Simplified Data-Reduction Method for Hybrid Propulsion
2024; American Institute of Aeronautics and Astronautics; Volume: 40; Issue: 3 Linguagem: Inglês
10.2514/1.b39387
ISSN1533-3876
AutoresYuji Saito, Shota Kameyama, Toshinori Kuwahara,
Tópico(s)Spacecraft and Cryogenic Technologies
ResumoOpen AccessTechnical NotesSimplified Data-Reduction Method for Hybrid PropulsionYuji Saito, Shota Kameyama and Toshinori KuwaharaYuji Saito https://orcid.org/0000-0003-2804-8076Tohoku University, Sendai 980-8578, Japan, Shota KameyamaElevationSpace, Inc., Sendai 980-8579, Japan and Toshinori KuwaharaTohoku University, Sendai 980-8579, JapanPublished Online:29 Feb 2024https://doi.org/10.2514/1.B39387SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionHybrid rockets are much safer than solid and liquid rockets because the fuel and oxidizer are physically separated and stored as different material phases. Data reduction in static firing tests of hybrid rockets presents unique challenges compared to those of using other types of chemical rockets [1]. During liquid-rocket firing, a feed system supplies liquid propellants to the combustion chamber, and the propellant mass-flow rates are directly measurable. During solid rocket firing, the oxidizer-to-fuel mass ratio (O/F) is a known value, and it is easy to calculate the regression rate of a solid propellant from the chamber pressure and nozzle throat area, assuming a constant characteristic exhaust velocity (c*). Hybrid rockets use a combination of solid and liquid propellants; however, a fuel is typically used in the solid state. Although the mass-flow rate of the liquid oxidizer is directly measurable, the one of the solid fuel is not easily measurable. Direct calculation of the flow rate from the chamber pressure and nozzle throat area, which is possible with solid rockets, is impossible for hybrid rockets because c* strongly depends on O/F. Kuo [1] indicated that the measurement of the fuel regression rate as a function of the operating conditions is a major challenge for hybrid rocket propulsion researchers. This is because the instantaneous fuel regression rate is a function of the motor operating conditions, firing duration, port diameter, and the occurrence of nozzle erosion. Here, nozzle erosion does not affect the fuel regression rate, but it may affect the results of the reconstruction technique (RT) described below because the starting points of the nozzle erosion and the rate of nozzle erosion are unknown.Many data reduction methods for hybrid rocket firing rely on the endpoint averaging method. The endpoint averaging method uses information on the initial and final fuel shapes with firing duration to determine the average fuel regression rate and the average fuel mass-flow rate. A difficulty arises because neither the fuel regression rate nor the fuel mass-flow rate is constant or a linear function of time during firing. Many combustion experiments have been designed to use short-firing durations to minimize errors introduced by averaging these nonlinear functions of time. In addition to requiring numerous short-firing tests, the errors associated with ignition and shutdown transients were significant in this case. For example, Frederick and Greiner reported errors of up to 8.7% in experimental studies [2].Several experimental techniques [3] have been used where the fuel grain thickness or port diameter is directly measured using specialized instrumentation, such as x rays [4], thrust stands [5], or ultrasonic sensors [6–8]. X rays provide the 2D resolution of a motor, which is helpful for basic research on boundary-layer development while being limited to the measurement of the most basic fuel shapes. Furthermore, the x-ray radiography equipment is expensive and requires skilled personnel. Thrust stands that measure fuel regression through weight reduction are also very costly, and their accuracy varies greatly with experimental conditions. Ultrasonic measurements are the least costly but can only measure fuel regression at a specific location in fuel grains with simple geometries. The fuel regression during firing was captured using a camera [9,10], and the fuel regression rate was successfully obtained at a spatiotemporal resolution. However, it is difficult to apply this method to rocket combustion tests because it uses rectangular fuel instead of a rocket chamber. By blending a solid fuel, that is, a nonconductive material with a conductive material, and measuring the physical quantity of the electrode network, it is possible to obtain time-resolved and partially space-resolved fuel regression rates using 3D printer modeling fuel [11–13]. However, 3D printer modeling is required to place electrodes within the fuel, and there are limitations to the fuel types that can be used.Researchers have developed data-reduction methods to obtain fuel consumption rates from measurable data such as chamber pressure Pc and oxidizer mass-flow rate m˙o. Wernimont and Heister introduced a data-reduction method that approximates the characteristic exhaust velocity c* as a constant during firing [14]. However, this assumption is unrealistic because c* varies significantly with changes in O/F during firing or underthrottling operations in hybrid rockets. George et al. developed the RT for determining the fuel consumption rate as a function of time from the histories of Pc and m˙o, assuming a constant c* efficiency [15]. However, Nagata et al. also reported that, under fuel-rich conditions, there is a region in the solution space of RT calculations where the instantaneous O/F has multiple solutions [16]. Several previous studies have reported multiple solutions [11,17–19]. Nagata et al. developed another RT method to avoid multiple solutions [20]. The RT accuracy was compared in actual and numerical firing tests, and the validity of the data reduction method was examined [21,22]. In addition, RT has also been extended to the thrust equation to improve the accuracy of the method [7], and it has been extended to nozzle throat-erosion history acquisition [18,23–25].RT is a powerful tool for a data reduction method; however, it has not been widely used in research. A possible reason for the lack of RT users is the complexity of the RT calculation. To perform RT, a preprepared program (e.g., Fortran [7] or MATLAB [22,26]) is required. It is not possible to immediately analyze the data obtained for the firing tests using spreadsheet software (e.g., Excel, Google Spreadsheets) because it requires a large number of convergence calculations. Therefore, in this study, we proposed a simplified RT that is computationally cost-effective without multiple solution problem and can be used with spreadsheet software. The proposed method was applied to simple numerical firing tests and associated data, after which it was compared in terms of accuracy and computational cost.II. MethodologyA. Simple Data Reduction1. Conventional Reconstruction TechniqueUsing a conventional RT [15,16,27], we can obtain the following equation by introducing the efficiency of the characteristic exhaust velocity ηc*: ηc*cth*=PcAtm˙o(1+1/(O/F))(1)where we assume a constant ηc* during firing and At is nozzle throat area. Assuming a value of ηc*, Eq. (1) can be used to calculate O/F at each time step. Solving Eq. (1) requires an iterative procedure because the theoretical characteristic exhaust velocity cth* is a function of O/F and Pc, where cth* is calculated using NASA-CEA [28]. After determining all the time steps for O/F and the fuel mass-flow rate m˙f, the integral of Eq. (2) provides the overall fuel mass consumption during firing, as follows: MfRT=∫tfm˙fdt=∫tfm˙oO/Fdt(2)where tf denotes the firing duration. The value of ηc* is adjusted such that MfRT agrees with the experimental value Mf: Mf=MfRT(ηc*)(3)As mentioned previously, the conventional RT incurs a large computational cost because of the complexity of the calculation. This is because it requires a convergence calculation of O/F at each time step for a specific ηc*, which requires calculations until Mf=MfRT. Let N be the number of time steps, O/F and ηc*, and πO/F and πηc* be the time required for convergence; then, the computational cost of the conventional RT is O(NπO/Fπηc*). It should be noted that more time is required for the O/F calculation if NASA-CEA [28] is calculated each time. This can be avoided using a polynomial approximation formula [16]; however, if the approximation formula itself is complex, a significant amount of time is required to obtain it. Figure 1 shows the relationship between the characteristic exhaust velocity c* and O/F at Pc=0.6 MPa, as calculated using NASA-CEA [28]. The propellants used were polymethylmethacrylate (PMMA)–GOx (gaseous oxygen), high-density polyethylene (HDPE)–GOx, PMMA–nitrous oxide (N2O), and HDPE-N2O. If these upward convex curves can be approximated using linear, power, or exponential functions, the O/F can be uniquely determined without convergent calculations, as shown in Eq. (1); however, approximating these functions is difficult.Fig. 1 Relationship between characteristic exhaust velocity c* and O/F at Pc=0.6 MPa calculated using NASA-CEA [28].2. Simplified Reconstruction TechniqueTo overcome the multiple-solution problem and the high computational cost of conventional RT, we proposed a simplified RT. From Eq. (1), the equation is transformed to obtain an approximate equation using the power function: PcAtm˙o=ηc*cth*(1+1/(O/F))=ηc*A(O/F)B(4)where A [m/s] and B are constants determined using propellant combinations. Figure 2 illustrates the relationship between c*(1+1/(O/F)) and O/F. There are multiple O/F values for some c*(1+1/(O/F)) with 0.5<O/F 1 in the case of ηc*=0.950, as shown in Fig. 14. In the case of ηc*=0.969, where MfRT/Mf=1, the O/F error is small in the relatively low O/F range. However, the O/F error is high in the relatively high O/F range. Owing to the aforementioned reasons, O/F obtained by simplified RT yields an error when O/F shifts during firing across the O/F range which lacks an effective approximated c*.Fig. 11 O/F's calculated by the conventional RT and the simplified RT in test 6.Here, we compare the computation times of conventional and simplified RTs. For conventional RT, the program was implemented using Fortran90, and NASA-CEA [28] was calculated sequentially at each O/F and Pc. For the computing environment, the adopted processor and random access memory (RAM) were Intel(R)/Xeon(R)/Gold/6240R/[email protected] GHz and 768 GB, respectively. The simplified RT was calculated using Microsoft/Excel (Microsoft Corp., Redmond, WA). By entering the formulas in advance, O/F and ηc* can be instantly obtained by simply pasting the firing test data. However, calculating test 6 with conventional RT required approximately 2 h of computation time. Because O/F can be obtained uniquely, a simple RT is suitable for immediate data reduction at the firing test site. This is because data analysis can be easily performed using a spreadsheet. In addition to significantly reducing the computational cost, the application of the simplified RT is particularly useful for preventing the occurrence of the multiple-solution problem.Fig. 12 O/F calculated by RT at ηc*=0.957 and simplified RT at ηc*=1.000, 0.969, 0.950, and 0.900.Fig. 13 Graph showing m˙f calculated by RT at ηc*=0.957 and simplified RT at ηc*=1.000, 0.969, 0.950, and 0.900.Fig. 14 MfRT/Mf calculated by RT at ηc*=0.957 and simplified RT at ηc*=1.000, 0.969, 0.950, and 0.900.IV. ConclusionsThe RT is a very powerful tool for the reduction of hybrid rocket combustion data; however, the lack of RT users depends on the complexity of the RT calculations. Therefore, in this study, we proposed a simplified RT that is computationally cost-effective and can be used with spreadsheet software to calculate the RT. The proposed method was applied to simple numerical firing tests and firing test data, after which it was compared in terms of accuracy and computational cost.In the proposed simplified RT using two power functions for calculating O/F and ηc*, the convergence calculation was eliminated, and O/F could be determined uniquely. This not only significantly reduced the computational cost but also overcame the multiple-solution problem. When applied to a simple numerical firing test, the results showed approximate agreement with the O/F used in hybrid rocket firing. The simplified RT can overcome the multiple-solution problem. The error produced by the approximation in the simplified RT is smaller than that produced in the case where the multiple-solution problem occurs in the conventional RT. When applied to an actual firing test, the results were similar to those of the simple numerical combustion test and approximately identical to those of the conventional RT. Here, O/F can be obtained instantly using spreadsheet software with simplified RT, whereas it took approximately 2 h using conventional RT. Based on these results, the simplified RT proposed in this study is a suitable method for obtaining O/F immediately at the site of a siring test and can significantly contribute to future research and the development of hybrid rockets.A. K. GuptaAssociate EditorAcknowledgmentThis work was supported by JSPS KAKENHI Grant Number 23H01598. References [1] Kuo K. K., "Challenges of Hybrid Rocket Propulsion in the 21st Century," Fundamentals of Hybrid Rocket Combustion and Propulsion, Vol. 218, AIAA, Reston, VA, 2007, pp. 593–638. 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All rights reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3876 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsComputing and InformaticsComputing, Information, and CommunicationData AnalyticsData ProcessingData ScienceHybrid-Propellant RocketRocket EngineRocketry KeywordsData ProcessingHybrid RocketData AnalyticsAcknowledgmentThis work was supported by JSPS KAKENHI Grant Number 23H01598.Digital Received2 September 2023Accepted13 January 2024Published online29 February 2024
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