Oblique detonation wave engine

Universidade do Minho Escola de Engenharia Tomás Pinto de Jesus Ferreira Oblique Detonation Wave Engine October 2024 Universidade do Minho Escola de Engenharia Tomás Pinto de Jesus Ferreira Oblique Detonation Wave Engine Specialization in Aerospace Engineering Design Work carried out under the guidance of the Professor Dr. Gustavo Rodrigues Dias Professor Dr. Francisco Brojo October 2024 ii COPYRIGHT AND CONDITIONS OF USE OF THE WORK BY THIRD PARTIES This is an academic work that can be used by third parties if the internationally accepted rules and good practices regarding copyright and related rights are respected. Thus, the present work can be used under the terms provided for in the license indicated below. If the user needs permission to be able to use the work under conditions not provided for in the indicated licensing, he should contact the author, through the RepositóriUM of the University of Minho. License Granted to Users of this Work Atribuição CC BY https://creativecommons.org/licenses/by/4.0/ iii AGRADECIMENTOS A realização desta dissertação só foi possível, graças ao apoio de todos os que me são próximos, e ajudaram a que esta caminha fosse possível. Agradeço desde já aos meus orientadores, Professor Gustavo Dias e o Professor Francisco Brojo, que me acompanharam e apoiaram ao longo da realização deste trabalho. Pela sua disponibilidade para discutir as minhas dúvidas, mesmo em horas tardias, para além das críticas que ajudaram a melhorar a dissertação realizada. A todos os professores que tive, quer na licenciatura, quer no mestrado, que fomentaram o crescimento do interesse já existente neste curso e nesta área. De salientar o Professor Pedro Dias, que não sendo orientador, também foi importante na realização desta dissertação, através da sua disponibilidade constante para resolver qualquer questão. A todos os meus amigos que fiz ao longo destes 5 anos, que tornaram esta experiência inesquecível, através do seu apoio quer a nível académico, quer a nível pessoal. E finalmente, aos meus Pais e ao meu Irmão, que através do seu apoio e amor incondicional, tornaram possível esta caminhada, mesmo quando eu não achava possível. Sem vocês não seria possível. iv INTEGRITY STATEMENT I declare that I have acted with integrity in the preparation of this academic work and confirm that I have not resorted to the practice of plagiarism or any form of misuse or falsification of information or results in any of the stages leading to its preparation. I further declare that I know and that I respected the Code of Ethical Conduct of the University of Minho. v Motor de detonação oblíqua RESUMO Nos últimos anos a indústria espacial e da aviação tem procurado soluções para resolver os principais problemas, principalmente ao nível dos motores utilizados. Alguns desses problemas estão ligados ao peso da aeronave e ao impulso específico para elevados números de Mach. A hypersonic airbreathing propulsion oferece diferentes soluções para este tipo de problemas nomeadamente a diminuição do peso, uma vez que não necessita de transportar no interior da aeronave o oxidante usado na combustão para além de também oferecer um impulso específico superior para elevados número de Mach. O principal tipo de motor utilizado deste tipo de propulsão foi durante muitos anos, e ainda o é, o scramjet embora cada vez mais sejam realizados estudos procuram descobrir novos tipos de configurações, que apresentem melhor desempenho, surgindo então os motores de detonação oblíqua. Esta dissertação, apresenta o desenvolvimento de um código, que procura calcular e comparar a performance dos motores scramjet com motor de detonação obliqua. Para efetuar essa comparação, é utilizada uma configuração de área variável para o caso do motor scramjet, ao contrário do que é mais comum, área constante. A comparação entre a área variável e a área constante permitirá perceber se o motor de detonação obliqua apresenta vantagens a nível de performance, sobre as configurações de scramjet apresentadas. O estudo será efetuado através da análise dos diversos sistemas de um motor, nomeadamente a admissão e consequente compressão do ar, a combustão e finalmente a respetiva expansão, sendo também definidos parâmetros que procuram definir as condições de operação deste mesmo motor, principalmente a altitude de operação e o número de Mach. PALAVRAS-CHAVE Detonações, motor de detonação oblíqua, propulsão hipersónica, scramjet de área variável vi Oblique detonation wave engine ABSTRACT In recent years, the space and aviation industry has been looking for solutions to solve the main problems, especially in terms of the engines used. Some of these problems are linked to the weight of the aircraft and the specific thrust for high Mach numbers. Hypersonic airbreathing propulsion offers different solutions to this type of problem, namely weight reduction, since it does not need to carry the oxidizer used in combustion inside the aircraft, in addition to also offering a higher specific impulse dust high Mach number. The main type of engine used in this type of propulsion was for many years, and still is, the scramjet, although more and more studies are being carried out to discover new types of configurations, which present better performance, then the ODWE (Oblique Detonation Wave Engine) emerged. This dissertation presents the development of a code, which seeks to calculate and compare the performance of scramjet engines with oblique detonation engines. To make this comparison, a variable area configuration will be used for the case of the scramjet engine, as opposed to what is more common, constant area. The comparison between the variable area and the constant area will allow us to understand if the oblique blasting engine has performance advantages over the scramjet configurations presented. The study will be carried out through the analysis of the various systems of an engine, namely the intake and consequent compression of the air, the combustion and finally the respective expansion, and defined parameters that seek to define the operating conditions of this same engine, mainly the operating altitude and the number of Mach. KEYWORDS Detonations, hypersonic propulsion, oblique detonation wave engine, variable area scramjet vii INDEX Agradecimentos ........................................................................................................................ iii Resumo ....................................................................................................................................... v Abstract ..................................................................................................................................... vi Index ......................................................................................................................................... vii List of Figures ............................................................................................................................. ix List of Tables .............................................................................................................................. xi List of Symbols, Greek letters, Subscripts and Abbreviations .................................................. xii 1. Introduction ........................................................................................................................ 1 1.1 Motivation ................................................................................................................... 1 1.2 Objectives .................................................................................................................... 1 2. Bibliographic Review ........................................................................................................... 3 2.1 Hypersonic Airbreathing Propulsion ........................................................................... 3 2.2 Scramjet ....................................................................................................................... 3 2.3 Oblique Detonation Wave Engine ............................................................................... 4 2.3.1 Combustion .......................................................................................................... 6 2.3.2 Chapman-Jouguet Theory .................................................................................... 8 2.4 Synthesis ...................................................................................................................... 9 3. Methodology ..................................................................................................................... 11 3.1 Atmosphere Model .................................................................................................... 11 3.2 Compression System ................................................................................................. 13 3.2.1 Inlet Type ............................................................................................................ 13 3.2.2 Isolator................................................................................................................ 14 3.2.3 Number of Oblique Shocks ................................................................................. 16 3.3 Combustion System ................................................................................................... 19 3.3.1 Post-combustion ................................................................................................ 20 3.3.2 Oblique Detonation Wave Engine ...................................................................... 21 3.3.3 Post-Detonation Properties ............................................................................... 22 viii 3.4 Expansion System ...................................................................................................... 23 3.4.1 Performance ....................................................................................................... 24 4. Results ............................................................................................................................... 27 4.1 Model Validation ....................................................................................................... 27 4.1.1 Compression System .......................................................................................... 27 4.1.2 Scramjet .............................................................................................................. 28 4.1.3 Oblique Detonation Wave Engine ...................................................................... 29 4.2 Parametric Studies ..................................................................................................... 31 4.2.1 Compression System .......................................................................................... 32 4.2.2 Oblique Detonation Wave .................................................................................. 33 4.2.3 Scramjet combustion ......................................................................................... 35 4.3 Case Study ................................................................................................................. 37 4.3.1 Input ................................................................................................................... 37 4.3.2 Performance ....................................................................................................... 38 5. Conclusion ......................................................................................................................... 44 5.1 Future Works ............................................................................................................. 45 References ................................................................................................................................ 46 Annexes .................................................................................................................................... 48 Annex 1 – Code input ........................................................................................................... 48 Annex 2 – Code ..................................................................................................................... 49 xv Abbreviations Description ODW Oblique Detonation Wave ODWE Oblique Detonation Wave Engine 1 1. INTRODUCTION Interest in hypersonic airbreathing propulsion has been growing in recent years, since this technology allows several advantages over traditional space and aviation engines, namely the possibility of flying at speeds greater than five times the speed of sound, but also the fact that it allows faster access to space and faster commercial flights. The reason why this work has been carried out, as well as what will be carried out in it, will be referred to in the Motivation and Objectives Chapters. 1.1 Motivation The most used type of Hypersonic airbreathing propulsion engines is the scramjet, which has been the target of intense studies for several years, with the aim of enhancing their performance. This type of motors has the advantage of having a high specific impulse, in addition to not having to transport the oxidizer, compared to scramjet engines. For this reason, it is possible to carry more payload weight, increasing cost efficiency. Although the scramjet has several advantages, it also has some disadvantages, namely the complexity of the combustion system being high. Due to this reason, the concept of oblique detonation engines emerged, which replaces the diffusion process in the combustion of scramjets, by oblique detonations, reducing the geometry and complexity of the combustion system. Even so, several studies will still be needed to make the construction and use of this type of engine viable. 1.2 Objectives The objective of this dissertation is to describe the process of developing a numerical tool, which aims to compare the performance of an oblique detonation engine with a scramjet engine. For this, the results obtained in this work has been compared with the results of other works, namely with works that perform combustion at constant pressure and constant area, since this project will address, in the case of the scramjet, variable area. 2 The numerical tool should be able, considering the inputs given, and the geometry of the chosen inlet, to calculate the performance for a given range of Mach numbers. In the end, and after the results are obtained by the numerical tool, they will be compared with the results of other studies, to understand the advantages and disadvantages of its use. 3 2. BIBLIOGRAPHIC REVIEW In this chapter, the preliminary concepts will be addressed, which will serve as a basis for the concepts and formulas, and the decisions made, in the methodology chapter, and the conclusions that will be drawn from the results obtained in the numerical tool. 2.1 Hypersonic Airbreathing Propulsion With the aim of making hypersonic respiratory propulsion more accessible and efficient, this technology has been increasingly studied, although there has been no practical evolution in the last 40 years in terms of propulsive systems [1], at least, in terms of vehicles. Therefore, the air-breathing propulsion systems most used now are ramjet and scramjet, which differ in the way in which the air flow enters the combustor, causing combustion in each case to be subsonic and supersonic, respectively [2]. In addition to these configurations, others have been proposed, such as oblique wave detonation engines (ODWE), rotary detonation engines (RDE), and pulse detonation engines (PDE) [2]. In the case of ODWE, its main peculiarity is the fact that combustion occurs by detonation and not by deflagration. The main disadvantage of ODWEs, and of scramjets, is the fact that they have a high initial Mach number, which makes them dependent on other engines for take-off and acceleration, up to the desired initial Mach number, increasing the complexity of the propulsive system. For this reason, much study is still needed in this area until its application is viable. 2.2 Scramjet To define what a scramjet engine is, we first must understand the bases of a gas turbine and a ramjet engine. The main difference between the two is the air compression process before combustion, where in the gas turbine this is done through a compressor, while in the ramjet, this is done through shock waves, which are normally caused by the front body of the vehicle. When looking more specifically at the operation of the ramjet engine, the air flow is slowed down to subsonic speeds after compression, making combustion also subsonic. According to Heiser and Pratt et al. [3], the operation of the ramjet engine is efficient for Mach numbers in the range 3-6. As the Mach number exceeds this range, the use of the ramjet engine is no 4 longer efficient, as it is no longer advantageous to reduce the speed of the air flow to subsonic speed. Therefore, it was necessary to create an engine that did not require this deceleration, and could perform supersonic combustion, originating the scramjet engine. Although this configuration offers this advantage, when compared to the ramjet engine, and has been the subject of numerous studies and evolutions over the years, there are still problems that need to be overcome to make this engine more efficient and viable, in addition of the high initial Mach number. Since the operating environment has high temperatures and pressures, which jeopardizes the structural integrity of the engine and vehicle, the need to guarantee stable and efficient mixing and combustion in supersonic regime, in addition to a combustion chamber of appropriate dimensions, are some of the sizing problems of this type of engine [3]. In the Figure 2.1, it is possible to visualize a two-dimensional diagram of the structure of a scramjet engine. Figure 2.1: Diagram of a scramjet engine [4] 2.3 Oblique Detonation Wave Engine As seen previously, the engines most used in the construction of hypersonic airbreathing propulsion vehicles are the scramjet, which are characterized by using deflagration as a combustion process. According to Pratt et [5], the use of normal detonation waves was proposed by Roy in 1946 [6], and later studies proved its feasibility, although it was found difficult to control the detonation wave, as well as the complex geometry of the chamber, which this process would require. The use of oblique detonation waves instead of normal detonations was proposed by Dunlap et al [1], since as happens in a scramjet engine, it allows the flow that passes through the burner to be supersonic. This meant that it was not necessary to implement a variable geometry to stabilize the wave, since this stabilization would be carried out by a wedge [7]. If we look at the geometries of a 5 scramjet and an ODWE, we see that the difference between them is the existence of this wedge in the combustor. As will be seen later, the existence of a wedge is necessary for the stabilization of shock waves, unlike what happens in scramjet engines. The position of the wedge depends on the type of compression that will be chosen for the work, representing great influence mainly on the level of drag generated. The typical model of an oblique detonation engine can be observed in the Figure 2.2. Figure 2.2: Diagram of an Oblique Detonation Wave Engine [8] Citing Heiser and Pratt et al. [3], “The immediate advantages of the ODW process are that the drag, convection heating, length, weight, cost, and maintenance of the combustor are almost entirely eliminated”. In this way, different studies were carried out to increase the efficiency and viability of ODWEs. One of them was the study carried out by Valorani et al. [9], which sought to find a solution to the high initial Mach number, with the possibility of this being reduced. This would result in a reduction in operating weight, since less oxidant would be needed to be transported, since the lower the Mach number, the smaller the amount of oxidant used, for shorter time. But it would also allow the use of shorter burners, when compared to those used in scramjets. Another advantage is the simplicity of the compression system, especially when compared to Ramjet engines. This is because, the temperature variation throughout the compression process is smaller, due to the existence of higher temperatures and pressures in the burner due to detonation, since the existence of shock waves, causes an increase in pressure and temperature, without using more fuel. One of the most important studies was carried out by Ashford et al. [10], where the performance of an oblique wave detonation engine was compared with that of a diffusive scramjet engine. Some considerations were made, such as constant area and constant pressure, as well as the fact that it is an ideal and perfect gas. For Mach 10, at which the study was carried out, it was concluded that the use of ODWEs can reduce the length of the engine up to 50%. 6 To replace diffusive burning, Jiang et al [11], proposed a prototype of a standing oblique detonation ramjet engine. The engine was successfully developed, concluding that the model works steadily, and confirming that oblique detonation can be stationary and controllable in the burner. This model can be viewed in Figure 2.3 Figure 2.3: Schematic of the Sodramjet proposed by Jiang [11] 2.3.1 Combustion To understand the combustion process, it is important to define deflagration and detonation, to understand their differences. Starting with deflagration, a combustion process characteristic of scramjet engines, this can be defined as the passage of a flame front as a subsonic wave through a fuel mixture, according to Diéguez et al [12]. According to Wolanski et al. [13], the detonation process was first described by Berthelot, Vieille, Mallard and Le Chatelier in 1881. It was only twenty years later that Chapman and Jouguet presented the zero-dimensional theory of detonation. Detonation is a combustion wave that propagates through a gaseous mixture of fuel and oxidizer, causing a reaction. This can be classified according to the Mach number that occurs after detonation. If the Mach number is subsonic, we have a detonation called overdriven, if it is sonic, we are facing the Chapman-Jouguet detonation (referred to in the following chapter), and finally, if the Mach number is supersonic, the detonation is called underdriven. As with oblique shocks, oblique detonations can have a weak or strong shock solution, for the same wedge angle. To evaluate the angle of the oblique detonation wave as a function of the angle of deflection and heat addition, a diagram was created, based on the upstream conditions, and considering the different possible conditions downstream. This diagram can be seen in Figure 2.4. 7 Figure 2.4: Oblique detonation wave angle as function of deflection angle and heat addition [3] According to the diagram, we can have strong or weak overdriven oblique detonation waves. In the case of strong overdriven detonations, associated increases in static pressure are so high that they cause detachment and are therefore unnatural. In the case of weak overdriven detonation waves, these are restricted by the Chapman-Jouguet angle and the detachment angle, which is why this is the region of interest for ODW engines. The Chapman-Jouguet angle is connected to the Chapman-Jouguet locus, which is the point of minimum wave angle at each locus of states for a given 𝑞>0, as can be seen in Figure 2.4. Detonations are not considered for these engines due to the downstream Mach number being supersonic, and it needs to be subsonic or sonic, to ensure that the oblique detonation wave is attached and stable. Upon detonation, the front propagates at speeds in the order of km/s in the air-fuel mixture, which produces a significant increase in pressure. According to Wolanski et al [13], if the detonation speed is above 1.8 km/s, there is a pressure increase of more than ten times. If the detonation speed reaches 3km/s, a pressure increase of up to twenty times is obtained. During deflagration, the speed of the flame is in the order of dozens of m/s, so the combustion will have to be organized in a stoichiometric ratio, at a higher burning speed, which results in high combustion temperatures and production of high NOx concentration. Due to these phenomena, in common aviation engines, which have a turbine, as is the case with turbofans, turbojets, among others, it is necessary to mix more air before the turbine, due to the high temperature, increasing the complexity of the system. 8 In detonation engines this complexity does not exist, since the combustion temperature is lower, in addition to not using ignition devices [1]. According to Rosato et al. [14], detonation can effectively increase the efficiency of the thermodynamic cycle by up to 20%, when compared to typical cycles that are based on diffusive burning, since the greater the speed of burning or conversion of the "material", typically tens of thousands of times faster than engines that use deflagration, resulting in more compact and efficient systems. 2.3.2 Chapman-Jouguet Theory As we saw previously, Chapman and Jouguet established the first theory about detonations, the zero-dimensional theory of detonation. This described the detonation wave as a hydrodynamic discontinuity, where energy would be released, assuming that it was stable, planar, and one-dimensional. Another important topic for this work is the Rankine-Hugoniot conditions, which describe the relationship between the states on both sides of a shock wave or combustion wave (deflagration or detonation), for one-dimensional flows in fluids or for one-dimensional deformations in solids. In the equation (2.1), Rankine-Hugoniot conditions are described, where the prefixes 1 and 2 refer to the state before and after detonation, respectively. Therefore, ℎ1 and ℎ2 represent, respectively, the specific enthalpies before and after detonation, 𝑝1 and 𝑝2, the pressures before and after detonation and, finally, 𝜌1 and 𝜌2, the flow densities before and after detonation. ℎ2−ℎ1=12(𝑝2−𝑝1)(1 𝜌1+1 𝜌2) (2.1) Using these conditions, it is possible to plot the Rankine Hugoniot curve, where it will be possible to determine the conditions of state 2, considering the conditions of state 1 that are provided. One of these curves can be visualized in Figure 2.5, where regions l and ll correspond to supersonic waves, hence detonations, while regions lV and V correspond to subsonic waves, also called deflagrations. 9 Figure 2.5: Rankine-Hugoniot curve [15] In the case of detonation, considerations of gas dynamics are sufficient to predict the propagation speed of the wave, through the shock wave, while in deflagration it is necessary to know the structure of the wave, in addition to the transport processes, whether turbulent or diffusive. Chapman and Jouguet suggested that detonations travel at the lowest speed for all solutions in the detonation branch. Observing Figure 2.5, we can see that there are two branches, which correspond to the Chapman-Jouguet points, the upper point, 𝐶𝐽𝑢, and the lower point, 𝐶𝐽𝑙. The lower one is in the deflagration branch, while the upper one is in the detonation branch, which corresponds to the point used for ODWEs, since this type of engine uses detonations and not deflagrations, in addition to corresponding to the point of lowest entropy and total pressure lost. 2.4 Synthesis According to the points addressed during the bibliographic review, it is perceived that there is a lack of knowledge of oblique detonation engines, especially in the possibility of them replacing the most used hypersonic engines today, the scramjet. These reasons led to this work, with variable area configuration, but also the work carried out by Pereirinha et al. [16], with constant area configuration, to be carried out. In them, a comparison is made between oblique detonation engines and different configurations of 16 where 𝛾1 and 𝛾2 are the specific heat ratios of the gas at the corresponding cross-section, and R is the gas constant. Then, the area at the isolator exit/combustor inlet can be calculated using the equation (3.14). 𝐴2=𝐴1 √𝛾2𝑅𝑇2[1 𝜋2(1+𝛾𝑀12)−1] (3.14) Finally, the total pressure at the exit of the isolator can be calculated, as well as the recovery coefficient of the total pressure of the isolator. These can be calculated using the equations (3.15) and (3.16), respectively. 𝑃2=𝑃1 [ 𝜋2(1+𝛾2−1 2𝑀22)𝛾2 𝛾2−1 (1+𝛾1−1 2𝑀12)𝛾1 𝛾1−1 ] (3.15) 𝜎2=𝑃2 𝑃1 (3.16) 3.2.3 Number of Oblique Shocks After choosing the type of inlet to use, it is necessary to determine the number of oblique shocks that will be generated by the inlet, according to the desired conditions, namely the desired cycle temperature ratio. The number of shocks must guarantee that this ratio is efficient and the increase in entropy is kept to a minimum, in addition to providing an acceptable performance. In this project, the sizing of the compression system has been based on the cycle temperature ratio 𝑇3/𝑇0, representing the temperature ratio between the burner inlet and the outside, as shown by the Figure 3.1. It is also important to ensure that all oblique shock waves transmit an equal amount of geometric rotation of the flow, in addition to ensuring that the shock-on-lip condition occurs [3]. According to Smart et al [21], the greater the number of shocks, the more the optimal total pressure recovery will increase, although this decreases with the increase in Mach number. This effect can be seen in Figure 3.3, that describes how the optimal total pressure recovery varies with the Mach number, for a different number of shocks. 17 Figure 3.3: Maximum total pressure recovery for two-dimensional scramjet inlets with up to five shocks [21] In Figure 3.4, it is analysed how the compressive efficiency varies with the number of Mach, for a different number of oblique shock waves. Observing the Figure 3.3 and Figure 3.4, it is concluded that a system of 4 oblique shocks is the most efficient choice, since a higher number of shocks leads to complications in system sizing, while a lower number leads to a reduction in the optimum total pressure recovery. Figure 3.4: Adiabatic compression efficiency as a function of freestream Mach number, static temperature ratio, and number of oblique shock waves [3] To calculate the flow properties after each shock, the oblique shock equations has been applied. In this way, equations (3.17), (3.18), (3.19), (3.20) and (3.21), taken from Pereirinha et al. [16], were implemented in the project code. 18 𝑝𝑟𝑎𝑡𝑖𝑜=1+ 2𝛾0 𝛾0+1(𝑀2sin2𝛽−1) (3.17) 𝑝𝑟𝑎𝑡𝑖𝑜=(𝛾0+1)𝑀2sin2𝛽 2+(𝛾0−1)𝑀2sin2𝛽 (3.18) 𝑇𝑟𝑎𝑡𝑖𝑜=𝑝𝑟𝑎𝑡𝑖𝑜 𝜌𝑟𝑎𝑡𝑖𝑜 (3.19) tan𝜃=2cot𝛽(𝑀2sin2𝛽−1) 𝑀2(𝛾0+cos(2𝛽))+2 (3.20) 𝑀𝑟𝑎𝑡𝑖𝑜=1 𝑀(2 𝛾0−1+𝑀2sin2𝛽)1/2(2𝛾0𝑀2sin2𝛽 𝛾0−1 −1)1/2 (3.21) In order to determine compressive efficiency and the kinetic energy (ƞ𝑐 and ƞ𝐾𝐸, respectively), the equations (3.22), (3.23) and (3.24), taken from [3], are applied. 𝜋𝑐=𝑝𝑡3 𝑝𝑡0=𝑝3 𝑝0(1φ)𝛾𝑐/(𝛾𝑐−1) (3.22) ƞ𝑐=φ−(1 𝜋𝑐)(𝛾𝑐−1)/𝛾𝑐 φ−1 (3.23) ƞ𝐾𝐸=1− 2 (𝛾𝑐−1)𝑀02[(1 𝜋𝑐)(𝛾𝑐−1)/𝛾𝑐−1] (3.24) where φ= 𝑇3/𝑇0. 19 3.3 Combustion System The next thing to pay attention to is the combustion study, where it is considered that it starts with the air and fuel already homogeneously mixed. To understand how combustion takes place, it can be described using a chemical equation (3.38), called the complete stoichiometric equation, described in the equation (3.25). 𝐶𝑥𝐻𝑦+(𝑥+𝑦4)(𝑂2+79 21𝑁2)→𝑥𝐶𝑂2+𝑦2𝐻2𝑂+79 21(𝑥+𝑦4)𝑁2 (3.25) In this case it represents the complete combustion of air and hydrocarbon fuel. Next, it is necessary to calculate the stoichiometric fuel/air ratio. This can be obtained through the equation (3.26). 𝑓𝑠𝑡=36𝑥+3𝑦 103(4𝑥+𝑦) (3.26) For the case of non-stoichiometric mixtures, it is important to define another property, called equivalence ratio, which represents the ratio between the real fuel/air ratio and the stoichiometric fuel/air ratio. The representation of the equivalence ratio is made in the equation (3.27). 𝜙=𝑓 𝑓𝑠𝑡 (3.27) The equivalence ratio, according to Heiser and Pratt [3], must be in the range of 0.2 to 2, so that combustion occurs within a reasonable time scale, which is the range considered in this work. Another important point for the project is the selection of fuel. For this purpose, several fuels were compared, in different aspects, as can be seen in the Table 3.1, where ρ is for standard conditions and 𝑇𝑖𝑔𝑛 for standard conditions and 1 atm. The choice ended up being 𝐻2, to obtain a more reliable comparison between the results of the works. When observing the Table 3.1, it can be seen that 𝐻2 has the highest heat of combustion value, in addition to a higher auto-ignition temperature. The main drawback of 20 using 𝐻2 is the need for a lot of storage space due to its very low density, although several studies and advances have been carried out to counter this problem. Table 3.1: Properties of several fuels [22] Fuel ℎ𝑝𝑟[𝑀𝐽/𝑘𝑔] 𝜌[𝑘𝑔/𝑚3] 𝑇𝑖𝑔𝑛[𝐾] 𝑓𝑠𝑡 𝐻2 119.96 0.08 845.15 0.02913 𝐶𝐻4 50.01 0.65 810.15 0.05825 𝐶2𝐻6 47.49 1.22 745.15 0.06241 𝐶3𝐻3 46.3 1.79 743.15 0.06408 𝐶4𝐻10 45.74 2.36 693.15 0.06497 3.3.1 Post-combustion It is now necessary to calculate the properties of the flow after combustion. Since we are dealing with area sections, and since the radius of the section after combustion is different from the inlet entrance radius, this difference must be considered. For this project, a section radius of 101.8 mm will be considered. To calculate these properties, it is necessary to apply the equations (3.28), (3.29) and (3.30), taken from the work of Gu et al. [20]. The variables represented with point 2 are calculated at the exit of the isolator. 𝑀4=𝑀2 √𝜏(𝑟)(1+𝛾4(𝑟)−1 2𝑀22)−(𝛾4(𝑟)−1 2𝑀22) (3.28) 𝑃3=𝑃2 [1+𝛾4(𝑟)−1 2𝑀22(1− 1 𝜏(𝑟))]𝛾4(𝑟) 𝛾4(𝑟)−1 (3.29) 𝑇3(𝑟)=𝑇2𝜏(𝑟) (3.30) where, 21 𝜏(𝑟)=Ҩ(𝑟)ℎ𝑝𝑟 34.32𝑐𝑝2𝑇2 (3.31) To simplify the calculations, certain values taken from other works were assumed. In case of Ҩ(𝑅), a value of 0.2 was assumed, according to Gu et al.[20], and for 𝛾4(𝑅) a value of 1.286 was assumed, according to Yang et al. [23]. 3.3.2 Oblique Detonation Wave Engine Previously, in Chapter 2, it was said that the types of detonations that could be applied in hypersonic airbreathing propulsion were weak overdriven detonations and Chapman-Jouguet detonations, although the latter correspond to those of minimum total pressure loss and the point of minimum entropy increase. Considering the types of detonations that can be used in this project, the developed program must be able to determine the wedge angle that generates the Chapman-Jouguet detonation, while for the remaining Mach numbers, the weak overdriven detonation must be guaranteed. To ensure that pre-ignition does not occur, in the specific case of burning 𝐻2, below stoichiometric conditions, the temperature recorded before detonation must be below 1000 K, while to guarantee ignition, the temperature after detonation must be greater than 1000 K. If these conditions are not met, the program will warn the user of the error that has occurred. To determine the Chapman-Jouguet detonation angle for the Mach number considered, the equation (3.32) must be applied. (𝑀1𝑛 2−1)2−2(𝛾+1)𝑀1𝑛 2𝑄=0 (3.32) where, 𝑀1𝑛=𝑀1sin𝛽 (3.33) 𝑄=𝑄𝑛𝑏 𝑐𝑝𝑇 (3.34) 22 To find the wedge angle that generates the Chapman-Jouguet detonation under the desired conditions, the equation (3.35) must be solved. 𝜃𝐶𝐻.𝐽.=𝛽𝐶𝐻.𝐽.−tan−1 [ 1+𝛾𝑀1𝑛𝐶𝐻.𝐽. 2 (𝛾+1)𝑀1𝑛𝐶𝐻.𝐽. 2√(𝑀1/𝑀1𝑛𝐶𝐻.𝐽. 2)2−1 ] (3.35) For the remaining Mach numbers of the range under study, and in order to guarantee weak overdriven detonation, the detonation angle will be calculated using the equation (3.36), since the wedge angle has already been defined in the previous equation (3.35). 𝑄=−𝛾+1 2𝑋2𝑀12sin2𝛽+(1+𝛾𝑀12sin2𝛽)𝑋−(1+𝛾−1 2𝑀12sin2𝛽) (3.36) The equations (3.32) and (3.35) were taken from Murthy et al.[24], and X has been obtained through the equation (3.39). 3.3.3 Post-Detonation Properties After calculating the conditions before detonation, mainly the detonation angles for each Mach number in the desired range, it is necessary to determine the conditions after detonation. For this it is assumed that the air and fuel are homogeneously mixed [5], although this is not completely real. Since the influence of the fuel temperature is neglected, the air temperature after the compression system is equal to the temperature of the air/fuel mixture. To calculate the pressure after detonation, the equation (3.37) is used, 𝑝2=𝑝1+𝜌1𝑢1𝑛 2(1−𝑋) (3.37) and the detonation temperature ratio can be calculated using equation (3.38). 𝑇2 𝑇1=1+ 𝑢1𝑛 2 2𝐶𝑝𝑇1(1−𝑋)2+𝑄ƞ𝑏 𝐶𝑝𝑇1 (3.38) while X is calculated by 23 𝑋≡𝜌1 𝜌2=𝑢2𝑛 𝑢1𝑛=tan(𝛽−𝜃) tan𝛽 (3.39) where, 𝑢1𝑛=𝑢1sin𝛽 (3.40) 𝑢1𝑡=𝑢1cos𝛽 (3.41) 𝑢2𝑛=𝑢2sin(𝛽−𝜃) (3.42) 𝑢2𝑡=𝑢2cos(𝛽−𝜃) (3.43) 3.4 Expansion System The last system to be assessed is the expansion system, whose main objective is to accelerate the flow to produce the maximum amount of thrust. It can operate in different ways, whether over-expanded, under-expanded or ideally, depending on the ratio between the engine inlet and outlet pressures. If this ratio is 1, we are facing an ideal configuration; while if it is greater than 1, we will have an over-expanded nozzle configuration, allowing a lower weight in the nozzle configuration; Finally, if the ratio is less than 1, we are dealing with an under-expanded nozzle, which causes a decrease in the thrust produced and an increase in the nozzle geometry and consequently in its weight. In this work an ideal nozzle will be considered, where the pressure at inlet of the engine and on the outlet is the same, making the ratio between them 1. The expansion process will be adiabatic and isentropic, in addition to being one-dimensional and calorically perfect flow. To calculate the properties of the expansion system, the equations (3.44), (3.45) and (3.46) will have to be applied. 𝑇5=𝑇4[1−ƞ𝑒(1−(𝑝5 𝑝0𝑝0 𝑝4))𝑅 𝑐𝑝𝑒] (3.44) 24 𝑉5=√𝑉42+2𝑐𝑝𝑒(𝑇4−𝑇5) (3.45) 𝐴5 𝐴0=(1+𝑓)∗𝑝0 𝑝5𝑇5 𝑇0𝑉0 𝑉5 (3.46) 3.4.1 Performance Finally, the last step is the definition of the equations that will evaluate the performance of the designed engine, which were taken from the work of Pratt et al. [3]. As previously mentioned, assumptions were made that are far from what happens in reality, at least it is necessary to understand and know how to interpret the results obtained in accordance with the assumptions made previously. The first performance factor to be evaluated is specific thrust, which is defined by the equation (3.47). Uninstalled thrust Inlet air mass flow rate=F m󰇗0 (3.47) When using the stream thrust function, there is another way to determine the specific thrust, from velocities and temperatures obtain at the entrance and exit of the engine, defined in the equation (3.48). 𝐹 𝑚󰇗0=(1+𝑓)𝑆𝑎5−𝑆𝑎0−𝑅0𝑇0 𝑉0(𝐴5 𝐴0−1) (3.48) where, 𝑆𝑎5=𝑉5(1+𝑅𝑇5 𝑉52) (3.49) 𝑆𝑎0=𝑉0(1+𝑅𝑇0 𝑉02) (3.50) 25 𝑆𝑎 represents the stream thrust function at inlet and exit of the engine (Point 0 and 5, respectively), depending on the flow velocities and temperatures recorded in these sections. Based on the equations described, it is possible to obtain the flow rate of the air mass, which can be obtained through the derivative of equation (3.48), which indicates not only the uninstalled thrust required for the operation, but also the specific thrust of the developed engine. The next factor to consider is the specific fuel consumption, which can be defined by the equation (3.51). Fuel mass flow rate Uninstalled thrust=S=m󰇗f F (3.51) The specific fuel consumption can also be defined through the specific thrust, using the equation (3.52). 𝑆= 𝑓𝐹 𝑚󰇗0 (3.52) Next, is one of the most important performance parameters to evaluate the performance of an engine, the specific impulse, representing the efficiency with which the engine produces thrust. This is defined by the equation (3.53). Uninstalled thrust Fuel weight flow rate=Isp=F g0m󰇗f (3.53) Next, it will be the propulsive efficiency, which represents the ratio of thrust power to the mechanical power of the engine, being defined by the equation (3.54), which can be applied due to the fact that an ideal nozzle has been selected, where the inlet and outlet pressure of the engine are the same. 32 4.2.1 Compression System Once again, the study begins with the compression system. The first study carried out seeks to demonstrate how the 𝑇𝑟𝑎𝑡𝑖𝑜, influences the efficiency of the compression system and kinetic energy. This influence can be seen in the Figure 4.3 and Figure 4.4, respectively. Figure 4.3 : Inlet compression system efficiency as function of 𝑀0 [16] Starting with Figure 4.3, we noticed that as the 𝑇𝑟𝑎𝑡𝑖𝑜 increases, there is a decrease in the efficiency of the compression system. On Figure 4.4, we noticed that the behaviour of the efficiency of kinetic energy is very similar to the behaviour of the efficiency of the compression system, since when 𝑇𝑟𝑎𝑡𝑖𝑜 increases, there is a decrease in the efficiency of kinetic energy. Therefore, when choosing the 𝑇𝑟𝑎𝑡𝑖𝑜, the negative impact that it has on both efficiencies should be considered, although a low value of 𝑇𝑟𝑎𝑡𝑖𝑜, negatively influences the overall performance of the engine, so it is necessary to choose the value of 𝑇𝑟𝑎𝑡𝑖𝑜. Figure 4.4: Kinetic energy efficiency as function of 𝑀0 [16] 33 Observing the Figure 4.5, it can be seen how the Mach number at the burner input varies with the freestream Mach number, for different values of 𝑇𝑟𝑎𝑡𝑖𝑜. As the 𝑇𝑟𝑎𝑡𝑖𝑜 increases the M3 decreases. This chart allows us to choose the value of 𝑇𝑟𝑎𝑡𝑖𝑜, which allows a supersonic flow at the burner inlet, although low Mach numbers at the burner inlet, have some advantages, namely in terms of combustion. In this case, low Mach numbers mean high values of 𝑇𝑟𝑎𝑡𝑖𝑜, this implies high temperatures inside the engine, implying specific materials in this region, increasing the cost of the developed engine. Figure 4.5: 𝑀3 as a function of 𝑀0 [16] 4.2.2 Oblique Detonation Wave The next study to be carried out was that of oblique detonation waves. This study seeks to understand how 𝑄 influences the wedge and detonation angles of a Chapman-Jouguet detonation. Both influences can be seen in Figure 4.6 and Figure 4.7, respectively. Figure 4.6: Wedge angle as a function of freestream Mach number [16] 34 Figure 4.7: Detonation angle as a function of freestream Mach number [16] As the value of 𝑄 increases, the wedge angle required to generate a Chapman-Jouguet detonation will also increase, also leading to an increase in the detonation angle. Next, it was studied how the detonation pressure and temperature ratio are influenced by the heat flux. This influence can be visualized in the Figure 4.8 and Figure 4.9. Figure 4.8: Pressure ratio as function of ODW heat flux [16] 35 Figure 4.9: Temperature ratio as function of ODW heat flux [16] It can then be concluded that as the heat flux of ODW increases, there is also a linear increase in the 𝑝𝑟𝑎𝑡𝑖𝑜 and 𝑇𝑟𝑎𝑡𝑖𝑜. 4.2.3 Scramjet combustion In this section, the influence of φ for a constant pressure burner will be studied. To carry out this study, the properties visible in Table 4.7. Table 4.7: Inputs of the study of influence of φ [16] Property Assigned Value 𝑂𝑛−𝑑𝑒𝑠𝑖𝑔𝑛 𝑀0 8 𝑂𝑛−𝑑𝑒𝑠𝑖𝑔𝑛 𝑇𝑟𝑎𝑡𝑖𝑜 2 𝑉𝑓𝑥 𝑉3 0.5 𝑉𝑓 𝑉3 0.5 𝐶𝑓𝐴𝑤 𝐴3 0 𝐶𝑃𝐵 1510 J/kgK 𝑇° 222 K ℎ𝑝𝑟 119.96 MJ/kg ƞ𝑏 1 36 The results obtained for the burner at constant pressure can be seen in the Figure 4.10, Figure 4.11 and Figure 4.12. As expected, as the equivalence ratio increases, there is an increase in the temperature at the output of the burner. Figure 4.10: Temperature at the exit of the burner as a function of 𝑀0 [16] A different behaviour is observed at the burner output speed, where low equivalence ratio values lead to higher burner exit speed values. At the level of the area along the burner, it is necessary that it increases, so that the pressure is maintained. Therefore, an increase in the equivalence ratio leads to higher variations in the area along the burner, making it easier to maintain pressure. For the engine in general, by increasing the cross-section area, you are increasing the drag generated by the engine, which is a negative aspect. Figure 4.11: Velocity at the exit of the burner as a function of 𝑀0 [16] 37 Figure 4.12: Area ratio as a function of 𝑀0 [16] 4.3 Case Study After the explanation, of the parametric studies obtained by Pereirinha et al. [16], and after carrying them out, to prove the results obtained by Pereirinha et al. [16], the case study intended in this work, can be developed. Based on the parametric studies, the inputs for the case study have been carefully selected, so that the case study is as accurate and real as possible. 4.3.1 Input Some considerations were made during the choice of inputs, namely the fact that the freestream number of Mach is 8, although in this work an isolator is implemented. The range of the Mach number implemented in the study should be small since some properties are considered constant along this range. Other points to be considered are the choice of the 𝑇𝑟𝑎𝑡𝑖𝑜, as seen in the chapter on parametric studies, in addition to the temperature before detonation being at least 1000 K, to prevent pre-ignition. If these conditions are not registered, the code will warn the user. In the case of the values, especially the specific heats, these must be different in the scramjet and in the ODWE, since the temperature and pressure recorded in the burner of the ODWE's is much higher when compared to those of the scramjet. The values used as input for the case study can be visualized in Table 4.8. 38 Table 4.8: Case study inputs Property Assigned Value h [m] 0 𝑀0𝐷 10 𝑀𝑢𝑏 15 𝑇𝑟𝑎𝑡𝑖𝑜 2 𝛾𝑐 1.362 Scramjet𝛾𝑏 1.238 𝑂𝐷𝑊𝐸 𝛾𝑏 1.170 Scramjet𝛾𝐶 1.238 𝑂𝐷𝑊𝐸 𝛾𝐶 1.170 𝐶𝑝0 1005 𝐶𝑝𝑐 1090 𝑆𝑐𝑟𝑎𝑚𝑗𝑒𝑡 𝐶𝑝𝑏 1510 𝑂𝐷𝑊𝐸 𝐶𝑝𝑏 2000 𝑆𝑐𝑟𝑎𝑚𝑗𝑒𝑡 𝐶𝑝𝑒 1510 𝑂𝐷𝑊𝐸 𝐶𝑝𝑒 2000 𝑉𝑓𝑥 𝑉3 0.5 𝑉𝑓 𝑉3 0.5 Scramjet ƞ𝑏 0.9 ODWE ƞ𝑏 0.9 Scramjet ƞ𝑒 0.9 ODWE ƞ𝑒 0.9 x of CxHy 0 y of CxHy 2 ℎ𝑝𝑟 [𝑀𝐽/𝑘𝑔] 119.96 4.3.2 Performance In this section, the results obtained will be analysed, after the implementation of the case study in the numerical tool developed. Primarily, the analysis of the specific thrust will be made as a function of the Mach number at the inlet, as shown in Figure 4.13. As expected, for all three cases, the maximum specific thrust is recorded for a Mach number of 10, and as the Mach number increases, there is a decrease 39 in specific thrust, due to the more expressive increase in the velocity of the flow outside in relation to the escape velocity. For scramjet engines, we found that the variable area configuration has a specific thrust higher than the constant pressure configuration. Even so, this value is much lower when compared to the specific thrust for the case of ODWE's. Another important point is the fact that the slope of the curve of the variable area configuration is quite pronounced, being in ODWE, lighter, and the constant pressure almost non-existent. Figure 4.13: Specific Thrust as function of freestream Mach number According to these results, it is expected that the engine with the highest specific thrust will have the lowest specific fuel consumption. This is proven by the Figure 4.14. It is also noted that the lowest specific fuel consumption is recorded for lower Mach numbers, and as the Mach number increases, so does the specific fuel consumption. Once again, the ODWE is the engine that has the lowest specific fuel consumption, being inferior to the variable area engine, and even more so to the constant pressure engine. 40 Figure 4.14: Specific fuel consumption as function of freestream Mach number Next, the specific impulse will be analysed, which can be observed in Figure 4.15. The motor that has the highest specific thrust is the ODWE, followed by the variable area and finally the constant pressure motor. It is also noted that the highest specific impulse values recorded are for lower Mach numbers. Figure 4.15: Specific impulse as function of freestream Mach number Observing the Figure 4.16, we realized that the next factor to be studied was propulsive efficiency. We realized that in this case the ODWE has the lowest propulsive efficiency, since it has the highest exhaust velocity, which can be proven by the equation (3.55). On the contrary, the constant pressure motor has the highest propulsive efficiency, since it has the lowest exhaust speed. An important point is the fact that the constant-pressure motor is the only motor that converts power, mechanical energy into thrust power more efficiently. 41 Figure 4.16: Propulsive efficiency as function of freestream Mach number In terms of thermal efficiency and observing the Figure 4.17, we realize that ODWE has the highest thermal efficiency compared to scramjet engines, since ordinary thermal machines do not have shock waves, other than burning. The shock waves allow an increase in pressure and temperature, without using more fuel, improving thermal efficiency. This value increases as the number of Mach in the inlet also increases. In the case of scramjet engines, the variable area engine has better thermal efficiency when compared to the constant pressure engine, although it decreases as the Mach number increases. Figure 4.17: Thermal efficiency as function of freestream Mach number The next study to be carried out is the overall efficiency study, as shown in Figure 4.18. This parameter demonstrates how efficiently the engine uses the energy that was initially stored 48 ANNEXES Annex 1 – Code input 49 Annex 2 – Code 50 51 52 53 54 55 56 57 64 65