Development and Testing of a Fast-Acting, 8-Bit, Digital Throttle for Hybrid Rocket Motors

Abstract

The potential for throttle control of hybrid rocket systems has long been known as a potential advantage for a variety of applications. Because only a single flow path is controlled, theoretically, hybrids should be significantly easier to throttle than bipropellant systems. Unfortunately, the slow response times and nonlinearity of traditional position-control valves have limited practical applications of hybrid throttling. This paper presents an alternative throttling system where the oxidizer flow path is broken into multiple streams, with each flow path controlled by a solenoid-operated on/off valve. The parallel paths allow significantly faster and more precise control than can be achieved using a single position-control valve. The achievable thrust levels are limited only by the size and number of components in the valve cascade. The 8-bit digital throttle system, developed by Utah State University’s Propulsion Research Lab, uses commercial, off-the-shelf components. The throttle system was tested using a 200-N hybrid rocket motor, burning gaseous oxygen, and ABS plastic as propellants. The testing campaign of more than 50 hot fires has demonstrated multiple profiles, including deep throttle ramps, multistep boxcars, and sine waves at frequencies varying from 0.25 to 1-Hz. Comparisons to analytical models are also presented, showing good agreement. Fourier-transform spectra demonstrating the total-system, frequency response are also presented.

1. Introduction

The inherent safety and environmental friendliness of hybrid rocket systems have been known for several decades [1]. Hybrids have the potential to act as an ideal “green” alternative for many of the current generation of toxic or hazardous propellants. Because hybrid systems only require a single fluid flow path, they are of similar complexity to monopropellant systems but with significantly higher performance. Hybrid rocket systems that employ nontoxic, nonexplosive propellants have the potential to fulfill a large role in the emerging commercial space market. Multiple applications for hybrid launch vehicle stages are currently being considered. These applications include sounding rockets [2,3], orbital insertion for SmallSats [4], upper stages for nanolaunchers, and surface launch systems for Lunar, Mars, and other sample return missions [5].

When compared to an equivalently sized, solid-propelled system, hybrid-propelled rocket stages have a higher dry mass fraction. However, hybrids offer capabilities that cannot be matched by solid-propellant systems. Hybrid rocket motors (HRMs) have a demonstrated capability for on-demand ignition, shutdown, and re-ignition. Examples of restartable HRMs include Chiba, et al. [6], Saraniero, et al. [7], Zakirov, et al. [8], Whitmore et al. (2015). [9]. HRMs can also be deeply throttled with little loss in performance. Glaser et al. [10] present an overview of the state of the art of hybrid rocket systems, including the ability for deep throttle. Whitmore, et al. (2014) [11] demonstrate a closed-loop HRE throttle system using a position-control ball valve. More recently, Reffin et al. (2022) [12] have demonstrated real-time throttling of a hydrogen peroxide-based HRM.

When these capabilities are properly optimized, several studies, including Casalino and Pastrone ([13], 2013, and [14], 2016), Casalino et al. [15], and Betts [16] have shown that hybrids, acting as “smart systems,” have the potential to significantly out-perform solid and monopropellant systems, and may provide equivalent performance to some significantly more complex and expensive bipropellant systems. Each of these studies exploits the capability for closed-loop throttle of the hybrid rocket system. A restartable and throttleable hybrid rocket motor could provide a safer alternative to liquid engines and a higher efficiency alternative to cold gas and monopropellant thrusters for a secondary-payload satellite while providing multiple-use capabilities for station keeping and orbit transfers. The ability for hybrid systems to deeply throttle also offers the potential for hybrid systems to perform both orbital insertion and onorbit maneuvering functions. Additionally, an optimal thrust profile can be implemented, such as a ramp up, for a rocket-assisted take off (RATO) of uncrewed aerial vehicles (UAVs), where high initial thrust could damage the airframe, or for minimizing the drag of a launch vehicle during endoatmoshperic flight.

2. Background

Unfortunately, due to some of the peculiarities of hybrid combustion, achieving this closed-loop precision control is not easily achieved in practice. Unlike solid rocket motors where the burn properties, including the rate of fuel regression and associated oxidizer-to-fuel (O/F) ratio, can be precisely controlled by the a priori propellant formulation, the O/F ratio and fuel regression rate vary continually throughout the burn of a hybrid rocket, as shown by Karabeyoğlu, et al. [17] The rate of fuel pyrolysis is driven by complex, often chaotic, and marginally understood fluid mechanical and heat transfer phenomena within the flame zone and boundary layer. Thus, when compared to solid propellant rockets, hybrid systems generally have a greater degree of burn-to-burn variability, including mean thrust level, specific impulse, and total delivered impulse. This high degree of motor-to-motor variability may be acceptable for experimental vehicles operated on a restricted test range but presents a potential impediment to securing a Federal Aviation Administration (FAA) operating permit for nonexperimental, commercial spaceflight operations. Furthermore, motor-to-motor variability has the potential to produce significant thrust asymmetries for clustered hybrid motor configurations. These asymmetries currently preclude using multiple, simultaneously burning hybrid motors for launch vehicles. One of the most effective means of reducing this burn-to-burn inconsistency is exploiting the ability of hybrid rockets to be easily and deeply throttled.

2.1. Hybrid Rocket Motor Throttling

Traditionally, a deep-throttled rocket system has been defined as any configuration capable of a thrust “turndown ratio” of 4:1 from its nominal thrust level. Deep throttling of a hybrid rocket is far simpler than for liquid bipropellant systems [18]. When a liquid bipropellant rocket is throttled, both the oxidizer and fuel mass flow rates must be directly controlled using the upstream feed pressure and injector flow area. Maintaining sufficiently high-pressure drops across the injectors is necessary to achieve the atomization levels required for stable combustion [19]. The required minimum injector pressure drop, an injector pressure ratio (IPR) of approximately 1.25 or greater, sets a practical lower limit to the depth of throttling that can be achieved by pressure modulation alone.

Theoretically, a liquid bipropellant rocket system can be throttled to any level by lowering the oxidizer and fuel feed pressures upstream of the injector. However, reducing propellant flow rates causes the upstream injector pressure to drop faster than the chamber pressure. At some point, the IPR becomes so low that coupling occurs between the chamber and the propellant feed system. This coupling limits the ability to throttle liquid rockets by using feed pressure, typically 60–70% of the nominal operating thrust level. As a point of reference, the space shuttle’s main engine was normally throttled within a turndown ratio of only 1.67:1 [20].

Other factors also contribute to the complexity of deep-throttle liquid rocket systems. Both the fuel and oxidizer valves are required to precisely maintain near-optimal O/F ratios over a wide range of propellant mass flow rates. The combustor L*—the ratio of combustor chamber volume to nozzle throat area—is typically configured for a near-optimal O/F ratio. In a deeply throttled engine, small variations in either propellant flow rate can result in a significantly skewed O/F ratio, and this off-design condition can interact with the chamber L* to produce either incomplete combustion or combustion instability. Either case will produce a suboptimal combustor performance compared with the full-throttle motor. Also, with the reduced propellant mass flow, turbopumps must also be designed to avoid cavitation, stalling, or surging. Turbomachinery components must have stable rotational and structural dynamics for a wide flow range. Regeneratively cooled liquid engines may also have insufficient heat transfer at high throttle turndown ratios.

Much of the throttling technology developed for liquid bipropellant rockets is applicable to hybrid rockets. However, there exists one major exception; when a hybrid motor is throttled, only the oxidizer mass flow rate is directly controlled, and the fuel mass flow rate is an indirect response to the change in oxidizer mass flow. The fuel pyrolysis rate is driven primarily by the oxidizer mass flux rate and when combined with the exposed area of the continuously variable fuel port burn surface, sets the motor O/F ratio response. Thus, the hybrid motor O/F ratio “self-adjusts” during throttling. This fortunate compensating effect makes hybrid rockets significantly less susceptible to combustion instabilities during throttling than liquid bipropellant systems.

2.2. Throttled Hybrid Rocket Development History

As reported by Spurrier [21], throttled hybrids using a variety of propellants have been under development since the 1940s. One of the earliest recorded throttled hybrids, Moore and Berman [22], started out as an augmented monopropellant hydrogen peroxide motor to which a polyethylene fuel grain section was added to increase specific impulse Isp. This work noted that throttling was easily accomplished through the use of a single valve, but that due to the thermal instabilities of peroxide, it was difficult to vary the burn rate to throttle the motor by more than a factor of two. In the mid-1960s, the ONERA Lithergol Experimental System (LEX), Duban [23], developed a throttled hybrid based around a hypergolic propellant combination of red fuming nitric acid and an amine fuel of meta toluene diamine/nylon propellant combination. By using a pneumatically actuated valve with a programmable timer, this motor was throttleable over a 5:1 turndown range, with thrust levels varying from 10,000 N down to 2000 N.

During the same time period, the United Technology Center and Beech Aircraft, Franklin et al. [24] and Jones [25], developed the Sandpiper, a target drone for the US Air Force. The motor for Sandpiper used a nitric oxide and nitrogen peroxide oxidizer (RON-25) and a polymethyl-methacrylate (PMMA)/magnesium fuel. Sandpiper was throttleable over an 8:1 turndown range with a peak thrust of 2300 N. To achieve this throttle, Sandpiper had two oxidizer feed lines; one that had a preset flow control valve that provided enough oxidizer for the vehicle to maintain a constant velocity and a second valve which allowed the motor to accelerate. The second valve was closed once cruise velocity had been reached.

A final Air Force throttled-hybrid project during the 1960s was the high altitude supersonic target (HAST), Penn Branigan [26]. Compared to Sandpiper, HAST had a larger thrust chamber, IRFNA-PB/PMM fuel, a cruciform port configuration, and the oxidizer was pressurized by a ram air turbine instead of a nitrogen top pressure. The HAST motor had a peak thrust of 5300 N with a 10:1 throttle range controlled by an on-command throttling valve consisting of a torque motor with a ball screw that actuated a pintle valve. In flight, the HAST motors were programmed to throttle from 50% to 100% over 20 s. After this point, the valve position could be manually adjusted remotely.

Development of hybrid rocket motors was scarce in the 1970s through the 1980s primarily due to the success of solid and liquid motors at the time. NASA started the Joint Government/Industry Research and Development (JIRAD) program in the mid-1990s, Boardman et al. [27] and Carpenter et al. [28] Two of the hybrid motors evaluated at this time were an 11-inch diameter and a 24-inch diameter motor designed for 13.3 kN and 178 kN thrust, respectively. Both of these motors operated in a binary thrust mode with a pressure-fed system achieving a 1.6:1 throttle turndown ratio and the pump-fed system achieving a 2.4:1 throttle turndown ratio.

More recently, several academic institutions have been involved in the development of throttled hybrid systems. The University of Arkansas at Little Rock (UALR), Wright et al. [29] developed an oxidizer delivery system that throttled the oxidizer mass flow between 18 and 37 g/s using a Teledyne-Hastings HFC307 control valve. Stanford and SPG have developed a custom throttling plate for the Peregrine sounding rocket that rotates to control the oxidizer mass flow rate to between 50% and 100%, Dyer et al. [30] and Doran et al. [31] report these results. The Purdue Zucrow laboratory, Austin et al. [32] has demonstrated a throttle-down profile with a square wave profile similar to a boost/sustain/boost profile used for a tactical missile flight. This motor throttled with a 10:1 turndown ratio using a ball-type, position-control valve.

Throttled-hybrid development began at the Utah State University’s Propulsion Research Lab in 2011. Whitmore and Peterson (2014) [11] and Whitmore et al. (2014) [33] developed an 800 N hybrid thruster burning nitrous oxide and hydroxyl-terminated polybutadiene (HTPB) as propellants. This system used a ball-type actuation valve, controlled by an electromechanical actuator (EMA) to throttle the motor across a turndown range of 67:1. The system used a proportional, integral, derivative (PID) control law, allowing the system to track prescribed step and linear ramp profiles. Later, Whitmore and Spurrier [34] modified the earlier motor of Refs. [11,33] to burn gaseous oxygen and acrylonitrile butadiene styrene (ABS) as propellants. The hybrid system was designed as a launch assist-motor for NASA’s Towed-Glider Launch (TGLAS) Platform. The motor used an upgraded valve control system, allowing a faster response time and a PID controller to track prescribed burn maneuvers. The throttle control features chamber pressure feedback with either open-loop or recursive servo-voltage output commands. A throttle turndown ratio of 5:1 was achieved with this system. This motor also featured an upgraded motor ignition system [9,35], allowing multiple system restarts. The prescribed thrust profiles were designed to allow the TGLAS to perform two critical flight maneuvers: (1) a 2.5 g pull-up to a 70-degree pitch angle, followed by a push-over to a level attitude, and (2) an “Immelmann” half-loop maneuver, a classic air combat maneuver named for World War I ace Max Immelmann [36].

2.3. Hybrid-Throttle Response Fidelity Issues Associated Analog Position-Control Valves

All motor systems described in the previous section achieved throttle by placing a movable position-control valve; either a ball- or pintle-type, in the oxidizer flow path. Ball valves require sizable actuation torque and have a very nonlinear flow coefficient (C_v) actuation profile. There exists a very narrow range over which the ball-valve system is effective. When first opened, the valve requires very large position changes in valve position to effect even minor flow control changes. Conversely, near full open, very small position changes result in a large change in the flow C_v. The result is poor response fidelity.

Pintle valves, a type of need-control valve, offer a more linear response than ball valves. The valve changes the flow area by moving a tapered pintle relative to a metering nozzle. The pintle has a hyperbolic profile so that a change in position is roughly proportional to a change in the effective flow area. A linear actuator, either a jack-screw actuated drive or a spring-solenoid configuration, is mounted to the pintle shaft and causes the pintle to move fore and aft. The jack-screw drive system allows for precise control of the pintle position but has a necessarily slow response rate. The spring-solenoid configuration features a fast response time but operates only in an on/off configuration. Thus, the response fidelity issue still remains for pintle valve systems.

Figure 1 shows the throttling behavior of the TGLAS system. Figure 1a shows the achieved deep throttle, down to less than 2.5% of nominal thrust. Figure 1b shows the closed-loop response of the ball-valve throttle to a commanded box-car step profile. Note that there is a large response latency when the first step-down is first commanded and a significant peak overshoot when the throttle steps back up. This behavior is a function of both the ball-valve control actuation latency and the C_v curve nonlinearity.

2.4. Digital Valve Concept as Alternative to Position Control Valves for Hybrid Throttling

The Utah State University’s Propulsion Research Lab has developed an 8-bit digital-valve alternative to the position control valve. The concept to be presented is inspired by earlier work performed at the NASA Marshall Space Flight Center using a lower-fidelity 6-bit digital valve system intended for liquid rocket systems fluidic flow control and adapted for HRM throttle control [37]. In this concept, shown in Figure 2, eight binary (on/off) solenoid-operated valves (SOVs) are placed in parallel between an inlet and an outlet flow manifold. Each SOV is connected to a flow restriction orifice that has a C_v roughly half that of the next largest valve. With eight SOVs in parallel, any of the 256 possible combinations of C_v set-points can be commanded. In Figure 2, the Blue Boxes represent inlet and outlet of the flowing oxidizer. The parallel paths allow significantly faster and more precise control than can be achieved using a single position-control valve. The achievable thrust levels are limited only by the size and number of components in the valve cascade.

3. System Hardware Design

This section presents the system’s hardware that was developed in order to support the development and testing campaign to be described later in the Results and Discussion section. The 8-bit digital valve hardware and actuation systems will be presented first, followed by the legacy 200 N motor thrust chamber design. Finally, the oxidizer flow and motor fire-control systems will be described.

3.1. Digital Valve Hardware Description

Figure 3 shows the as-built valve hardware assembly. For this configuration, the eight SOVs are mounted in parallel to inlet and outlet flow manifolds. This configuration allows for SOVs 5–8 to be placed slightly above SOVs 1–4; leading to a more compact control valve design. Pressure transducers installed on the inlet and outlet manifolds measure the inlet and outlet pressure of the digital throttle valve. The total collected Cv value of the 8-bit digital throttle valve was selected to accommodate the anticipated mass flow rates necessary to achieve the full 200-N thrust level of the legacy motor thrust chamber. Each flow-control orifice was selected so that the next larger valve orifice has a Cv that is precisely two times larger. The calculations for selecting these orifices sizes will be presented in the Analytical Methods section of this report. The actual hardware Cv values resulted from the “closest” available online hardware choices (http://catalog.okeefecontrols.com, accessed on 1 September 2024). The end-to-end Cv values for each flow path were later calibrated to give the true values. The calibration results will be presented later in the Results and Discussion section of this report.

3.2. Digital Valve Electrical Interface

The electrical interface for the digital valve features rail-mounted, solid-state relays that convert the SOVs from alternating current (A/C) input power to direct current (D/C) control logic; Figure 4 shows this system schematic. Each SOV’s control logic is commanded from a National Instruments^® (NI) data acquisition and control (DAQ) module (https://www.ni.com, accessed on 1 October 2024), connected via universal series bus (USB) to a laptop running the LabVIEW^® control software, Ver. 2022. The DAQ device also measures the voltage signals produced from the pressure transducers attached to the inlet and outlet valve mixing plenums. Because of the asynchronous USB communication across long used lines, greater than 20 ft, data sampling and control of the system were limited to 50 samples per second.

3.3. Hybrid Rocket Test Article

As described earlier, a well-characterized, legacy 75-mm diameter hybrid motor was repurposed for the digital throttle testing campaign. This motor burns gaseous oxygen (GOX) and acrylonitrile butadiene (ABS) as propellants, operates at a nominal chamber pressure of approximately 140 psia (965 kpa), and produces a nominal thrust level in the range from 200 to 220 N.

3.3.1. Thrust Chamber

Figure 5a shows the thrust chamber assembly layout, and Figure 5b shows the fuel grain assembly. Major components are (1) the nozzle assembly, (2) the nozzle retention ring, (3) the motor case, (4) a 3D printed ABS ignitor cap with embedded electrodes, extruded main fuel grain section, (5) an insulating phenolic liner, (6) chamber pressure fitting, and (7) a motorcap with a single-port injector. The motor case is constructed from T-6061 aluminum alloy and was procured commercially (http://www.pro38.com/products/pro75/pro75.php, accessed on 1 October 2024).

3.3.2. Motor Ignition System

The motor is fully restartable, with up to 10 restarts available, without motor refurbishment. Motor ignition relies on USU’s patented low-wattage, arc-ignition system Whitmore et al. (2015) [9] and (2020) [35]. The ignition system power processing is based on the UltraVolt^® line of high-voltage power supplies (HVPS) (https://www.advancedenergy.com, accessed on 1 October 2024). An NI Digital control module is used to initiate ignition power to the thruster by sending a transistor–transistor logic (TTL)-level activate logic bit to the HVPS. The HVPS provides a current-limited (60 mA) high voltage output of up to 30 watts total output. Depending on the impedance of the arc path between the ignitor electrodes, the dissipated voltage typically varies between 100 and 250 volts. For a typical ignition cycle, the HVPS would be energized 1 s prior to the opening of the run valve and would remain energized for an additional 1 s after the run valve opened. Each ignition cycle draws approximately 10 watts of power and consumes less than 20 joules of electrical energy. Figure 6 shows the ignition system interface to the test article thrust chamber. The HVPS is also depicted. The full function of the HVPS is described by https://www.mouser.com/datasheet/2/863/ENG_HV_AASeries_230_Y-1137750.pdf, (accessed on 1 October 2024). On Figure 6 the red and black lines represent positive and returnpower paths. The orange, blue, and brown lines represent digital logic paths.

As a time and cost-saving measure for this test series, the ABS fuel-grain was manufactured with two pieces. The head-end section was 3-D printed using commercial ABS feedstock, and printed at full in-fill density. The head-end section has 3-D printed slots where the ignition electrodes are inserted. Printed inserts protect the embedded electrode wires from the combustion flame. The lower fuel section was machined from extruded ABS, procured commercially from one of several industrial supply houses. Printing of the ignitor section of the fuel grain changes the electrostatic breakdown properties of the ABS material and enables the inductive spark that initiates combustion.

3.4. Motor Test Support Systems

Figure 7 shows the piping and instrumentation diagram (P&ID) of the experimental apparatus used for this test series. The motor was mounted to a calibrated thrust-stand with flexible mounts that allowed thrust transmission in the axial direction. Test stand measurements include Venturi and the flow-orifice-based oxidizer mass flow rate, load-cell-based thrust, chamber pressure, GOX tank pressure, injector feed pressure, digital valve inlet and outlet pressures, and multiple thermocouples mounted at various points along the flow path. The test cart was configured to allow both analog (ball valve) and digital throttle control. A hand-operated 3-way valve selects between digital and analog control. Custom fire control, data acquisition, and processing software were pre-programmed to ensure run-to-run test consistency. Connection from the motor instrumentation pallet to the control/data logging laptop was via a single universal serial bus (USB). All tests were performed in the ballistics and survivability limits testing (BLAST) Lab on the USU campus. Full details of the motor and fire-control systems are presented by Whitmore et al. (2020) [38].

3.5. Digital Valve Control Software

The digital valve control software, written with the LabVIEW graphical programming language, discretizes the commanded throttle level into 8 bits (256 parts). The associated 8-bit unsigned integer is converted to a binary number representing the commanded throttle. This binary array multiplies the valve array C_v set values to calculate a total valve C_v. The binary throttle equivalent is run through a software shift register to spread the bits into eight discrete elements, and these bits are used to drive the on/off channels of the digital control module, activating the appropriate valves. The throttle setting is converted into an 8-bit binary number. Figure 8 shows examples of some of the duty cycles that were programmed and exercised for this testing campaign. These cycles include (1) deep-throttle ramp-down and ramp-up, (2) a complex variable deep-throttle ramp, (3) a multiple-step “boxcar” throttle cycle with gradually deepening throttle cycles, and (4) a sine wave with variable amplitude and frequency. Clearly, such complex command cycles cannot be achieved by a position control valve.

4. Analytical Modeling

In support of the experimental campaign, a one-dimensional ballistic model of the system was developed in order to understand separate response-fidelity effects that are due to the mechanical response of the oxidizer flow path control and those effects that are due to the fuel pyrolysis.

4.1. Calculating Chamber Pressure

As developed by Eilers and Whitmore [39] and modified for compressible flow by Whitmore (2019) [40], by applying the compressible-flow conservation equations across the hybrid fuel combustion port, the longitudinal mean of the rate of change of the combustion chamber pressure P_0c is written as

$$\begin{array}{c}{\displaystyle \frac{\partial P_{0c}}{\partial t}}={\displaystyle \frac{A_{burn}{\dot{r}}_L}{V_c}}\left[{\rho }_{fuel}{R}_g{T}_{0c}-P_{0c}\right]-P_{0c}\left[{\displaystyle \frac{A^{\ast }}{V_c}}\sqrt{\gamma R_g{T}_{0c}{\left({\displaystyle \frac{2}{\gamma +1}}\right)}^{\frac{\gamma +1}{(\gamma -1)}}}\right]+\\ {\displaystyle \frac{R_{g_c}{T}_{0_c}}{V_c}}\cdot \left\{\left(K_n\cdot C_d\cdot A_{inj}\right)\cdot {\displaystyle \frac{P_{0inj}}{\sqrt{R_{g_{ox}}\cdot T_{0_{inj}}}}}\right\}\end{array}.$$

(1)

In Equation (1), the first term on the right-hand side of the equation results from the rate of fuel regression and the associated growth in the combustion chamber volume V_c, the second term results from the nozzle throat choke point, and the final term results from the flow across the oxidizer injector. For the first two terms, the symbols A_burn, and A* are the fuel surface burn area and the nozzle throat choking area. V_c is the total instantaneous combustion chamber volume. The parameter ρ_fuel is the density of the solid fuel material. The combustion parameters R_g, T_oc, and γ are the gas-constant, flame temperature, and ratio of specific heats associated with the combustion chamber combustion products. In the last term of Equation (1), the symbols P_0inj and T_0inj represent the injector feed pressure and temperature, R_g is the oxidizer ratio of specific heats, and C_d·A_inj is the effective discharge area of the oxidizer injector. The parameter K_n depends on whether the injector flow is choked or unchoked. For an unchoked injector flow,

$$K_n=\sqrt{\left({\displaystyle \frac{2\cdot {\gamma }_{ox}}{{\gamma }_{ox}-1}}\right)\cdot {\left({\displaystyle \frac{P_{0_c}}{P_{0_{inj}}}}\right)}^{{\displaystyle \frac{2}{{\gamma }_{ox}}}}\cdot \left(1-{\left({\displaystyle \frac{P_{0_c}}{P_{0_{inj}}}}\right)}^{{\displaystyle \frac{{\gamma }_{ox}-1}{{\gamma }_{ox}}}}\right)},$$

(2a)

and for choked injector flow

$$K_n=\sqrt{{\gamma }_{ox}\cdot {\left({\displaystyle \frac{2\cdot }{{\gamma }_{ox}+1}}\right)}^{\frac{{\gamma }_{ox}+1}{{\gamma }_{ox}-1}}}.$$

(2b)

The associated oxidizer mass flow rate is calculated as

$${\dot{m}}_{ox}={\left(K_n\cdot C_d\cdot A\right)}_{inj}\cdot {\displaystyle \frac{P_{0inj}}{\sqrt{R_{g_{ox}}{T}_{0_{inj}}}}},$$

(3)

The total mass flow through the (choked) nozzle is calculated by

$${\dot{m}}_{total}=C_d\cdot A^{\ast }\cdot {\displaystyle \frac{P_{0c}}{\sqrt{T_{0_c}}}}\cdot \sqrt{{\displaystyle \frac{{\gamma }_c}{R_g}}\cdot {\left({\displaystyle \frac{2}{{\gamma }_c+1}}\right)}^{{\displaystyle \frac{{\gamma }_c+1}{{\gamma }_c-1}}}}.$$

(4)

The combustion thermodynamic and transport properties necessary to perform the ballistic calculations of Equations (1)–(3) were modeled using the NASA Chemical Equilibrium with Applications (CEA) code [41]. Tables of properties with O/F ratio and combustion chamber pressure P_0c s independent look-up variables were developed from the CEA calculations. Once the combustor chamber pressure is calculated, the thrust, mass flow, and other associated performance parameters are calculated using the standard one-dimensional De Laval flow equations (Anderson [42]).

4.2. Calculating Fuel Mass Flow and Linear Regression Rate

Early studies by Marxman and Gilbert [43] and Marxman [44] demonstrated that combustion processes for hybrid rockets and the associated fuel pyrolysis rates are mostly driven by viscous heat transfer within the boundary layer. Thus, the fuel regression rate is strongly correlated with the mass flux through the combustion chamber fuel port. Marxman and Gilbert posed a length-dependent “Saint-Roberts” type of power-law model of the form,

$${\dot{r}}_L=a\cdot G_{ox}^n\cdot L^m,$$

(5)

where, in Equation (5), L is the fuel grain length, measured as the longitudinal distance down the fuel port axis, ${\dot{r}}_L$ is the longitudinal mean of the fuel regression rate, G_ox is the oxidizer mass flux down the fuel port, and {a, n, m} are empirically-determined coefficients. For cylindrical fuel ports, Marxman determined that the burn exponent parameter set can accurately be reduced by one degree of freedom by assuming m ~ n − 1. Thus, Equation (5) can be approximated by

$$\dot{r}=a\cdot {G_{ox}}^n\cdot L^m\approx a\cdot {G_{ox}}^n\cdot L^{n-1}.$$

(6)

Equation (6) can be linearly curve-fit to regression rate time-history data by multiplying Equation (6) by the fuel grain length, L,

$$\dot{r}\cdot L=a\cdot {\left(G_{ox}\cdot L\right)}^n,$$

(7)

and taking the natural logarithm of both sides,

$$\ln (\dot{r}\cdot L)=\ln \left(a\cdot {\left(G_{ox}\cdot L\right)}^n\right)=\ln (a)+n\cdot \ln \left(G_{ox}\cdot L\right).$$

(8)

Writing Equation (8) in matrix form,

$$\ln (\dot{r}\cdot L)=\left[\begin{array}{c}1 \ln \left(G_{ox}\cdot L\right)\end{array}\right]\cdot \left[\begin{array}{c}\ln (a)\\ n\end{array}\right].$$

(9)

The evaluating Equation (9) for all N data points in the time history,

$$\left[\begin{array}{c}\ln (\dot{r}\cdot L{)}_1\\ \ln (\dot{r}\cdot {)}_2\\ \cdot \\ \cdot \\ \cdot \\ \ln (\dot{r}\cdot L{)}_N\end{array}\right]=\left[\begin{array}{cc}1& \mathrm{l}\mathrm{n}(G_{o{x}_1}\cdot L)\\ 1& \mathrm{l}\mathrm{n}(Go{x}_2\cdot L)\\ \cdot & \cdot \\ \cdot & ,\\ \cdot & ,\\ 1& \mathrm{l}\mathrm{n}(G_{o{x}_N}\cdot L)\end{array}\right]\cdot \left[\begin{array}{c}\ln \left(a\right)\\ n\end{array}\right].$$

(10)

Equation (10) now allows a linear least-squares solution for the regression-rate burn parameters (ln(a), n) using the pseudo-inverse method (Beckwith [45], Chapter 5).

Given the regression rate burn parameters, the fuel mass flow rate is calculated from the fuel regression rate by

$${\dot{m}}_{fuel}={\dot{m}}_{total}-{\dot{m}}_{ox}={\rho }_{fuel}\cdot A_{burm}\cdot {\dot{r}}_L.$$

(11)

Marxman and Gilbert [43] and Marxman [44] predicted values for n~0.8 and m~0.2 (~n − 1) for the parameters of Equation (5). The author has verified that these exponent values agree well with hybrid combustion from data collected by Eilers and Whitmore [39] nitrous oxide and HTPB as propellants. However, as shown by Zilliac and Karabeyoglu [46], these parameters do not fit as well for other combinations of propellants. Although the results of many regression rate tests have proven that the power-law form of Marxman’s regression rate law is valid for nonerosive burning, to date, there exists no comprehensive, first-principal theory that can be used to reliably predict this quantity over a range of propellants and motor sizes. Curve-fit data, not the Marxman parameters, will be used for this analysis.

4.3. Calculating Oxidizer-to-Fuel (O/F) Ratio, and O/F-Shift

Unlike solid-rocket motors, where the propellant oxidizer-to-fuel ratio (O/F) is pre-set by the propellant formulation and remains constant throughout the burn, hybrid rockets experience a continual O/F shift throughout the burn lifetime. A simple Linear Analysis demonstrates this effect. Writing the O/F ratio as the ratio of oxidizer and fuel mass flow rates,

$$O/F={\displaystyle \frac{{\dot{m}}_{ox}}{{\dot{m}}_{fuel}}}.$$

(12)

For a cylindrical fuel port,

$${\dot{m}}_{fuel}={\rho }_{fuel}\cdot \left(A_{burn}\right)\cdot \dot{r}={\rho }_{fuel}\cdot (2\pi \cdot r_{port}\cdot L)\cdot \dot{r}.$$

(13)

Assuming the power-law regression rate model of Equation (5) and allowing m = n − 1, Zilliac and Karabeyoglu [46] and Whitmore and Merkley [47] show for a cylindrical hybrid fuel port, the instantaneous longitudinal mean of the O/F ratio can be written as,

$$O/F={\displaystyle \frac{{\dot{m}}_{ox}}{{\rho }_{fuel}\cdot (2\pi \cdot r_{port}\cdot L)\cdot a{\left(\frac{{\dot{m}}_{ox}}{\pi \cdot r_{port}^2}\right)}^n\cdot L^m}}={\displaystyle \frac{{\dot{m}}_{ox}^{1-n}\cdot r_{port}^{2n-1}}{{\rho }_{fuel}\cdot (2{\pi }^{1-n}\cdot a)L^{1+(1-n)}}}={\displaystyle \frac{{\dot{m}}_{ox}^{1-n}\cdot r_{port}^{2n-1}}{{\rho }_{fuel}\cdot (2{\pi }^{1-n}\cdot a)\cdot L^n}}.$$

(14)

From Equation (15), it is observed that as the port radius grows during the fuel burn, when the burn exponent n > 1/2, the O/F ratio experiences a positive shift, and the motor burns increasingly leaner with time. When the burn exponent is exactly equal to 1/2, the burn is neutral, and the motor experiences no O/F shift. Finally, when the burn exponent n < 1/2, the motor burns increasingly fuel rich with time, and the O/F shift is negative. Similarly, for a burn exponent n > 1/2, the rate of the O/F shift rate will increase with time, and for n > 1/2 O/F shift rate will decrease with time.

By calculating the derivative of Equation (14) with respect to time, the rate of the O/F shift can be written as,

$$\begin{array}{c}{\displaystyle \frac{\partial (O/F_{\left(t\right)})}{\partial t}}=\left({\displaystyle \frac{{\dot{m}}_{ox}^{1-n}}{{\rho }_{fuel}\cdot (2{\pi }^{1-n}\cdot \alpha )L^n}}\right){\displaystyle \frac{\partial \left(r_{port}^{2n-1}\right)}{\partial t}}=\left({\displaystyle \frac{{\dot{m}}_{ox}^{1-n}}{{\rho }_{fuel}\cdot (2{\pi }^{1-n}\cdot \alpha )L^n}}\right)\cdot ((2n-1)\cdot r_{port}^{2n-2}\cdot \dot{r})=\\ \left({\displaystyle \frac{(2n-1)\cdot {\dot{m}}_{ox}^{1-n}}{{\rho }_{fuel}\cdot (2{\pi }^{1-n}\cdot \alpha )L^n}}\right)\cdot r_{port}^{2(n-1)}\cdot [\alpha {\left({\displaystyle \frac{{\dot{m}}_{ox}}{\pi \cdot r_{port}^2}}\right)}^n\cdot L^{n-1}]=\left({\displaystyle \frac{(2n-1)\cdot {\dot{m}}_{ox}}{{\rho }_{fuel}\cdot (2\pi \cdot r_{port}^2)\cdot L}}\right)=\left({\displaystyle \frac{(2n-1)\cdot {\dot{m}}_{ox}}{{2\cdot \rho }_{fuel}\cdot {\left(V_{ol}\right)}_{port}}}\right)\end{array}.$$

(15)

In contrast to Equation (14), Equation (15) shows that the O/F shift rate is negative for all burn exponents, and because the port volume grows over time, the O/F shift rate inherently slows with time. The O/F shift effects, as described by Equations (14) and (15), have a major effect on the observed motor burn profiles. Assuming a constant oxidizer mass flow rate, a positive O/F shift motor will generally see a drop off in thrust as the motor burns and a negative O/F-shift motor will experience an increase in thrust with time. The effects of O/F shift upon the digital throttling characteristics of the test motor will be described later in the Results and Discussion section. However, regardless of the O/F shift direction, the shift rate will always slow with time, eventually leading to a zero-shift condition for sufficiently large fuel grain.

4.4. Calculating the Orifice Flow Coefficient, C_v

For the orifice flow calibration, the orifice C_v is calculated using industry-standard formulae [48,49] as derived from the compressible mass flow equations, where for unchoked flow,

$$C_v={\displaystyle \frac{Q_{STP}}{961.679}}\cdot \sqrt{{\displaystyle \frac{s_{grav}\cdot T}{(p_1^2-p_2^2)}}},$$

(16)

and for choked flow

$$C_v={\displaystyle \frac{Q_{STP}}{816.532\cdot p_1}}\cdot \sqrt{s_{grav}\cdot T}.$$

(17)

In Equations (16) and (17)

• Q_stp = Volumetric flow rate at standard temperature and pressure, ft³/h
• p₁ = Pressure upstream of the orifice, psia, (0.145 kPa)
• p₂ = Pressure downstream of the orifice, psia, (0.145 kPa)
• T = Local flow stagnation temperature, °R, (1.8 K)
• S_grav = Specific gravity of the working gas with respect to air.

Contrary to common belief, C_v is not a nondimensional quantity. As expressed by Equations (16) and (17), the units of C_v are

$${\displaystyle \frac{f{t}^3/hr\cdot \sqrt{°R}}{lbf/i{n}^2}}.$$

The volumetric flow rate is calculated from the oxidizer mass flow rate as

$$Q_{STP}=\left({\displaystyle \frac{\dot{m}}{{\rho }_{ST{P}_{gas}}}}\right)=\left({\displaystyle \frac{\dot{m}}{{\rho }_{ST{P}_{air}}}}\right)\cdot \left({\displaystyle \frac{{\rho }_{ST{P}_{air}}}{{\rho }_{ST{P}_{gas}}}}\right)=\left({\displaystyle \frac{\dot{m}}{{\rho }_{ST{P}_{air}}}}\right)\cdot \left({\displaystyle \frac{M_{w_{air}}}{M_{w_{gas}}}}\right)=\left({\displaystyle \frac{\dot{m}}{{\rho }_{ST{P}_{air}}}}\right)\cdot \left({\displaystyle \frac{1}{S_{gra{v}_{gas}}}}\right).$$

(18)

where at a standard temperature of 60 °F (315.4 K) and pressure (1 atms),

$${\rho }_{ST{P}_{air}}={1.2229}_{kg/m^3}={0.076343}_{\mathrm{l}\mathrm{b}\mathrm{m}/{\mathrm{f}\mathrm{t}}^3},$$

and the specific gravity of GOX is 1.1044.

5. Results and Discussion

This section presents preliminary results from the digital throttle testing campaign. Results from the digital valve calibration will be presented first. These calibrations were used to refine the mass flow measurements from the digital valve system. The calibration test results will be followed by the results from the motor regression rate characterization tests. Finally, the digital throttling tests were performed. To date, more than 50 successful throttle tests have been performed, including deep throttle-ramp tests and boxcar and sine wave duty cycle tests. Results from deep-throttle tests using the analog ball-valve and digital throttle controls will be presented and compared first. Next, results from the duty cycle tests will be presented. End-to-end response spectra will be examined, and sources for throttle latency and response damping will be identified.

5.1. Digital Valve C_v and Throttle Mass Flow Calibration

A series of cold flow tests using gaseous nitrogen (N₂) were performed in order to calibrate the digital valve. Calibration validates the accuracy of the orifice C_v values and mass flow as a function of the digital valve inlet and outlet pressures in preparation for experimental testing.

Table 1. Comparing nominal and measured digital valve C_v values.

Valve Number	Orifice Diameter, mm	Design Cv	Actual Cv	Digital Channel
SOV-1	0.2540	0.0025	0.0016	0
SOV-2	0.4064	0.0050	0.0032	1
SOV-3	0.5334	0.0096	0.00660	2
SOV-4	0.7366	0.019	0.0123	3
SOV-5	1.0160	0.0360	0.0224	4
SOV-6	1.3970	0.0680	0.0452	5
SOV-7	2.0828	0.1550	0.0929	6
SOV-8	4.3688	0.3500	0.1818	7
SOV-Total		0.6451	0.3659

compares the measured C_v values to the nominal (design) values based on the manufacturer’s specifications (https://www.mcmaster.com/ accessed on 1 September 2024). Figure 9 presents bar graphs comparing the design and actual C_v values, along with error bars showing the measurement uncertainty.

The as-installed C_v mean values and associated error bars were evaluated statistically. For the cold-flow calibration, each of the 8-bit valves was individually actuated at inlet pressure levels varying from 100 to 500 psig in increments of 100 psig. For each valve flow path, the tests were repeated 3 times at each pressure level, and mean C_v values and standard deviations were calculated from the 18-member ensembles. The plotted error bars are the 95% confidence levels based on a student’s-t distribution [45],

$${\mu }_{C_v}={\displaystyle \frac{{\sum }_{i=1}^N{C}_{v_i}}{N}}\pm {\displaystyle \frac{{\delta }_{95\mathrm{\%}}\cdot S_{C_v}}{\sqrt{N}}},$$

(19)

where N = 18 (the number of ensemble members), S_Cv is the sample standard deviation over the 18-member set,

$$S_{C_v}=\sqrt{{\displaystyle \frac{1}{N-1}}{\sum }_{i=1}^N{(C_{v_i}-{\displaystyle \frac{j{\sum }_{i=1}^N{C}_{v_j}}{N}})}^2},$$

(20)

and δ_95% is the student-t multiplier corresponding to 17 (N − 1) degrees of freedom, approximately 2.108.

The mean valve N₂ mass flow during each cold-flow test was calculated from lost mass, calculated from pre- and post-tank mass measurements. Thermocouple measurements at the valve inlet and outlet ports allowed flow density and volumetric flow calculations. The differences in molecular weight between the N₂ gas used for the calibrations and the gaseous oxygen (O₂) were accounted for in the C_v calculations using the specific gravity term of Equations (16) and (17).

The calibration results show that the tested C_v values of each SOV are approximately 56% of the design values, including the end-to-end total flow C_v. The author believes that there are two primary reasons for the lower as-installed flow path C_v values. (1) The 8-bit digital throttling valve used 1.93 mm (0.11 in.) inner diameter (ID) tubing for system compactness, resulting in significant friction losses along the eight flow paths. (2) Aerospace-grade components were beyond the budget for this project, and the low-cost flow-control orifices, ordered from an industrial supply house (https://mcmaster.com, accessed 1 October 2024), were used for this testing campaign. These orifices were primarily intended for water flow control, not for the flow of gaseous oxygen. It is likely that compressibility losses contributed to the lowered C_v values.

Figure 10 compares the total C_v and achieved throttle levels against the commanded throttle for the design, actual, and “ideal” digital valve flow paths. For this Figure, the “ideal” system starts with a C_v of 0.0025 for SOV-0 and doubles that value for each subsequent valve. The design condition uses the C_v values from column 1 of Table 1. Finally, the “actual” uses the calibrated C_v values from column 4 of Table 1. Note that although the actual valve C_v curves lie significantly below the design and ideal curves, they also do not exhibit the discontinuities that occur with the design C_v conditions. In other words, the actual flow-path C_v values are very nearly multiples of two for each successive orifice size increase. When the actual throttle level is scaled by a factor of 1.77, the actual throttle curve lies nearly on top of the commanded ideal curve.

As described earlier, reduced total system C_v is very likely due to compressible losses across the orifices, and differences in fluid viscosity between water and gaseous oxygen. However, the resulting C_v and throttle curves are remarkably linear (Figure 10). The effect of the “as installed” system C_v was easily overcome by increasing the regulator set value for the system to achieve the desired maximum mass flow and throttle levels.

5.2. Test Motor Fuel-Regression Rate Calibration

Based upon the earlier O/F shift discussion of Section 4.3, it becomes clear that the motor fuel regression rate variability will have a strong influence on the achieved accuracy of the end-to-end motor throttle levels. As described in the introduction section, the fuel pyrolysis rate is driven primarily by the oxidizer mass flux rate and when combined with the exposed area of the continuously variable fuel port burn surface, sets the motor O/F ratio response. Thus, the hybrid motor O/F ratio “self-adjusts” during throttling. Because of the interdependence between fuel mass flow, oxidizer mass flux, and the instantaneous O/F ratio, the fuel regression of hybrid motors rate typically varies nonlinearly with time, and time-resolved fuel regression rates are difficult to measure. Over the years, various techniques have been used to collect time-resolved measurements. Whitmore (2020) [50] presents a detailed summary of the available regression rate methodologies and presents a time-resolved method that uses measurements of oxidizer mass flow and chamber pressure to calculate the instantaneous fuel regression rate.

For this experiment, where the individual burns each exceeds 15 s in duration, the fuel regression rate is calculated using a simple mass-depletion model. Using the depleted fuel and oxidizer masses to calculate fuel regression rate and oxidizer mass flux over the course of the total burn duration, though tedious, is generally accurate for time-averaged calculations. With this model, the consumed oxidizer and fuel weights are measured pre- and post-test, and then divided by the estimated burn times to calculate the mean mass flow rates over the duration of the burn. The time-averaged longitudinal mean regression rate over the burn duration t_burn, is calculated from the consumed fuel mass ΔM_fuel, as

$${\bar{\dot{r}}}_L={\displaystyle \frac{\mathsf{\Delta }{M}_{\mathit{fuel}}/t_{\mathit{burn}}}{2\pi ·{\rho }_{\mathit{fuel}}·\left[\frac{{\bar{r}}_L\left(t_{\mathit{burn}}\right)+r_0}{2}\right]·L}}.$$

(21)

In Equation (21), the parameter r₀ is the initial fuel port radius, and ${\bar{r}}_L$(t_burn) is the mean measured fuel port radius at the end of the burn. The instantaneous fuel port radius is estimated as

$${\bar{r}}_L\left(t\right)=r_0+{\bar{\dot{r}}}_L·t.$$

(22)

In order to measure the variation in fuel regression rate over time, a set of three steady-state burns were performed on a single fresh ABS fuel grain, with each burn performed at a commanded 80% full-throttle level, approximately 160 Newtons. For each test, the pre- and postburn fuel masses were recorded to calculate ΔM_fuel. Figure 11 shows the data for each of the 100% throttle three burns. Plotted are the thrust time-history profile as directly measured by the test-stand load cell and the thrust as calculated from chamber and nozzle exit pressure using the De Laval flow thrust Equation (23) (Anderson [42]), Chapter. 5),

$$\mathit{Thrust}=P_0{A}^{*}·\gamma \sqrt{{\displaystyle \frac{2}{\gamma -1}}{\left({\displaystyle \frac{2}{\gamma +1}}\right)}^{{\displaystyle \frac{\gamma +1}{(\gamma -1)}}}}{[1-{\left({\displaystyle \frac{p_{\mathit{exit}}}{P_0}}\right)}^{{\displaystyle \frac{\gamma -1}{\gamma }}}]}^{1/2}+{\displaystyle \frac{A_{\mathit{exit}}}{A^{*}}}[{\displaystyle \frac{p_{\mathit{exit}}}{P_0}}-{\displaystyle \frac{p_{\infty }}{P_0}}].$$

(23)

For the testing campaign leading up to the digital throttle tests, a series of 6 regression-rate calibration tests were performed. The first three burns were performed on a single, new fuel grain using a 100% commanded throttle level. Next, the test series was repeated using a new fuel grain with a 50% commanded throttle level. As a visual example, Figure 11 shows the test results from the 100% throttle regression-rate calibration tests. Plotted are the measured thrust and the thrust level calculated from chamber and nozzle exit pressure using the De Laval flow equations. Figure 11b–d plots the time histories of the oxidizer, total, and fuel mass flow rates. The total mass flow and fuel mass flow rates are calculated using the methods, as described in the previous section. Note, that there is a strong rise in the thrust level for the first burn, a modest increase in thrust for the second burn, and very little thrust increase for the third and final burn. The reason for this thrust increase is shown by the mass flow plots of Figure 11b,d,f. Note, that although the oxidizer mass flow remains essentially constant for each burn, the fuel mass flow, and hence the total mass flow, gradually increases for each burn before leveling off on the final burn. This fuel mass flow shift is indicative of a strong negative O/F shift of the motor, as previously described in Section 4.3 of this paper.

Figure 12 plots the fuel regression rates associated with the burn data of Figure 11. Here, the fuel regression rate, calculated using the previously described mass-depletion method of Section 5.2, is plotted as a function of the fuel-port oxidizer mass flux, with the data for burns 1, 2, and 3, concatenated and plotted end-to-end. The associated least-squares curve fit parameters, as calculated using the method of Equation (10) and the associated error bands, are also plotted. For this calculation, the effective fuel port length is estimated to be approximately 26 cm, as shown in Figure 5b.

Table 2. Motor fuel regression rate test data power-law curve fit coefficient summary.

Throttle Level	50%			100%
Fit coefficients	a, ${\displaystyle \frac{\left(\frac{cm}{\mathrm{s}}\right)·{cm}^{n+1}}{{\left(\frac{g}{sec}\right)}^n}}$	n	RMS fit error, cm/s	a, ${\displaystyle \frac{\left(\frac{cm}{\mathrm{s}}\right)\cdot {cm}^{n+1}}{{\left(\frac{g}{sec}\right)}^n}}$	n	RMS fit error, cm/s
Burn 1	0.51059	0.2536	±0.0125	0.54327	0.2836	±0.0225
Burn 2	0.35053	0.3396	±0.0135	0.31378	0.3796	±0.0155
Burn 3	0.26083	0.4265	±0.0141	0.34427	0.4065	±0.0161
Concatenated data	0.35736	0.3400	±0.0112	0.48695	0.3239	±0.0121
Mean values				0.41911	0.3320	±0.0117

summarizes the curve-fit data for each of the burns in the two-burn series, i.e., three burns each at 50% throttle and three burns each at 100% throttle. The b regression rate burn parameters (a, n) are estimated for each individual burn as well as for the concatenated data sets. Table 2 also calculates the mean values for the burn parameters, evaluated over the entire data ensemble.

The “odd” units for the burn-rate multiplier, a, result from the units of Equation (9). This dimensional analysis is shown by Equation (23) for cm-k-s units,

$$a={\displaystyle \frac{\dot{r}·L}{{(G_{ox}·L)}^n}}~{\displaystyle \frac{\frac{cm}{\mathrm{s}}·cm}{{(\frac{g}{sec}·\frac{1}{{cm}^2}·cm)}^n}}=({\displaystyle \frac{cm}{\mathrm{s}}}){\displaystyle \frac{cm}{{\left(\frac{g}{sec}\right)}^n·{\left(\frac{1}{cm}\right)}^n}}={\displaystyle \frac{\left(\frac{cm}{\mathrm{s}}\right)·{cm}^{n+1}}{{\left(\frac{g}{sec}\right)}^n}}.$$

(24)

The estimated burn exponents all lie clearly in the “negative O/F shift” range. The curve fit coefficients, and the associated fit errors for each of the individual burns, as well as the curve fit for the concatenated data sets, are listed. When averaged over all of the six burns in the data set, the resulting curve fit parameters for the mean fuel regression rate are listed in the last column of Table 2.

Substituting the mean burn exponent from Table 2, n = 0.332, into Equations (14) and (15)

$$O/F_{\left(t\right)}={\displaystyle \frac{{{\dot{m}}_{ox}}^{0.668}}{4.297\cdot {\rho }_{fuel}\cdot r_{port}^{0.336}\cdot L^{0.332}}}$$

(25a)

$${\displaystyle \frac{\partial (O/F_{\left(t\right)})}{\partial t}}=\left({\displaystyle \frac{-0.336\cdot {\dot{m}}_{ox}}{{2\cdot \rho }_{fuel}\cdot {\left(V_{ol}\right)}_{port}}}\right)$$

(25b)

Then, from Equation (25a) for a constant oxidizer mass flow, it is observed that the O/F ratio decreases with time as the fuel port burns and the port radius increases. However, from Equation (25b), it is also observed that the O/F shift rate will diminish with time as the fuel port radius and internal volume grow during the burn lifetime. This predicted behavior is exactly the fuel-rich O/F shift, diminishing with time, as exhibited by the plots in Figure 11.

5.3. Deep Throttle Test and Analysis

A series of throttle tests were performed to investigate the motor response to a deep-cycle ramp. The commanded throttle starts at 100% for 4 s after the run valve opens, then ramps down to 1% throttle for 4 s, ramps back up to 100% throttle for 4 s, and finally remains at 100% for an additional 4 s, 16 s total burn time. Each fresh fuel grain was burned twice in order to look for the effects of the negative O/F shift, as described in the previous section. Figure 13 plots the thrust and O/F ratio time histories for two consecutive deep-throttle burns of a single fuel grain. Figure 13a,b plots thrust measured by the cart load cell, thrust calculated from chamber pressure, and the prescribed throttle thrust profile. Figure 13c,d plots the Burn O/F ratio time history, along with the stoichiometric and optimal O/F ratios for GOX/ABS combustion.

Note that the burns show moderate response latency and roll-off of the command “shoulders”, but also note that the first burn exhibits uneven thrust “humps”, and a significant fuel-rich shift in O/F ratio from the beginning to the end of the burn. For the second burn, the peak thrust levels and O/F ratios are more evenly distributed. During the first burn, the change in the fuel port diameter and the associated changes of mass flux and burn area change significantly more rapidly than occurs with the second burn. Thus the second burn exhibits a greater response fidelity to the commanded throttle level. Finally, note that both thrust curves exhibit curve “ticks” in the profile, occurring at approximately 7.5 and 9.5 s into each burn. These “ticks” correspond to the unchoking of the thrust chamber injector during deep throttling, allowing coupling of the combustor and throttle valve outlet pressures.

5.4. Deep Throttle, Multiple-Step, Boxcar Duty-Cycle, Test and Analysis

Multiple tests were performed using the “boxcar” duty cycle of Figure 8c. Figure 14 and Figure 15 present typical results. Figure 14 plots the time history data for the first burn of a fresh fuel grain for five key parameters: (a) injector and chamber pressure, (b) thrust, (c) mass flow, and (d) O/F ratio. Also, plotted are the 1-D model predictions for each of these parameters. The hybrid model assumes the regression rate power-law model with burn parameters derived from the regression rate burn calibrations, as shown in Table 2. Figure 15 plots the same parameters but for the second burn of the same fuel grain.

Generally speaking, the injector and chamber pressure peak amplitudes closely match the thrust responses. In contrast, even though the oxidizer mass flow curves for both burns are nearly identical, the first-burn time histories of total mass flow and O/F ratio exhibit significantly different behaviors when compared to the second burn. Note that the total mass flow for the first burn grows with each boxcar pulse; in contrast, the total mass flow peaks for the second burn remain relatively flat. The behaviors for O/F ratios are similar but with the peaks for the first burn decreasing with time and the peaks for the second burn remaining relatively flat. Also, the mean O/F ratio for the second burn is lower, indicating a fuel-rich O/F shift. This behavior is best explained by comparing the fuel mass flow rates or the two burns. Figure 16 shows this comparison, where the fuel mass flow rates are calculated as the difference between the total and oxidizer mass flow. For the first burn, the thrust and fuel mass flow exhibit peaks that grow over time, leading to the O/F ratio that significantly decreases with time, as shown in Figure 14d. In contrast, for the second burn, the fuel mass flow peaks have mostly leveled off, leading to flat O/F ratios, as shown in Figure 15d.

5.5. Sine Wave Duty-Cycle Test and Analysis

Finally, the digital valve system was exercised using a series of sine wave profiles commanded at various frequencies. Figure 17, Figure 18, Figure 19 and Figure 20 compare typical test results for throttle sine waves commanded at 0.25, 0.5, and 1.0 Hz, respectively. Plotted are (a) throttle command, (b) thrust, and (c) digital valve inlet/outlet pressure and injector feed pressure. For these figures, the “achieved” throttle level plotted in Figure 17a, Figure 18a, Figure 19a and Figure 20a is calculated by dividing the measured thrust level by the nominal full-throttle motor thrust, 200 N. As shown in Figure 17b, Figure 18b, Figure 19b and Figure 20b, note that the thrust “buckets” (at lowest throttle) are highly attenuated, and this attenuation grows with increasing sine wave frequency. Also, as shown in Figure 17c, Figure 18c, Figure 19c and Figure 20c, note that at the lowest pressure levels, the digital valve upstream pressure exhibits an inverse correlation to the injector feed pressure. Typically, due to injector choking, the upstream is mostly independent of the thrust-chamber response. The displayed correlation behavior indicates that the injector is unchoking at the lowest pressure levels. The unchoked injector leads to injector-feed coupling (Karabeyoglu [51]), which is a likely contributor to observed attenuation and other distortions observed at the lowest throttle levels. Figure 19 and Figure 20 also plot the 1-Hz sine response for two burns of the same fuel grain. Note that for the second burn, the data exhibit a much more constant thrust level; this behavior is the same as was previously exhibited by the boxcar burns of Figure 14 and Figure 15. These previously described O/F shift trends were observed for all the multistep, boxcar, and sine-wave throttle tests. The observed anomalous steady-state deviations observed at the end of the sine-wave burns are due to the nitrogen purge used to cool the motor and prevent nozzle erosion due to free air burning after the test firing.

5.6. Frequency Response Analysis

As shown in the previous section, there exists significant latency and roll off in the end-to-end digitally throttled motor response. This section investigates the source of this fidelity loss. In order to develop a good understanding of the system frequency response, the following frequency spectra are compared; (1) digital valve inlet and outlet pressures, (2) digital valve outlet and injector pressures, (3) injector and chamber pressures, and (4) chamber pressure, thrust measured from the load cell and 8-bit throttle command. The last comparison plot represents the end-to-end frequency response of the system. This frequency-response analysis was performed for the data previously shown in Figure 15 (multistep boxcar, burn 2), and Figure 20 (1-Hz sine wave, burn 2). For this analysis, the “burn 2” data are used to investigate the system frequency response in order to eliminate the effects of the previously presented O/F shift as a variable in the analysis.

5.6.1. Multistep Boxcar Frequency Response Analysis

Figure 21 shows the system frequency response for the multistep boxcar data of Figure 15. Plotted are (1) Figure 21a, the digital valve inlet-to-outlet pressure frequency response, (2) Figure 21b, the digital valve outlet-to-injector pressure frequency response, (3) Figure 21c, the injector pressure-to-chamber pressure frequency response, and (4) Figure 21d, the commanded throttle-to-thrust and -chamber pressure frequency response. On each plot the spectrum of the commanded throttle is also included for reference purposes. Magnitudes are plotted on the upper graphs, and the phase angles are plotted on the lower graphs. All magnitudes have been normalized to zero bias values in order to account for the different signal amplitudes. In the figures, the colors are included for presentation clarity. The specific traces are labeled by the figure annotations.

From Figure 21a it is observed that both the inlet and outlet pressures exhibit significant harmonic ringing, mostly likely resulting from the mixing plenums. Also, notice in Figure 21a that, as expected, the digital-valve outlet pressure exhibits slightly greater attenuation when compared to the inlet pressure. The throttle valve outlet pressure begins to roll off from the commanded throttle signal at approximately 5 Hz. From Figure 21b, the injector feed pressure spectrum basically tracks the valve outlet pressure up to 10 Hz, indicating little attenuation due to the piping and running valve downstream of the throttle valve. Also, notice that phase angles for the digital valve and injector feed pressures are significantly higher than the phase angle corresponding to the commanded throttle. This difference is indicative of the response latency caused by the mixing chambers. Finally, the injector feed pressure spectrum does not display the strong, harmonic oscillations that are apparent in the digital valve outlet signal, which shows the harmonic oscillations caused by the digital valve have mostly been damped before entering the motor, likely due to the transmission due to the piping.

From Figure 21c,d, the motor chamber pressure rolls off considerably when compared to the injector feed pressure and motor thrust signal. These considerable magnitude differences are very likely due to the chamber pressure transmission tubing. In order to protect the chamber pressure transducer from radiation heating due to the combustion chamber flame, a small-ID (2 mm) pressure transmission tube was used, and this tube was bent at a right angle so the transducer did not “look” directly into the flame. This sensing arrangement induces considerable attenuation into the chamber pressure measurements. Finally, based on the data of Figure 21d, the end-to-end system response, as represented by the load-sensed thrust measurement, is actually quite good. Although the thrust spectrum begins to roll away from the commanded throttle magnitude frequencies below 5 Hz, the roll off is slow, and the end-to-end frequency response generally tracks the commanded sine wave.

5.6.2. 1-Hz Sine Wave Frequency Response Analysis

Figure 22 presents the same analysis methods as the previous section but now analyzes the 1 Hz sine wave data of Figure 20. The observations are mostly the same; however, there are a few valuable items to point out for this frequency response analysis. First, the 1-Hz sinusoid peak is clearly evident in all of the magnitude lots. Next, the injector feed pressure exhibits a lower roll off rate than does the throttle valve outlet pressure. Since the injector lies downstream of the throttle valve, this result is physically unrealistic. Thus, it must be reasoned that the observed throttle valve outlet pressure roll-off results from destructive harmonics inside the outlet mixing plenum. The chamber pressure and thrust measurements exhibit a strong harmonic at approximately 2 Hz. This harmonic likely results from a low-frequency injector feed coupling in the thrust chamber. Finally, as with the Boxcar throttle input, the end-to-end system response, as represented by the load-sensed thrust measurement, is actually quite good. As with the data of Figure 21, the thrust roll off is slow, and the end-to-end frequency response generally tracks the commanded sine wave.

5.6.3. Discussion of System Response Fidelity

From the previously presented frequency response data, it becomes clear that the current digital valve design is the limiting factor in the overall system response, with a transfer function that begins to roll off at approximately 2 Hz. The slow response of the valve is due primarily to the large size of the input and output mixing chambers and attenuation in the valve system piping, which used 8 mm (3/16″) ID tubing for all of the flow paths. Fortunately, the frequency roll off is fairly slow, and considerable response fidelity exists at frequencies of up to 10 Hz. Thus, the system is still able to reasonably capture both the multiple-step boxcar and 1-Hz sine wave responses. This prototype system was designed to demonstrate the feasibility of the concept, and no system response-performance optimization was attempted for this testing campaign. With the acquired knowledge from this work, it would be easily possible to increase the fluidic response fidelity of the system by an order of magnitude or more. For example, nonuniform diameter piping can be used with small-diameter piping used for the lowest mass flow paths and larger-diameter piping applied to the higher mass flow paths. Similarly, the inlet mixing plenum can be entirely eliminated, and the outlet flow plenum can be reduced significantly in size.

6. Future Work: System Response Scaling and Compensation

The hot fire testing campaign reported in this paper uses only open-loop throttle control, where a prescribed throttle profile is fed into the control software, and the 8-bit digital valve responds by actuating the appropriate SOVs to alter the oxidizer flow during a burn. This throttling technology, however, opens up the possibility of developing a compensation/control scheme that allows the system to overcome observed system latencies, commanded throttle distortions, and O/F shifts in the motor. Having the ability to compensate for O/F shift in hybrid rocket motors has the potential to eliminate the significant obstacles, as described in Section 2, that keep hybrids from being used in a wide range of spaceflight applications. The digital throttling valve technology is a key component of this project, and clearly, this topic is rich for future research. Figure 23 illustrates this concept, demonstrated using the analytic model of Section 4, and boxcar throttle data of Figure 14. Here, the growing thrust peaks caused by the negative O/F shift are used to estimate the appropriate scale of the injector feed pressure, using the output-error maximum likelihood estimation method [52].

Figure 23 shows the results of this calculation. Figure 23a,b plot the original thrust and chamber pressures, along with the initial injector feed pressure. Using the output error approach, the injector feed pressure is scaled over successive iterations until each of the boxcar peaks reaches the desired peak throttle settings, 200 N. The maximum likelihood method minimizes the total accumulated error for each of the duty-cycle peaks. Figure 23c plots the maximum likelihood estimate of the scaling factor, and Figure 23d compares the scaled injector pressure to the original value. The scaled injector pressure is subsequently submitted to the calibrated motor ballistic model, and the resulting chamber pressure and thrust profiles are calculated. Figure 23e,f compares these results to the original chamber pressure and thrust data. Each of the individual boxcar peaks has now achieved the desired 200 N thrust level.

Obviously, using this method, the system must have excess mass flow capacity in order to scale the throttle command, but the useful potential is illustrated by this example. Also, the output-error approach is not suitable for real-time applications. Applications of advanced machine learning techniques may provide a better real-time solution. Clearly this topic is rich for future development.

7. Summary and Conclusions

The results of a development program of an 8-bit digital throttling valve for a hybrid rocket motor are presented. The control valve technology utilizes a series of eight individually operated control valves set in parallel, with each valve being individually operated. Since the system divides the achievable total mass flow across all the valves, the required mass flow of any single valve remains low enough for solenoid-operated valves (SOVs) to be used with flow restriction orifices attached downstream in order to alter the flow coefficient. This design allows for rapid response times and precise control of the achievable total mass flow and thrust levels, which is only limited by the number of components in the digital valve design. An 8-bit prototype configuration was tested using a previously well-characterized 98 mm hybrid rocket motor that burns gaseous oxygen (GOX) and ABS plastic as propellants. Calibration tests performed on the 8-bit digital valve system show that the corrected orifice sizes on each SOV caused a reasonably linear mass flow and commanded throttle response.

Multiple throttle profiles, including steady-state, deep throttle ramp, stepped-boxcar throttle, and sine waves at various frequencies were demonstrated. Steady-state regression rate tests demonstrate that the motor exhibits a negative, that is a fuel-rich oxidizer-to-fuel shift with time. The effect of this O/F shift means that the motor thrust and fuel mass flow rate grows as a function of time, even when the oxidizer mass flow rate is constant. The O/F shift is most apparent when a new fuel grain is burned. As the fuel port burns and opens up, the effects of the O/F shift diminish. Multistep boxcar throttle tests and the associated spectral analysis, demonstrate that the fundamental response frequency of the end-to-end motor systems is approximately 10 Hz. Two factors contribute to this behavior; the mechanical and acoustical responses of the systems’ components, and the limited bandwidth capability of the asynchronous software driving software and the control laptop. Comparisons to analytical models of the throttle control are also presented, showing good agreement.

In the pretest design stage of this project, there was some concern that the switching of the electromagnetic valves would induce combustion oscillations in the engine system, adversely affecting the system’s performance. However, during the more than 50 hot-fire tests that made up this testing campaign, not a single case of motor combustion instability was observed; even at the lowest throttle levels where the injector was clearly unchoked. It must be noted that the maximum mass flow and thrust levels investigated for this campaign, roughly 200 N and 10 g/s, are relatively small, and it is possible that at significantly higher thrust and mass flow levels, throttle-induced instabilities may occur. The author does not consider this low-to-moderate mass flow range a significant limitation. The author’s intent for this technology was not for the control of large-high, high-mass flow, hybrid rocket systems. Instead, the system is intended for small-to-intermediate motor sizes in the range of 100–400 N, such as upper-stage systems where throttling can be most advantageous for trajectory optimization, realizing the full potential of hybrid rocket systems.

Finally, follow-on efforts are needed to investigate throttle-command shaping, allowing system response latencies and other distortions, including O/F shift to be compensated. An analytical example using the developed ballistic model is presented. The resulting scaled command generates a chamber response that diminishes over time, resulting in a nearly constant thrust profile. Obviously, using this method the system must have excess mass flow capacity in order to scale the throttle command, but the useful potential is illustrated by this example. Clearly this topic is rich for future development.