled. The second conclusion is that the use of N

_{2}O in place of O

_{2}increased the threshold for Π. This is expected, since the presence of nitrogen in the combustion gas reduces the concentration of oxidizing species. Similar findings are reported by Bianchi and Nasuti regarding the difference in erosion rates when using N

_{2}O as an oxidizer versus O

_{2}[21].

## 4. Conclusions

_{2}O) and high-density polyethylene (HDPE) shows promise for use in hybrid rocket apogee kick motors for the attributes of storability, non-toxicity, and stable burning with minimal combustion oscillations. However, there has been very little research published on the performance of hybrid rockets employing this propellant combination, and the research that has been made available to the public is limited in scope and inadequate for serious design considerations. This study introduces an improvement to well-tested non-intrusive data collection and analysis methods called ballistic reconstruction techniques in order to cost-effectively collect data on hybrid rockets using this propellant combination. The concept of this improvement is to introduce time-resolved fuel mass consumption into the governing equations of characteristic exhaust velocity and thrust, making it possible to simultaneously collect and evaluate time histories of the fuel regression rate, combustion efficiency, and nozzle throat erosion. The results were a highly cost-effective test series. With just six firing tests, the fuel regression rate could be correlated (a = 2.52 × 10

^{−5}; n = 0.76; such that the regression rate is in units of m/s), the combustion efficiency transients and trends became known (2 s transient; steady state values of 87% to 94%), and the erosion onset factor threshold was shown to be 50% larger than in the case where O

_{2}is used as the oxidizer. Moreover, now it is clearer how the chamber pressure, equivalence ratio, and nozzle heating transients at the beginning of a burn dictate when nozzle erosion is likely to begin.

_{2}O/HDPE, including the apogee kick motor being developed by the authors. Essentially, all key aspects of combustion can be examined with a limited number of trials. This method may be especially helpful when studying the behavior of long-duration firing tests, for which neither time-averaged analysis methods nor high-resolution direct measurement techniques, such as X-ray radiography, are practical.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a,n | empirical constants of Equation (9) |

A_{e} | nozzle exit area, m^{2} |

A_{t} | nozzle throat area, m^{2} |

C_{or} | dimensionless orifice flow coefficient |

c_{p} | specific heat capacity of the nozzle, J/kg-K |

${c}_{th}^{\ast}$ | theoretical characteristic exhaust velocity, m/s |

d_{e} | nozzle exit diameter |

D_{fu} | fuel outer diameter, m |

d_{or} | orifice hole diameter, m |

d_{t} | nozzle throat diameter, m |

E_{Π},n_{Π} | empirical constants of Equation (22) |

F | thrust, N |

G_{ox} | oxidizer port mass flux, kg/m^{2}-s |

I_{sp} | specific impulse, s |

k | thermal conductivity of the nozzle, W/m-K |

L_{fu} | fuel length, m |

M_{fu} | fuel mass, kg |

m_{fu} | fuel mass consumption, kg |

$\dot{m}$ | propellant mass flow rate, kg/s |

${\dot{m}}_{ox}$ | oxidizer mass flow rate, kg/s |

P_{a} | atmospheric pressure, Pa |

P_{dw} | orifice downstream pressure, Pa |

P_{e} | nozzle exit pressure, Pa |

P_{t} | nozzle throat pressure, Pa |

P_{up} | orifice upstream pressure, Pa |

P_{v} | vapor pressure (of N_{2}O), Pa |

r_{fu} | fuel port radius, m |

${\dot{r}}_{fu}$ | fuel (port) regression rate, m/s |

t | time, s |

t_{b} | burn time, s |

T_{w} | nozzle wall temperature |

U | experimental uncertainty |

u_{e} | nozzle exit velocity, m/s |

x,y | arbitrary input, output |

Δr | radial mesh spacing, m |

Δt | time step or data recording interval, s |

α | thermal diffusivity of the nozzle, m^{2}/s |

η^{∗} | efficiency of ${c}_{th}^{\ast}$ |

γ | specific heat ratio of combustion gas |

λ | thrust correction factor |

Φ | equivalence ratio |

Π | erosion onset factor |

ρ_{ox} | oxidizer density, kg/m^{3} |

ψ_{d} | nozzle throat diameter residual |

ψ_{F} | thrust residual |

ψ_{T} | thermocouple temperature residual |

ξ | oxidizer-to-fuel-mass ratio |

Additional Subscripts | |

i | radial node index in Equations (14) & (15) |

o,f | initial, final |

w | nozzle wall position |

5mm, 8mm, 10mm | thermocouples positioned 5 mm, 8 mm, and 10 mm from the nozzle throat |

Π | specifies property at the onset of nozzle erosion |

Additional Superscripts | |

j | time index in Equations (14) & (15) |

Acronyms | |

CAMUI | Cascaded Multistage Impinging-jet |

CEA | (NASA) Chemical Equilibrium with Applications |

GFRP | Glass Fiber-Reinforced Plastic |

GTO | Geostationary Transfer Orbit |

HDPE | High-Density Polyethylene |

ISAS | Institute of Space and Astronautical Science (Japan) |

JAXA | Japan Aerospace Exploration Agency |

NASA | National Aeronautics and Space Administration (USA) |

NIST | National Institute of Standards and Technology (USA) |

NTRT | Nozzle Throat Reconstruction Technique |

PMMA | Polymethyl Methacrylate |

TTRT | Throat Temperature Reconstruction Technique |

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**Figure 5.**The experimental apparatus (

**a**) test setup; (

**b**) cut-away view of the motor assembly; (

**c**) detailed schematics of the fuel grain and nozzle (units of mm).

**Figure 6.**Screenshots of the firing test sequence (

**a**) nichrome wire heating (0 s to 10 s); (

**b**) O

_{2}ignition assist (10 s to 12 s); (

**c**) main burn (12 s to 12 + t

_{b}s); (

**d**) N

_{2}purge/cooling (12 s + t

_{b}s + 30 s).

**Figure 7.**Unprocessed static firing test data of all tests; (

**a**) the first 40 s of the firing sequence, (

**b**) expanded graph of the first 4 s of ignition and start-up transients.

**Figure 8.**Fuel consumption analysis (

**a**) fuel mass consumption trendline; (

**b**) fuel regression by block.

**Figure 10.**The NTRT+ analysis (

**a**) propellant flow rates; (

**b**) throat erosion and efficiency histories.

**Figure 11.**The TTRT analysis (

**a**) nozzle temperature histories; (

**b**) confirmation of temperature profile histories in Test-3s.

Test | Local | Δd_{t} | ΔM_{fu} | $\overline{\mathit{F}}$ | ${\overline{\dot{\mathit{m}}}}_{\mathit{o}\mathit{x}}$ | ${\overline{\mathit{P}}}_{\mathit{c}}$ | ${\overline{\mathit{T}}}_{10\mathit{m}\mathit{m}}$ | ${\overline{\mathit{T}}}_{8\mathit{m}\mathit{m}}$ | ${\overline{\mathit{T}}}_{5\mathit{m}\mathit{m}}$ |
---|---|---|---|---|---|---|---|---|---|

Label | Name | mm | g | N | g/s | MPa | K | K | K |

3s | (CBX-5) | 0.05 ± >100% | 30 ± <1% | 74 ± 4% | 41 ± 4% | 4.1 ± 1% | 518 | 551 | 661 |

4s | (CBX-4) | 0.20 ± 50% | 35 ± <1% | 78 ± 4% | 41 ± 4% | 4.1 ± 1% | 520 | 570 | 717 |

10s-I | (CBX-6) | 0.53 ± 20% | 74 ± <1% | 81 ± 4% | 39 ± 4% | 4.2 ± 1% | 551 | >582 | >745 |

10s-II | (CBX-8) | 0.75 ± 10% | 73 ± <1% | 89 ± 4% | 38 ± 4% | 4.4 ± 1% | 710 | 776 | >874 |

10s-III | (CBX-9) | 1.00 ± 10% | 74 ± <1% | 83 ± 4% | 39 ± 4% | 4.1 ± 1% | 556 | 600 | >764 |

15s | (CBX-7) | 0.80 ± 13% | 110 ± <1% | 88 ± 4% | 37 ± 4% | 4.3 ± 1% | 658 | >723 | >859 |

Test | Local | λ | ${\overline{\mathit{\eta}}}^{\ast}$ | ${\overline{\mathit{T}}}_{\mathit{w}}$ | t_{Π} | Φ | P_{Π} | T_{Π} |
---|---|---|---|---|---|---|---|---|

Label | Name | K | s | MPa | K | |||

3s | (CBX-5) | 0.97 ± 10% | 0.87 ± 11% | 1243 ± 9% | 3 | 2.03 ± 4% | 2.6 ± 2% | 1778 ± 10% |

4s | (CBX-4) | 1.00 ± 10% | 0.86 ± 10% | 1464 ± 10% | 3 | 2.02 ± 4% | 2.6 ± 2% | 2030 ± 12% |

10s-I | (CBX-6) | 0.97 ± 10% | 0.88 ± 10% | 1467 ± 10% | 4 | 1.95 ± 4% | 2.6 ± 2% | 2076 ± 9% |

10s-II | (CBX-8) | 0.98 ± 9% | 0.94 ± 9% | 1455 ± 11% | 5 | 1.96 ± 4% | 2.7 ± 2% | 1811 ± 7% |

10s-III | (CBX-9) | 0.90 ± 8% | 0.91 ± 9% | 1538 ± 10% | 3 | 2.05 ± 4% | 2.7 ± 2% | 1963 ± 11% |

15s | (CBX-7) | 0.94 ± 8% | 0.94 ± 10% | 1527 ± 8% | 5 | 1.95 ± 4% | 2.7 ± 2% | 1905 ± 7% |

^{∗}were taken after the first 3 s into the burn time to avoid startup transients.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kamps, L.; Sakurai, K.; Saito, Y.; Nagata, H. Comprehensive Data Reduction for N_{2}O/HDPE Hybrid Rocket Motor Performance Evaluation. *Aerospace* **2019**, *6*, 45.
https://doi.org/10.3390/aerospace6040045

**AMA Style**

Kamps L, Sakurai K, Saito Y, Nagata H. Comprehensive Data Reduction for N_{2}O/HDPE Hybrid Rocket Motor Performance Evaluation. *Aerospace*. 2019; 6(4):45.
https://doi.org/10.3390/aerospace6040045

**Chicago/Turabian Style**

Kamps, Landon, Kazuhito Sakurai, Yuji Saito, and Harunori Nagata. 2019. "Comprehensive Data Reduction for N_{2}O/HDPE Hybrid Rocket Motor Performance Evaluation" *Aerospace* 6, no. 4: 45.
https://doi.org/10.3390/aerospace6040045