# Hybrid Rockets as Post-Boost Stages and Kick Motors

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}envelopes of varying height-to-base ratios. Theoretical maximum ΔV are evaluated first, assuming constant O/F and no nozzle erosion. Of the four common liquid oxidizers: H

_{2}O

_{2}85 wt%, N

_{2}O, N

_{2}O

_{4}, and LOX, H

_{2}O

_{2}85 wt% is shown to result in the highest ΔV, and N

_{2}O is shown to result in the highest density ΔV, which is the ΔV normalized for motor density. When O/F shift is considered, the ΔV decreases by 9% for the N

_{2}O motor and 12% for the H

_{2}O

_{2}85 wt% motor. When nozzle erosion is also considered, the ΔV decreases by another 7% for the H

_{2}O

_{2}85 wt% motor and 4% for the N

_{2}O motor. Even with O/F shift and nozzle erosion, the H

_{2}O

_{2}85 wt% motor can accelerate itself (916 kg) upwards of 4000 m/s, and the N

_{2}O motor (456 kg) 3550 m/s.

## 1. Introduction

#### 1.1. State of Hybrid Rocket Development

#### 1.2. Post-Boost Stages and Kick Motors

#### 1.3. Considerations for Hybrid Rocket-Based Post-Boost Stages and Kick Motors

_{sp}losses on change in velocity ΔV, are the focus of this research.

## 2. Materials and Methods

_{propellant}; total wet mass, M

_{wet}; and (effective) time-averaged specific impulse I

_{sp}(see Equation (1)). This is done by carefully framing the problem at hand. First, the governing equation of ΔV is re-examined to highlight the key concerns of the post-boost stage and kick motor design in context. Next, as is the case for actual post-boost stages and kick motors, the envelope is defined. Then, a representative set of oxidizers are selected for comparison of theoretical I

_{sp}, and the effect of motor configuration on maximizing I

_{sp}. Note that hybrid rocket fuels have relatively little effect on theoretical I

_{sp}, and for this reason high-density polyethylene (HDPE) is chosen as the fuel material throughout. Next, to allow for a realistic estimation of masses, a standardized formulation for all key components of the motor is made. Regarding the estimation of time-varying I

_{sp}, this requires the simulation of internal ballistics in time, including the effects of O/F shift and nozzle throat erosion, which is accomplished by creating a new computational algorithm. Each of these steps are explained in detail through framing the problem in the following subsections.

#### 2.1. Maximizing the Change in Velocity, ΔV

_{sp}(in s), is constant, this ΔV can be calculated according to Tsiolkovsky’s rocket equation:

^{2}), M

_{wet}is wet mass in kg, and M

_{propellant}is propellant mass in kg. Based on the definition for post-boost stages and kick motors in Section 1.2, the M

_{wet}is a constraint of the launch vehicle, and thus Equation (1) can be considered predominately a function of I

_{sp}and M

_{propellant}, as shown by the contours in Figure 2. For the purposes of demonstrating the sensitivities of ΔV to I

_{sp}and M

_{propellant}, M

_{wet}has been defined for convenience to be 100 kg. Thus, the values of M

_{propellant}shown as the abscissa in kg, can also be interpreted as percentages of M

_{wet}—i.e., propellant mass fractions. The two key takeaways from Figure 2 are that: (1) both increasing the I

_{sp}and M

_{propellant}of the propulsion system greatly improves ΔV; and (2) that increasing the propellant mass has an even greater impact on ΔV than increasing the I

_{sp}. The latter point will be the explanation for some of the surprising, or counterintuitive results of this paper. Note that this takeaway is obvious from Equation (1), since ΔV is linearly proportional to I

_{sp}, while M

_{propellant}is in the denominator of the logarithm term. However, the non-linearity of the sensitivity to M

_{propellant}is difficult to intuit, and combined with lack of a standardized mass formulation makes quantitative comparisons of propulsion systems regarding M

_{propellant}challenging. Lastly, I

_{sp}is changing in time in most hybrid rockets, due to a mixture ratio shift, typically referred to as “O/F shift.” Accordingly, the analysis in this paper will consider the effect of both terms when evaluating performance (potential).

#### 2.2. The Envelopes and Their Parametrization

#### 2.3. The Oxidizers, Their Theoretical I_{sp}, and Effect on Motor Configuration

_{2}O

_{2}); nitrous oxide (N

_{2}O); dinitrogen tetroxide (N

_{2}O

_{4}); and (liquid) oxygen (LOX). Most previous research, both theoretical and experimental, has been carried out using one of these oxidizers, or a close cousin of these oxidizers. A summary of the basic properties of these oxidizers is listed in Table 1. The theoretical vacuum I

_{sp}dependencies on O/F are shown by the plots in Figure 4. These oxidizers are diversified in terms of density and specific impulse, which makes them appropriate for comparison of ΔV performance. High-density and high-specific impulse are expected to lead to high-performance, and vice versa. Some other important considerations which relate to cost, handling, and feasibility for use in a space environment have been listed as either positive or negative in Table 1.

_{sp}between oxidizers, 333 s for N

_{2}O to 357 s for LOX, or ±4% of the median, is much less than the variation in density, 550 kg/m

^{3}for N

_{2}O to 1440 kg/m

^{3}for N

_{2}O

_{4}, or ±80% of the median. Thus, these variations in storage density, which affect storage vessel size and mass, will have a greater impact on ΔV than variations of I

_{sp}. Secondly, the sensitivity of maximum I

_{sp}to O/F is greater for LOX than the other oxidizers. This means that although the LOX system has the greatest I

_{sp}potential, it is uniquely vulnerable to performance losses due to O/F shift. Lastly, the location of optimum O/F will determine the appropriate proportion of fuel mass, which means the appropriate allocation of volume to the thrust chamber. Since the optimum O/F of LOX is one-third that of N

_{2}O or H

_{2}O

_{2}85 wt% (2.5 versus 7.5), the thrust chamber will be larger in size, while the oxidizer pressure vessels will be smaller. Thus, if the mass per unit volume of the thrust chamber is larger than the oxidizer vessels, the non-propellant mass of the system will be larger and ΔV smaller. It will be shown in the following sections that this is in fact the case.

_{O}

_{/F}is the ratio of the cross-sectional area available for oxidizer vessels to that available for the thrust chamber (containing the fuel). Granted that the area ratio does not translate directly to propellant mass ratio, it is a primary factor. Other factors include the height of the vessels, and the densities of the propellants. For example, referring to the optimum O/F of the LOX case as an example, i.e., 2.5, and if the fuel and oxidizer have similar densities, we can expect the optimum motor configuration to be closer to the middle of Figure 5, when D/B = 1/2, which results in an A

_{O}

_{/F}= 2.3. Another insight from Figure 5 regarding the design of motors which require pressurizer gas is that larger oxidizer vessels lead to smaller pressurizer vessels. Either a larger volume or greater pressure of pressurizer gas is necessary for a larger oxidizer vessel volume. Thus, for the post-boost/kick motor configuration in which oxidizer vessel volume comes at the expense of pressurizer gas volume, increasing the pressurizer pressure is the only way to satisfy the requirements for pressure-fed systems.

#### 2.4. Standardized Formulation of Motor Component Size and Mass

_{yield}is yield stress in Pa. The MEOP of the pressure vessels depends on the storage pressure requirements, which is determined for each motor given the volume available for the motor components. However, MEOP for the thrust chambers would require advanced knowledge of maximum chamber pressure during firings, and in this research it needed to be defined in advance of computations. For this reason, it was set at a fixed value of 5 MPa. A summary of material properties assumed for mass estimation are listed in Table 2.

_{2}O and H

_{2}O

_{2}motors which have smaller thrust chambers.

_{2}O. Due to the combination of a low storage density and large optimum O/F, it is best to take advantage of the relatively high vapor pressure as a means for self-feeding to the thrust chamber. The self-fed N

_{2}O system is depicted in the upper half of Figure 8. Only one valve is required to control the flow, and no pressure regulation is necessary. The remaining three oxidizers have very low vapor pressures, and are incapable of using the self-fed configuration. Instead, a pressurizer gas, i.e., “pressurizer,” is used to force feed the liquid oxidizer into the thrust chamber. The gas-fed supply system is depicted in the lower half of Figure 8. Since (pressurizer) gas has a much lower density than the liquid (oxidizer), it must be stored at a relatively high pressure, and regulated to the appropriate feed pressure. Thus, two valves and a regulator are required for the gas-fed system. The mass breakdown of these components is shown in Table 3. Due to a lack of open-source information on high-flow rate regulators, the regulator mass is neglected in this research. The tube mass is based on Swagelok 3/4” stainless steel tubing, where tube length is set to the perimeter (i.e., 4B) of the envelope for each pressure vessel set. In other words, for the self-fed N

_{2}O system, the tubing length is one envelope perimeter, whereas it is two for the gas-fed systems. The valve mass is extrapolated from Marotta CoRe series valves for the larger tube diameter. The valve mass is double for the LOX downstream valve to act as a penalty for the cryogenic temperature conditions, which the valves are not designed for. Lastly, a structure for holding the components in place and mounting solar panels etc. is considered, such that the frame of gas-fed case is 50% heavier than the self-fed case. This frame is a reference from a recent paper [28], which based this mass estimation on CAD models.

#### 2.5. Internal Ballistics Simulation Algorithm

_{t}[m

^{2}], fuel port radius r

_{fu}[m], fuel density ρ

_{fu}[kg/m

^{3}], burning surface area A

_{bs}[m

^{2}], and an educated guess for oxidizer mass flow rate ${\dot{m}}_{ox}$ [kg/s], the fuel mass consumption rate ${\dot{m}}_{fu}$ [kg/s], chamber pressure P

_{c}[Pa], and characteristic exhaust velocity c

^{*}[m/s], can be determined by the system of Equations (3)–(5):

_{D}[-] and A

_{D}[m

^{2}], the pressure cascade can be determined according to the functional dependencies listed as Equation (6) or related fluid mechanical correlations.

_{c}. Accordingly, it will be assumed that nozzle wall temperature T

_{w}[K], is 80% of the chamber pressure T

_{c}, which can be output from NASA CEA simultaneously with c

^{*}as done in Equation (4). This assumption will result in an exaggeration of the nozzle erosion rates because according to [30,31], nozzle erosion does not begin until the surface reaches temperatures around 1500 K to 2000 K, and it takes a non-negligible amount of time for the nozzle (throat) surface to reach and surpass these temperatures. However, the exaggerated effect of nozzle erosion on motor performance is acceptable for the sake of understanding the impact on ΔV.

_{n}(m) is nozzle (inner) diameter, D

_{CO,w}[m

^{2}/s] is the binary mass diffusivity of CO in air calculated at the nozzle wall temperature, P (Pa) is (local) pressure, Re (-) is Reynolds number, Sc

_{CO}(-) is the Schmidt number of CO as calculated at the nozzle wall temperature, Φ (-) is the equivalence ratio of combustion gas, ρ

_{n}(kg/m

^{3}) is nozzle material density, and ρ

_{w}is the density of combustion gas calculated at the wall temperature (kg/m

^{3}). Note that this equation was only recent proposed, and before there was no correlation which included a nozzle temperature term. Currently, the empirical constants β

_{1,2,3}, b and n, as listed in the example formula of Equation (7), have only been determined for graphite nozzle erosion under the use of oxygen [30] and nitrous oxide [31]. The empirical constants of Equations (5) and (7) as they pertain to this research are summarized in Table 4.

_{t}and A

_{e}[m

^{2}], are defined within the initial conditions at the start of the algorithm, whereas ${\dot{m}}_{ox}$, ${\dot{m}}_{fu}$, and P

_{c}are determined by calculations completed earlier in the algorithm. NASA CEA or a similar chemical equilibrium solver is necessary for determining the specific heat ratio of combustion gas, γ, but it is worth mentioning that NASA CEA can be used to carry out Equations (8) and (9) as well.

_{2}O motor, whether liquid N

_{2}O remains and how much remains, is determined by treating the oxidizer vessel as a control volume and tracking the total enthalpy of this control volume. Here, the enthalpy values are a reference from the NIST online database, and are obtained by using the equilibrium method outlined in reference [34]. After liquid N

_{2}O has been depleted, the remaining gas N

_{2}O is supplied to the thrust chamber similar to a gas blowdown feed system.

## 3. Results

_{2}O

_{2}85 wt%, N

_{2}O, N

_{2}O

_{4}, and LOX—under the assumption that O/F and A

_{t}are constant during combustion. This means that the algorithm in Section 2.5 is unnecessary, and ΔV can be determined according to Equation (1) alone. The second part focuses on the two highest performing cases to investigate to what degree O/F shift and nozzle throat erosion are likely to degrade ΔV. In both parts, the independent variable is the ratio of thrust chamber diameter to envelope base, D/B, which also determines the space available within the envelope for the oxidizer vessels, and in turn the pressurizer vessels for the gas-fed motors. Moreover, ΔV is calculated based only on the masses of the thrust chamber, oxidizer supply system, and structure as outlined in Section 2.4. For this reason, the results of ΔV are larger than should be expected in practice, which would include the additional non-propellant masses of the payload, bus etc.

#### 3.1. Ideal Post-Boost Stage and Kick Motor Results

_{sp}is assumed to be constant in time based on the values as presented in Figure 4. This assumption entails that the O/F is also constant in time, which is determined according the standardized motor designs in Section 2.4, configured as shown in Section 2.3, constrained to the envelopes introduced in Section 2.2. The only independent parameter is the thrust chamber diameter, which once defined, constrains the pressure vessel designs as well. Simply put, the thrust chamber diameter determines the O/F for a given envelope. Since O/F determines I

_{sp}, the thrust chamber diameter determines the I

_{sp}. Consequently, the oxidizer mass flow rate and related equations from Section 2.5 are not necessary for calculating ΔV as determined by Equation (1). However, when referencing I

_{sp}from NASA CEA, chamber pressure is required as an input. For the sake of identifying the ideal performance of each motor, this pressure was set to a relatively high value of 5 MPa for all calculations. The results of this simplified analysis are used to inform us of the maximum possible performance of the standard motor designs of this paper.

_{2}O

_{2}85 wt%, N

_{2}O, N

_{2}O

_{4}, and LOX, are shown in the plots in Figure 10 for each of the three envelopes: H/B = 1/2, H/B = 1, H/B = 3/2.

_{2}O

_{2}85 wt% motor has the highest ΔV for the given envelope constraints, while the LOX motor has the lowest. Based on I

_{sp}and storage density alone, this conclusion is rather unexpected. When considering the mass penalty for requiring a large thrust chamber, LOX-based post-boost stages need to be highly optimized for thrust chamber mass to become competitive with the other oxidizers. Another key takeaway is that when the mechanism of achieving a given O/F is not considered, the tall and skinny motors perform better than the short and flat ones, for all oxidizers. This is also a consequence of the mass penalty of the thrust chamber. The thrust chamber mass increases with the square of the diameter, but proportionally to height, meaning that taller and skinnier thrust chambers have greater propellant mass fractions which lead to greater ΔV. Figure 11 shows that the thrust chamber diameter which optimizes ΔV in the cubic (i.e., H/B = 1) LOX motor is less than that which optimizes I

_{sp}. In the ideal case where all propellants are consumed, this would mean that the motor is operating at an O/F slightly more O-rich than the optimum value. In other words, the LOX motor should be operated in more fuel rich conditions, but it cannot be because the mass penalty of increasing the thrust chamber size is too great.

_{2}O

_{2}85 wt% outperforms the others in terms of ΔV by a margin of more than 15%, with N

_{2}O

_{4}as the next best performing, and LOX as the least performing. However, there is one more consideration that is essential for understanding all the tradeoffs at hand. In the context of an actual launch, where these motors are added to the payload compartment at the expense of the space and mass of the payload, the wet mass of these motors must also be considered. To account for this important factor, a “density change in velocity,” $\Delta {\mathrm{V}}_{\rho}$ ((m/s)/(kg/m

^{3})), term is introduced, and is shown in Figure 12b. The density change in velocity is determined by dividing the ΔV by the motor density, defined as the wet mass divided by the envelope volume. This calculation is shown as Equation (10) for clarification:

_{2}O motor outperforms the other motors by a margin of 50% in terms of density change in velocity. In other words, the N

_{2}O motors are significantly lighter than the others when taking up the same envelope space, and have a similar self-ΔV.

_{envelope}= 1, dividing by M

_{wet}alone would result in the same values for $\Delta {\mathrm{V}}_{\rho}$. However, if volume is not included in Equation (10), the $\Delta {\mathrm{V}}_{\rho}$ would scale with mass for all other cases where the volume is not one. Thus, as a term which is applicable to extrapolation, the motor density must be used. In this sense, the density change in velocity contains all the information important to the end user, who understands both the envelope volume and mass limitations of the launch vehicle. For an end user with little space yet ample mass budget, the H

_{2}O

_{2}motor is the best option. For an end user with ample space yet a strict mass budget, the N

_{2}O motor is best option. Answering the question of where exactly the line is drawn in selecting between these two types of systems is outside the scope of this paper.

#### 3.2. Impact of O/F Shift and Nozzle Erosion on Ideal Performance

_{w}is always 80% of the combustion gas temperature at the throat, which is the upper end of the values that the authors have experienced in practice.

_{2}O

_{2}85 wt% for maximizing ΔV; and N

_{2}O for maximizing $\Delta {\mathrm{V}}_{\rho}$. Moreover, the empirical formulations for the H

_{2}O

_{2}85 wt% are not available in the literature, and so the empirical coefficients for the LOX case in Table 4 were implemented instead. Since the mass fraction of hydrogen content in H

_{2}O

_{2}85 wt% is roughly 7%, meaning that the remaining 93% is oxygen, this approximation seems appropriate enough for the purposes of the investigation at hand.

_{D}and nozzle throat area A

_{t}need to be treated as independent variables. Due to the nature of the algorithm, which contains multiple iterative loops as well as decisions to switch between mass flow rate equations, a brute force method was used to find the greatest ΔV within a 2D domain of A

_{D}and A

_{t}for every value of D/B. This domain was created by defining the injector as having 10 holes, and varying the size of the injector hole diameters d

_{inj}, and nozzle throat diameters d

_{t}, to ensure a wide range of flowrates, chamber pressures, and thrusts would be considered. These domains of calculations are summarized in Table 5. Note that the authors originally tried several optimization techniques to reduce the number of calculations required to find the maximum ΔV, however no method consistently converged on a value that matched or bettered the brute force method described above, which calculated ΔV along a grid of fixed values. In other words, the landscape of the solution space is changing drastically for every value of D/B and between oxidizers.

_{2}O motor in Figure 13b, although not easy to replicate with simple algebra, is much more straight forward than the solution space of the H

_{2}O

_{2}85 wt% motor in Figure 13a. In a positive way, there is not such a narrow range of injector hole diameters—roughly 1.5 to 3 mm—that result in ΔV > 3500 m/s when the throat diameter is 30 mm. Furthermore, there is an even larger range, two separate ranges in fact, of such values for the H

_{2}O

_{2}85 wt% motor where ΔV > 4000 m/s. However, of some concern for applying these motors in practice is that in both cases these regions reside next to cliffs of meaningless solutions (ΔV < 500 m/s). Further investigations regarding this landscape require an analysis outside of the purview and objectives of this research. The results of ΔV for each type of motor without nozzle erosion for each envelope are plotted in Figure 14, and the corresponding results of $\Delta {\mathrm{V}}_{\rho}$ are plotted in Figure 15. One takeaway from Figure 14 and Figure 15 is that the thrust chamber diameter ratio that corresponds to the maximum ΔV of the H

_{2}O

_{2}85 wt% motor is only slightly affected by the envelope shape, and more so for the H

_{2}O

_{2}85 wt% motor than the N

_{2}O motor when internal ballistics simulations are used to estimate ΔV. This is because there is simply much more oxidizer mass than the nitrous oxide motor due to the higher density. Recall from Figure 4 that the optimum O/F are close in value for both oxidizers, yet since the density of H

_{2}O

_{2}85 wt% is more than twice that of N

_{2}O, a larger thrust chamber containing twice as much fuel is necessary to achieve this optimum O/F. Meanwhile, the fuel length is fixed, so variation in burning surface area during firing is exacerbated and leads to more sporadic results. Another takeaway is that plots of both motors skew to the left—i.e., lower D/B—for lower values of H/B. Once again, this trend reflects the dependence of fuel and oxidizer mass on thrust chamber diameter when the height is fixed. Since the skew is to the left, it means that the shorter the H/B, the smaller the motor diameter which is needed to maximize performance, regardless of the oxidizer used. Lastly, the magnitude of the skew is greatest for the $\Delta {\mathrm{V}}_{\rho}$ plots in Figure 15b, which show that the most mass efficient motor of this study is the short and flat—i.e., H/B = ½—N

_{2}O motor, with $\Delta {\mathrm{V}}_{\rho}$ > 8 m

^{4}/kg-s.

_{2}O

_{2}85 wt% motor; and a decrease of roughly 400 m/s, or 10% from 4100 m/s to 3700 m/s in the N

_{2}O motor. It is also clear that nozzle erosion is more detrimental to the H

_{2}O

_{2}85 wt% motor, leading to an additional departure from theoretical ΔV of 300 m/s or 7%, versus 150 m/s, or 4% for the N

_{2}O motor.

_{2}O

_{2}85 wt% motor and N

_{2}O motor result in similar values for initial throat diameter and combustion time when nozzle erosion is neglected, making this comparison particularly relevant. It is worth mentioning that this similarity is unintentional. Recall that all possible sets of nozzle throat diameter and injector hole diameter are considered for both motors (see Table 5 and Figure 13), and the motors which resulted in the highest ΔV are selected from this body of results. No other selection criteria are employed, and no other constrains are placed on the calculations outside of the shutdown criteria (see Section 2.5 final paragraph).

_{2}O

_{2}85 wt% motors, and that the main difference between motors with and without nozzle erosion is the range over which O/F shift occurs: from 16 to 9 without nozzle erosion; and from 12.5 to 10.5 with nozzle erosion. Comparing Figure 19a,c shows that the impact of O/F shift on I

_{sp}is significant. The value of I

_{sp}quickly surpasses 340 s in the H

_{2}O

_{2}85 wt% motor without nozzle erosion, however never reaches this value when erosion occurs.

_{2}O motors with and without nozzle erosion. In the motor without nozzle erosion, O/F increases from six to seven, before decreasing back to six by the end of the firing. In the motor with nozzle erosion, O/F increases from 4.5 to 8.5, where it remains relatively steady until the end of firing. The reason O/F shift is positive at first is related to the mass flux exponents n of Equation (5) and Table 3. When n > 0.5—as in the reference value for N

_{2}O—and oxidizer mass flow rate is constant, O/F shift is positive and vice versa. However, during firing, the N

_{2}O supply mode switches from flashing liquid flow to choked gas flow, leading to a rapid drop followed by a gradual decrease in flow rate until the end of the firing. This explains why the O/F shift switches from being positive to negative during firing. When nozzle erosion occurs, the chamber pressure decreases, leading to a decrease in fuel mass consumption flow rate (see Equation (5)). As a result, the O/F remains relatively constant, even though both propellants’ flow rates are decreasing in time.

_{2}O in the oxidizer vessels and the self-fed supply procedure. The N

_{2}O is stored at 550 kg/m

^{3}so that it remains as a saturated liquid even at the critical temperature of 309 K. However, the self-feeding supply of N

_{2}O refrigerates the mass remaining in the vessel, which means that the density of the liquid portion of oxidizer increases in time, taking up less and less volume per unit mass. At some time, no liquid remains in the oxidizer vessels and the N

_{2}O is supplied as a coked gas in a blow-down configuration

_{sp}is significant in the N

_{2}O motor as well. The I

_{sp}of the motor without nozzle erosion remains between 328 s and 333 s, or 1.5% of the mean for the entire duration of firing, whereas it varies between 295 s and 325 s, or 4.8% of the mean in the motor with nozzle erosion. Furthermore, the thrust of the N

_{2}O motor rapidly drops roughly 2800 N in value about 1/3 into firing.

## 4. Discussion

#### 4.1. Technologies for Reducing O/F Shift and Nozzle Erosion

_{2}O

_{2}85 wt% and N

_{2}O motors, O/F shift alone results in departures from theoretical ΔV of around 9–12%, which are further exacerbated by nozzle erosion by an additional 4–7%. Thus, eliminating O/F shift and nozzle erosion completely will lead to improvements in ΔV of 14–19%.

#### 4.2. Nonintuitive Nature of the Results

_{ρ}), N

_{2}O, which has the lowest density, would have the worst performance. Only after grasping the impact of requiring the pressurizing systems of the other oxidizers to fit into the limited space surrounding the oxidizer vessels was it clear to what extent the pressurizing feed systems reduce ΔV

_{ρ}. This outlook is reflected in previous research as well. Nitrous oxide appears to have been neglected from an otherwise groundbreaking analysis in previous research, based on the viewpoint that the relatively low storage density would prevent it from being comparable in performance to other oxidizers [48]. Lastly, LOX is commonly selected for use as an oxidizer for its high I

_{sp}. For this reason, we expected the LOX motor to match or perhaps surpass the other oxidizers in ΔV. In fact, LOX resulted in the overall worst theoretical ΔV and ΔV

_{ρ}. The reason is that LOX performs best at an O/F of around two, which is at least twice as low as the optimum O/F of the other oxidizers. Since the thrust chamber can never be made as light as the surrounding pressure vessels, there is a non-negligible penalty to require larger fuel grains. In the rocket design conditions of this study, this penalty was so large as to completely negate the advantage that LOX has in I

_{sp}.

## 5. Conclusions

_{2}O

_{2}(85wt%), N

_{2}O, N

_{2}O

_{4}, and LOX, H

_{2}O

_{2}is shown to outperform the others in theoretical ΔV by more than 10%, even though it does not have to the highest I

_{sp}. In terms of theoretical density ΔV, which is a term for normalizing ΔV by the density of the thruster, N

_{2}O outperforms the other oxidizer by 50%, even though it has the lowest storage density. Furthermore, LOX has both the worst theoretical ΔV and density ΔV, even though it has the highest I

_{sp}. This is due primarily to the mass penalty of requiring a larger thrust chamber or pressurizing system, and having to fit the pressurizing system in the limited space around the oxidizer vessels and thrust chamber. Further analysis of the top-performing motors, the H

_{2}O

_{2}85 wt% and N

_{2}O motors, shows that they lose 9–12% in ΔV due to O/F shift. Moreover, the H

_{2}O

_{2}85 wt% motors lose another 7% in ΔV due to nozzle throat erosion, and the N

_{2}O motors another 4%. Nonetheless, even with O/F shift and nozzle erosion, a one-cubic-meter H

_{2}O

_{2}85 wt% motor can accelerate itself (916 kg) upwards of 4000 m/s, and a N

_{2}O motor (456 kg) 3550 m/s.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a, m, n | fuel regression rate constants, Equation (5) |

A | area, m^{2} |

A_{O/F} | oxidizer-to-fuel cross-sectional area ratio |

b, E, n, β_{1}, β_{2,}β_{3} | nozzle erosion rate constants, Equation (7) |

c^{*} | characteristic exhaust velocity, m/s |

C_{D}, C_{G} | dimensionless flow coefficients |

d | inner diameter, m |

D | outer diameter, m or binary mass diffusivity, m^{2}/s |

D/B | (thrust chamber) diameter-to-base ratio |

F | thrust, N |

F.S. | factor of safety |

g | (Earth) gravitational acceleration, ≅9.8 m/s^{2} |

H/B | (envelope) height-to-base ratio |

H/D | (envelope) height-to-(thrust chamber) diameter ratio |

I_{sp} | specific impulse, s |

MEOP | maximum expected operation pressure, Pa |

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

M | (initial) propellant mass, kg |

M | (initial) wet mass, kg |

O/F | oxidizer-to-fuel-mass ratio |

P | pressure, Pa |

r | nozzle throat or fuel port radius, m |

Re | Reynolds number |

Sc | Schmidt number |

t | (wall) thickness, m |

T | temperature, K |

ΔV | change in velocity, m/s |

ΔV_{ρ} | density change in velocity, m^{4}/kg-s |

ε | nozzle expansion ratio |

γ | specific heat ratio |

Φ | equivalence ratio, Φ > 1 is fuel-rich |

ρ | Density, kg/m^{3} |

σ | tensile strength, Pa |

Subscripts | |

bs | burning surface |

c | thrust chamber (position) |

dw, up | (orifice) downstream, upstream (position) |

e, t, w | (nozzle) exit, throat, wall (position) |

fu, n, ox | fuel, nozzle, oxidizer |

inj | injector |

## Appendix A

## References

- Altman, D.; Holzman, A. Overview and History of Hybrid Rocket Propulsion. In Fundamentals of Hybrid Rocket Combustion and Propulsion, 1st ed.; Chiaverini, M.J., Kuo, K.K., Eds.; American Institute of Aeronautics and Astronautics: Renton, VA, USA, 2007; Progress in Aeronautics and Astronautics; Volume 218, pp. 1–36. [Google Scholar]
- Chiaverini, M. Review of Solid-Fuel Regression Rate in Classical and Nonclassical Hybrid Rocket Motors. In Fundamentals of Hybrid Rocket Combustion and Propulsion, 1st ed.; Chiaverini, M.J., Kuo, K.K., Eds.; American Institute of Aeronautics and Astronautics: Renton, VA, USA, 2007; Progress in Aeronautics and Astronautics; Volume 218, pp. 37–125. [Google Scholar]
- Story, G.; Arves, J. Flight Testing of Hybrid-Powered Vehicles. In Fundamentals of Hybrid Rocket Combustion and Propulsion, 1st ed.; Chiaverini, M.J., Kuo, K.K., Eds.; American Institute of Aeronautics and Astronautics: Renton, VA, USA, 2007; Progress in Aeronautics and Astronautics; Volume 218, pp. 553–591. [Google Scholar]
- Ansari X-Prize. Available online: https://www.xprize.org/prizes/ansari (accessed on 25 July 2021).
- Brinkmann, P. Rocket Crafters Pivots with New Patents for 3D-Printed Fuel. Available online: https://www.spacedaily.com/reports/Florida_space_startup_Rocket_Crafters_pivots_with_new_patents_for_3D_printed_fuel_999.html (accessed on 25 July 2021).
- Vaya Space. Available online: https://www.vayaspace.com (accessed on 25 July 2021).
- Gilmour Space. Available online: https://www.gspacetech.com (accessed on 25 July 2021).
- Cecil, O.; Majdalani, J. Several Hybrid Rocket Technologies Hit Advanced Test Stages. Aerospace America. Available online: https://aerospaceamerica.aiaa.org/year-in-review/several-hybrid-rocket-technologies-hit-advanced-test-stages (accessed on 25 July 2021).
- Chen, Y.S.; Wu, B. Development of a Small Launch Vehicle with Hybrid Rocket Propulsion. In Proceedings of the AIAA Propulsion and Energy 2018 Forum, Cincinnati, OH, USA, 9–11 July 2018. AIAA-Paper 2018-4835. [Google Scholar]
- Chen, Y.S. Development of Hapith Small Launch Vehicle Based on Hybrid Rocket Propulsion. In Proceedings of the AIAA Propulsion and Energy 2019 Forum, Indianapolis, IND, USA, 19–22 August 2019. AIAA-Paper 2019-3837. [Google Scholar]
- Faenza, M.G.; Boiron, A.; Haemmerli, B.; Verberne, O. Development of Nucleus Hybrid Propulsion System: Enabling a Successful Flight Demonstration. In Proceedings of the AIAA Propulsion and Energy 2019 Forum, Indianapolis, IND, USA, 19–22 August 2019. AIAA-Paper 2019-3839. [Google Scholar]
- Schmierer, C.; Kobald, M.; Tomilin, K.; Fischer, U. Low Cost Small-Satellite Access to Space Using Hybrid Rocket Propulsion. Acta Astronaut.
**2019**, 159, 578–583. [Google Scholar] [CrossRef] - Henry, C. DLR Spinoff HyImpulse Plans Small Launcher Debut. Available online: https://spacenews.com/dlr-spinoff-hyimpulse-plans-small-launcher-debut-in-2022 (accessed on 25 July 2021).
- Firehawk. Available online: https://www.firehawkaerospace.com (accessed on 25 July 2021).
- Gamal, H.; Matusiewicz, A.; Magiera, R.; Hubert, D.; Karolewski, L.; Zielinski, K. Design, Analysis and Testing of a Hybrid Rocket Engine with a Multi-port Nozzle. In Proceedings of the AIAA Propulsion and Energy 2018 Forum, Cincinnati, OH, USA, 9–11 July 2018. AIAA-Paper 2018-4666. [Google Scholar]
- Space Forest. Available online: https://spaceforest.pl/sir-suborbital-inexpensive-rocket/ (accessed on 25 July 2021).
- T4i Hybrid Propellant. Available online: https://www.t4innovation.com/hybrid-propellant (accessed on 25 July 2021).
- Eilers, S.D.; Whitmore, S.; Peterson, Z. Multiple Use Hybrid Rocket Motor. U.S. Patent US20140026537A1, 30 January 2014. [Google Scholar]
- Whitmore, S.A. Nytrox as “Drop-in” Replacement for Gaseous Oxygen in SmallSat Hybrid Propulsion Systems. Aerospace
**2020**, 7, 43. [Google Scholar] [CrossRef][Green Version] - Jens, E.T.; Cantwell, B.J.; Hubbard, G.S. Hybrid Rocket Propulsion Systems for Outer Planet Exploration Missions. Acta Astronaut.
**2016**, 128, 119–130. [Google Scholar] [CrossRef] - Jens, E.T.; Karp, A.C.; Rabinovitch, J.; Nakazono, B. Hybrid Propulsion System Enabling Orbit Insertion Delta-Vs within a 12 U Spacecraft. In Proceedings of the 2019 IEEE Aerospace Conference, Big Sky, MT, USA, 2–9 March 2019. [Google Scholar]
- Kamps, L.; Molas-Roca, P.; Uchiyama, E.; Takanashi, T.; Nagata, H. Development of N2O/HDPE Hybrid Rocket for Microsatellite Propulsion. In Proceedings of the 70th International Astronautical Congress, Washington, DC, USA, 21–25 October 2019. [Google Scholar]
- 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. [Google Scholar] [CrossRef][Green Version] - Karp, A.C.; Nakazono, B.; Shotwell, R.; Benito, J.; Vaughan, D. Technology Development Plan and Preliminary Results for a Low Temperature Hybrid Mars Ascent Vehicle Concept. In Proceedings of the 53rd AIAA/SAE/ASEE Joint Propulsion Conference, Atlanta, GA, USA, 10–12 July 2017. AIAA Paper 2017-4900. [Google Scholar]
- Story, G.; Karp, A.; Nakazono, B.; Evans, B.; Whittinghill, G. Mars Ascent Vehicle Hybrid Propulsion Effort. In Proceedings of the AIAA Propulsion and Energy 2020 Forum, Virtual Event, 24–28 August 2020. AIAA Paper 2020-3727. [Google Scholar]
- Process for Limiting Orbital Debris. NASA-STD-8719.14A. 2011. Available online: https://explorers.larc.nasa.gov/HPMIDEX/pdf_files/10_nasa-std-8719.14a_with_change_1.pdf (accessed on 1 July 2021).
- Marks, P. Dodging Debris. Aerospace America. Available online: https://aerospaceamerica.aiaa.org/features/dodging-debris (accessed on 25 July 2021).
- Molas-Roca, P. Design of Scalable Hybrid Rocket Motor for Space Propulsion Applications. In Proceedings of the 70th International Astronautical Congress, Washington, DC, USA, 21–25 October 2019. [Google Scholar]
- Gordon, S.; McBride, B. Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications; National Aeronautics and Space Administration: Lewis Research Center, OH, USA, 1994; NASA RP-1311. [Google Scholar]
- Kamps, L.; Hirai, S.; Sakurai, K.; Viscor, T.; Saito, Y.; Guan, R.; Isochi, H.; Adachi, N.; Itoh, M.; Nagata, H. Investigation of Graphite Nozzle Erosion in Hybrid Rockets Using Oxygen/High-Density Polyethylene. J. Propuls. Power
**2020**, 36, 423–434. [Google Scholar] [CrossRef] - Kamps, L.; Sakurai, K.; Ozawa, K.; Nagata, H. Investigation of Graphite Nozzle Erosion in Hybrid Rockets Using N2O/HDPE. In Proceedings of the AIAA Propulsion and Energy 2019 Forum, Indianapolis, IN, USA, 19–22 August 2019. AIAA Paper 2019-4264. [Google Scholar]
- Ito, S.; Kamps, L.; Nagata, H. Fuel Regression Characteristics in Hybrid Rockets Using Nitrous Oxide/High-Density Polyethylene. J. Propuls. Power
**2020**, 37, 342–348. [Google Scholar] [CrossRef] - Carmicino, C.; Sorge, A.R. Role of Injection in Hybrid Rockets Regression Rate Behavior. J. Propuls. Power
**2005**, 21, 606–612. [Google Scholar] [CrossRef] - Zimmerman, J.E.; Waxman, B.S.; Cantwell, B.J.; Zilliac, G.G. Review and Evaluation of Models for Self-Pressurizing Propellant Tank Dynamics. In Proceedings of the 49th AIAA/SAE/ASEE Joint Propulsion Conference, San Jose, CA, USA, 14–17 July 2013. AIAA Paper 2013-4045. [Google Scholar]
- Nagata, H.; Teraki, H.; Saito, Y.; Kanai, R.; Yasukochi, H.; Wakita, M.; Totani, T. Verification Firings of End-Burning Type Hybrid Rockets. J. Propuls. Power
**2017**, 33, 1473–1477. [Google Scholar] [CrossRef][Green Version] - Saito, Y.; Yokoi, T.; Yasukochi, H.; Soeda, K.; Totani, T.; Wakita, M.; Nagata, H. Fuel Regression Characteristics of a Novel Axial-Injection End-Burning Hybrid Rocket. J. Propuls. Power
**2018**, 34, 247–259. [Google Scholar] [CrossRef] - Hitt, M.; Frederick, R. Testing and Modeling of a Porous Axial-Injection, End-Burning Hybrid Motor. J. Propuls. Power
**2016**, 32, 834–843. [Google Scholar] [CrossRef][Green Version] - Hitt, M.; Frederick, R. Experimental Evaluation of a Nitrous-Oxide Axial-Injection, End-Burning Hybrid Motor. J. Propuls. Power
**2017**, 33, 1555–1560. [Google Scholar] [CrossRef] - Saito, Y.; Kimino, M.; Tsuji, A.; Okutani, Y.; Soeda, K.; Nagata, H. High Pressure Fuel Regression Characteristics of Axial-Injection End-Burning Hybrid Rockets. J. Propuls. Power
**2019**, 35, 328–341. [Google Scholar] [CrossRef] - Suzuki, S.; Tsuji, A.; Soeda, K.; Kamps, L.; Nagata, H. Influence of Port Manufacturing Accuracy on Backfiring in Axial-Injection End-Burning Hybrid Rocket. In Proceedings of the AIAA Propulsion and Energy 2021 Forum, Virtual Event, 9–11 August 2021. AIAA Paper 2021-3516. [Google Scholar]
- Nakagawa, I.; Kishizato, D.; Koinuma, Y.; Tanaka, S. Demonstration of an Alternating-intensity Swirling Oxidizer Flow Type Hybrid Rocket Function. In Proceedings of the AIAA Propulsion and Energy 2019 Forum, Indianapolis, IND, USA, 19–22 August 2019. AIAA-Paper 2019-4093. [Google Scholar]
- Ozawa, K.; Kitagawa, K.; Aso, S.; Shimada, T. Hybrid Rocket Firing Experiments at Various Axial-Tangential Oxidizer-Flow-Rate Ratios. J. Propuls. Power
**2019**, 35, 94–108. [Google Scholar] [CrossRef] - Kamps, L.; Saito, Y.; Kawabata, R.; Wakita, M.; Totani, T.; Takahashi, Y.; Nagata, H. Method for Determining Nozzle-Throat-Erosion History in Hybrid Rockets. J. Propuls. Power
**2017**, 33, 1369–1377. [Google Scholar] [CrossRef][Green Version] - D’Elia, R.; Bernhart, G.; Hijlkema, J.; Cutard, T. Experimental Analysis of SiC-based Refractory Concrete in Hybrid Rocket Nozzles. Acta Astronaut.
**2016**, 126, 168–177. [Google Scholar] [CrossRef][Green Version] - Whitmore, S.; Babb, R.; Stephens, J.; Horlacher, J. Further Development of Low-Erosion Nozzle Materials for Long-Duration Hybrid Rocket Burns. In Proceedings of the AIAA Propulsion and Energy 2021 Forum, Virtual Event, 9–11 August 2021. AIAA Paper 2021-3514. [Google Scholar]
- Ito, S.; Kamps, L.; Yoshimaru, S.; Nagata, H. Evaluation of the Thermal Onset of Graphite Nozzle Erosion. In Proceedings of the AIAA Propulsion and Energy 2020 Forum, Virtual Event, 24–28 August 2020. AIAA Paper 2020-3755. [Google Scholar]
- Bianchi, D.; Nasuti, F. Numerical Analysis of Nozzle Material Thermochemical Erosion in Hybrid Rockets. J. Propuls. Power
**2013**, 29, 547–558. [Google Scholar] [CrossRef] - Chandler, A.; Cantwell, B.; Hubbard, S. Hybrid Propulsion for Solar System Exploration. In Proceedings of the 47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, San Diego, CA, USA, 31 July–3 August 2011. AIAA Paper 2011-6103. [Google Scholar]

**Figure 10.**Ideal performance of hybrid rocket-based post-boost stages or kick motors using: (

**a**) H2O2 85wt%, (

**b**) N

_{2}O, (

**c**) N

_{2}O

_{4}, or (

**d**) LOX as oxidizers.

**Figure 15.**Realistic $\Delta {\mathrm{V}}_{\rho}$ solutions spaces for all three envelopes without nozzle throat erosion.

Oxidizer | Density, kg/m^{3} | Positive Attributes | Negative Attributes |
---|---|---|---|

H_{2}O_{2} 85wt% | 1380 | catalytic ignition possible | unstable, dilutes in time |

N_{2}O | 550 ^{1} | safe, self-pressurizing | high ignition threshold |

N_{2}O_{4} | 1440 | space heritage, stable | highly toxic |

LOX | 1140 | nontoxic, easy ignition | cryogenic storage required |

^{1}Density for safe storage up to worst hot case temperature of 309 K.

Category | Material | Density, kg/m^{3} | Yield Stress, MPa | Components, # ^{1} |
---|---|---|---|---|

case | aluminum | 2700 | 260 | 1–6, 12, 13, 16, 17 |

fuel | HDPE | 955 | not considered | 10 |

insulator | laminate | 1300 | not considered | 7, 8, 11, 14, 18 |

liner, mixing | graphite | 1850 | not considered | 9, 15, 19 |

vessel (wall) | FRP | 1370 | 800 | 21, 23 |

^{1}refers to the component number(s) in Figure 6.

Component | Reference | Mass Estimation | Quantity |
---|---|---|---|

valve | Marotta CoRe | 1.6 kg/valve ^{1} | 2 (self-fed), 4 (gas-fed) |

tube | Swagelok ½” | 0.25 kg/m | 4B (self-fed), 8B (gas-fed) |

structure | [28] | 60 × B^{2}H (self-fed),90 × B ^{2}H (gas-fed) | 1 |

^{1}the mass of valves downstream of LOX valves are penalized to be: 3.2 kg/valve.

Propellants ^{1} | a, m, n of Equation (5) | β_{1}, β_{2}, β_{3}, b, E, n of Equation (7) |
---|---|---|

N_{2}O | 8.1 × 10^{−8}, 0.43, 0.53 [32] | 0.10, 2.09, 3.77, −0.07, 2996, 0.15 [31] |

LOX ^{a} | 1.5 × 10^{−6},0.26,0.36 [33] | 4.6 × 10^{−5}, 5.43, 6.36, −0.02, 6436, 1.02 [30] |

^{1}The fuel is HDPE in both cases;

^{a}for single-port fuel.

Independent Variable | Increment | Lower Value | Upper Value |
---|---|---|---|

Thrust chamber diameter ratio, D/B | 1/40 | 1/40 | 36/40 |

Injector hole diameter, d_{inj} ^{1} | 0.5 mm | 0.5 mm | 3.5 mm |

Nozzle throat diameter, d_{t} | 5 mm | 10 mm | 50 mm |

^{1}The injector is assumed to have 10 holes.

Solution | H_{2}O_{2} 85wt%No Erosion | Erosion | N_{2}ONo Erosion | Erosion |
---|---|---|

Thrust chamber diameter ratio, D/B | 43% | 43% | 35% | 35% |

Injector hole diameter, d_{inj} ^{1} | 1.5 mm | 1.0 mm | 3.0 mm | 2.5 mm |

Nozzle throat diameter, d_{t} | 30 mm | 30 mm | 30 mm | 20 mm |

Change of velocity, ΔV | 4379 m/s | 4053 m/s | 3698 m/s | 3545 m/s |

Wet Mass, M_{wet} | 916 kg | 916 kg | 465 kg | 456 kg |

Density-ΔV, $\Delta {\mathrm{V}}_{\rho}$ | 4.8 m^{4}/kg-s | 4.4 m^{4}/kg-s | 8.0 m^{4}/kg-s | 7.8 m^{4}/kg-s |

^{1}The injector is assumed to have 10 holes.

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**MDPI and ACS Style**

Kamps, L.; Hirai, S.; Nagata, H.
Hybrid Rockets as Post-Boost Stages and Kick Motors. *Aerospace* **2021**, *8*, 253.
https://doi.org/10.3390/aerospace8090253

**AMA Style**

Kamps L, Hirai S, Nagata H.
Hybrid Rockets as Post-Boost Stages and Kick Motors. *Aerospace*. 2021; 8(9):253.
https://doi.org/10.3390/aerospace8090253

**Chicago/Turabian Style**

Kamps, Landon, Shota Hirai, and Harunori Nagata.
2021. "Hybrid Rockets as Post-Boost Stages and Kick Motors" *Aerospace* 8, no. 9: 253.
https://doi.org/10.3390/aerospace8090253