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Study of thermal throat of RBCC combustor based on one-dimensional analysis
Acta Astronautica 117 (2015) 130–141
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Study of thermal throat of RBCC combustor based on one-dimensional analysis Ya-jun Wang, Jiang Li, Fei Qin n, Guo-qiang He, Lei Shi Science and Technology on Combustion, Internal Flow and Thermal-Structure Laboratory, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China
a r t i c l e in f o
abstract
Article history: Received 8 March 2015 Received in revised form 14 July 2015 Accepted 31 July 2015 Available online 10 August 2015
An analysis model was developed to better understand the formation mechanism and variation law of the thermal throat in a rocket-based combined-cycle (RBCC) combustor. This analysis model is based on one-dimensional flow equations and consideration of the variation in factors such as the area, exothermic distribution, and the fuel-rich jet of the rocket. The influence law for the thermal throat under the interaction of the exothermic distribution and the variation of the area is consistent with the heat release models for a gaseous jet and liquid kerosene. The effective cross-sectional area of the jet was calculated and incorporated into the model. The results calculated using the one-dimensional model were found to be consistent with those obtained from a three-dimensional numerical simulation. The position of the thermal throat was predicted with an error of 0.36%. The maximum relative errors of the static pressure among the corresponding points were 7.4% and 9.3% for the static temperature and total pressure, respectively. The one-dimensional model and three-dimensional numerical simulation were validated using experimental data obtained in direct-connect testing. Except for the cavity region, the maximum relative error of the corresponding points between the simulation results and test results was less than 8.9%, and that between the model results and test results was 10.4%. Compared to the fuel equivalence ratio, the expansion ratio, injection location, and exothermic rate have a significant impact on the position of the thermal throat. An optimization study of the RBCC combustor for the ramjet mode was conducted by adjusting the thermal throat. The thrust performance improved by 31.6% at Ma3 after optimization. These results indicate the important role that the one-dimensional model can play in analyzing the thermal throat and guiding the preliminary design of an RBCC combustor. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Rocket-based combined-cycle One-dimensional model Thermal throat Performance optimization
1. Introduction In the development of hypersonic flight systems, several types of air-breathing engines for different application layers, such as scramjets [1,2], dual-mode scramjets [3,4], and combined-cycle engines [5,6], have long been studied.
n
Corresponding author. Tel./fax: þ 86 29 88494163. E-mail address: [email protected] (F. Qin).
http://dx.doi.org/10.1016/j.actaastro.2015.07.034 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.
A rocket-based combined-cycle (RBCC) combustor [7,8] is yet another type of advanced propulsion system. This type of system has the widest flight envelope and application scope of those mentioned, and it can satisfy the various requirements of future aircraft, such as high altitude, high speed, and flight maneuverability. RBCC combustors are considered to be among the most promising propulsion systems for reusable space transportation and hypersonic cruise vehicles [9–11]. A large amount of reference data is needed to develop a preliminary engine design, predict the
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performance of the engine, and plan various trajectories in accordance with various mission requirements. In the project demonstration stage, the use of one-dimensional analysis can greatly reduce time and cost expenditures. An RBCC engine typically has an ejector mode, a ramjet mode, a scramjet mode, and a rocket mode, all of which must operate in the same expanding duct. In the ejector mode, which is approximately from Ma 0 to 2.5, the incoming air is introduced into the duct by the highspeed jet of the rocket, which operates at a high flow rate with high combustor pressure. Extra thrust can be gained through afterburning of the fuel-rich jet or the mixing and combusting downstream fuel with the introduced air. In the ramjet mode, which is approximately from Ma 2.5 to 5.5, and scramjet mode, which is approximately from Ma 5.5 to 8, the ignition and stable combustion of fuel can be achieved with the rocket operating at a relatively low flow rate and under fuel-rich conditions. High-efficiency combustion is achieved by adjusting the injection position of the fuel depending on the incoming flow conditions. As the flow Mach increases, the inlet is shut, and the rocket operates at a high flow rate again to provide the engine the required power for speeding or climbing into orbit. To obtain satisfactory engine performance during the ejector mode at high Mach numbers as well as during the entire ramjet mode, a thermal throat formed by heat release is needed to choke, expand, and accelerate the inflow. Therefore, it is essential to investigate the formation mechanism and variation law of the thermal throat to be able to adjust and control it in an expanding duct. Traditional (quasi) one-dimensional analysis methods are mainly of two types. In the first type of method [12,13], inputs that can reflect the mass injection, chemical reaction, variation in area, and wall friction are added to the one-dimensional Euler equation, differential equations are solved iteratively from the entry to the exit in the flow direction, and numerical simulation and experimental results for various engines are obtained to improve and correct the inputs [14]. Methods of this type require one to calculate a complex chemical reaction; consider the method and interval of mass injection, combustion efficiency, and other factors; and modify the inputs so that the calculation results approximate the actual flow field of the engine. In the second type of method [15,16], a onedimensional flow equation is used, a distribution function of parameters such as the variation in area and exothermic regularity is obtained, the Mach number distribution is solved, and other related flow parameters are calculated based on the Mach number distribution. This type of method requires the distribution function of the parameters, which cannot be determined easily. However, the general functional form of the parameter distribution can be obtained based on expert knowledge, and the onedimensional analysis can be greatly simplified. In analyses focusing on only one certain parameter, such as the thermal throat in the present study, the one-dimensional analysis method is very simple and effective. Previous studies have mainly focused on supersonic combustion in a scramjet or dual-mode scramjet. However, in an RBCC engine, considering the wide operating range of subsonic combustion, it remains necessary to investigate
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the variation and adjustment law of the thermal throat. The present study was based on the one-dimensional analysis approach described above. Findings from previous studies were used to identify the general laws of exothermic distribution that describe the characteristics of an RBCC combustor. Accordingly, the generation condition and influencing factors of the thermal throat were obtained by solving the Mach number distribution based on the exothermic distribution and variation in area, and the influence of the thermal throat position on the performance of the RBCC combustor in ramjet mode was analyzed. 2. Establishment of one-dimensional model 2.1. One-dimensional flow theory Many factors influence the combustion and flow in an RBCC combustor, in which complicated physical and chemical processes occur constantly. To simplify the analysis of the thermal throat, one-dimensional theory was used in this study to describe the flow process. The differential equation for the axial variation in the Mach number is given as Eq. (1) [15]. For brevity and clarity, the effects of wall friction, drag of internal struts, cavity or fuel injection, heat convection, and mass addition due to fuel injection are not considered herein; these effects are of secondary importance compared to the strong interaction between the axial variation of the cross-sectional area A(x) and that of the total temperature Tt(x). Moreover, the present study was focused mainly on the ramjet mode, during which the low flow velocity in the combustor makes the influence of friction and other drag forces negligible. 9 !8 1 þ γ M 2 1 dT t = 1 þ γ 2 1M 2 < dM 1 dA ¼M þ : dx A dx T t dx ; 2 1 M2 ð1Þ For the expanding duct of an RBCC combustor, the axial variation of the cross-sectional area can be expressed as follows: AðxÞ ¼ 1 þax Ain
ð2Þ
where a ¼Aout/Ain 1 ; a is a function of the expansion ratio, Aout and Ain are the area of the combustor's exit and entry, respectively; and x is a dimensionless coordinate of length. For a scramjet combustor, the empirical expression of exothermic regularity obtained from the literature [15] can usually be represented in nondimensional form as follows: T t ðxÞ bcx ¼ 1þ T tin 1 þ ðc 1Þx
ð3Þ
where b¼Ttout/Ttin 1; b is a function of the total temperature rise ratio Ttb ¼Ttout/Ttin ; Ttout and Ttin are the total temperature of the combustor's exit and entry, respectively; and c is an empirical constant on the order of 1 to 10 that depends on the mode of fuel injection and fuel–air mixing. As Fig. 1 shows, the value of b determines the exothermic quantity and c, the exothermic rate; the
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noted that the range of one-dimensional analysis considered in this study is from the exit of the strut rocket nozzle (combustor entry) to the combustor exit, and the length coordinates x range from 0 to 1. 2.3. Exothermic distribution model
Fig. 1. Exothermic regularity expressed by Eq. (3) with different parameters b and c. (a) c ¼5 and (b) b ¼1.5.
greater the value of c is, the shorter the heat release distance is. 2.2. RBCC combustor configuration Fig. 2 shows the studied center strut of an RBCC combustor. The strut contains two expanding ducts, II and III. The rocket engine is embedded in the center strut in isolator I. Two pylons for fuel injection and a cavity for the flame holder are located in expanding duct II. When the RBCC engine operates in ramjet mode, the strut rocket operates at a relatively low flow rate and in fuel-rich conditions, and its high-temperature jet ensures the reliable ignition and stable combustion of fuel. Thus, compared to a traditional ramjet or scramjet engine, the studied RBCC combustor contains the combustion zone of the fuel-rich jet of the rocket, except for the zone of fuel combustion. The cut-off point between these two zones is the fuel injection location (pz). Because the fuel-rich jet mainly contains the gaseous component CO and the fuel is liquid kerosene, there are obviously many differences in the exothermic regularity between these two fuels. Therefore, it is necessary to study the exothermic distribution between the partitions in the RBCC combustor. It should be
With data obtained from a three-dimensional numerical simulation of a typical point Ma3, 12 km in the ramjet mode, Eq. (3) was used to fit the data for the total temperature distribution in the combustor. The numerical simulation was carried out using the software FLUENT. Three-dimensional unsteady Reynolds-averaged Navier– Stokes (N–S) equations were chosen as the governing equations, and the shear stress transport (SST) k–ω model for viscous flow was used. This model has a short calculation period and high calculation accuracy for the turbulence in free shear flow and for flows in which separation occurs. A simplified three-step kinetic model, shown in Table 1 for JP-10 kerosene, for which C10H16 was used as a surrogate, was used to calculate the chemical reaction rate. The Lagrange two-phase flow method was used for liquid fuel injection, and the Taylor analogy breakup (TAB) model was used for droplet breakup. The data fit the model well in the combustion zone of the fuel-rich jet but badly in the fuel combustion zone, as shown in Fig. 3. The main reason for this is that the exothermic regularity expressed by Eq. (3) has a faster rate, allowing it to complete the main heat release within a short distance, as shown in Fig. 1, which is closer to the exothermic regularity of a gaseous fuel. This is consistent with the use of gaseous hydrogen fuel in early studies on ramjet and scramjet engines [17,18]. In comparison, because of the longer ignition delay time, mixing, and cracking interval, the exothermic rate and reaction interval of liquid kerosene are much greater. Apparently, it is not appropriate to express the exothermic regularity of kerosene using Eq. (3). Thus, the total temperature distribution of fuel combustion was fitted again. The results show that the exothermic distribution of kerosene is closer to that described by Eq. (4), shown below. T t ðxÞ ¼ dxe þ f T tin
ð4Þ
Table 2 shows the precision of the exothermic distribution model fitted in different zones by Eqs. (3) and (4). Using Eqs. (3) and (4) to express the exothermic regularity of rocket jet and kerosene fuel, respectively, conforms to the characteristics of the flow field in an RBCC combustor. To verify the exothermic regularity of kerosene, the total temperature data obtained from the numerical simulation for other typical points (Ma4, 17 km and Ma5, 21 km) in ramjet mode were fitted using Eq. (4). The results shown in Fig. 4 and Table 2 indicate that Eq. (4) can be used to characterize the exothermic regularity of kerosene in an RBCC combustor. According to the mathematical characterization of Eq. (4), the total amount of heat release is determined by the parameters d and f, and the exothermic rate is determined by the parameters d and e. In a preliminary analysis, for simplicity, parameter d was assumed to be unchanging,
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Fig. 2. Configuration of RBCC combustor with center strut.
Table 1 Three-step kinetic model for JP-10 kerosene. Reaction
A (cm3/(mol s))
B
Ea (J/(kg mol))
C10H16 þ5O2 ¼10COþ 8H2 2COþO2 ¼2CO2 2H2 þ O2 ¼ 2H2O
23500 3.48E8 3.0E17
1 2 1
1.633E8 8.42E7 0
Table 2 Fitted precision of different exothermic distribution models in fuel and jet combustion zones according to Eqs. (3) and (4). SSE¼ Sum of Squared Errors, RMSE¼ Root Mean Squared Error. Precision
SSE RMSE
Fuel combustion zone
Jet combustion zone
Eq. (3)
Eq. (3)
Eq. (4)
0.0025% 0.25%
0.013% 0.66%
10.65% 8.42%
Eq. (4) Ma3
Ma4
Ma5
1.25% 2.99%
0.95% 2.6%
0.12% 0.93%
and parameters e and f were changed simultaneously. A similar characterization for expressing different exothermic quantities, as shown in Fig. 1(a), was obtained, as shown in Fig. 5(a). The total amount of heat release was assumed to be unchanging, and parameters d, e, and f were changed simultaneously. A similar characterization for expressing different exothermic rates, as determined mainly by the parameter e, as shown in Fig. 1(b), was obtained, as shown in Fig. 5(b). 2.4. Generation condition of thermal throat Previously, one-dimensional analysis of the Mach number distribution could be conducted only when Tt(x) was clear in the RBCC combustor. This is mathematically represented in Eq. (1) by the requirement that as M goes to 1 at the thermal throat, the singularity due to the (1 M2) term in the denominator must be offset by the polynomial in brackets going to 0 at exactly the same location. Therefore, the position of the thermal throat can be obtained from Eq. (5), as shown below, where the asterisk * denotes the thermal throat, at which M¼1. 1 þ γ M 2 1 dT t 1 dA þ ¼0 ð5Þ A dx n T t dx n 2
Fig. 3. Curves fitted with the data of total temperature distribution at Ma3, 12 km. (a) Fuel combustion zone and (b) jet combustion zone.
Apparently, the thermal throat in an RBCC combustor is located between x¼ pz and 1. The value of xn can be determined from Eq. (5) for a given pair of functions A(x) and Tt(x) when the equation has a solution between x¼pz and 1; when there is no such solution, it means that there is no thermal throat.
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Fig. 4. Curves fitted with the data of total temperature distribution in fuel combustion zone. (a) Ma4, 17 km and (b) Ma5, 21 km.
The expressions A(x) and Tt(x) can be substituted into Eq. (5) to obtain the following expression: γ þ1 γ þ1 e 1 e 1 xe þ dex f ðxÞ ¼ ad af 0 2 2 pz r x r 1 ð6Þ As is known from the characteristics of the fitted curve for Tt(x), do0, eo0, and f40; therefore, f(pz)4 0 and f(1) r0 are necessary requirements to ensure that Eq. (6) has a root between x ¼pz and 1. This is the first criterion for thermal throat formation. From Eq. (1), ðdM=dxÞn ¼ 0=0 is algebraically indeterminate at xn . Thus, L’Hôpital’s rule is used to evaluate Eq. (1) at this critical point [19]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dM 1 ¼ Ω 7 Ω2 4Ψ ð7Þ dx n 4 where
"
2
d ðlnAÞ Ψ ≡ ð1 þ γ Þ dx2
Ω
γ ðγ þ 1Þ 1 dT t 2
T t dx
# ⁎
" # 2 ð1 þγ Þ2 d ðlnT t Þ þ 2 dx2
ð8Þ ⁎
ð9Þ n
Fig. 5. Exothermic regularity expressed by Eq. (4) with different parameters d, e, and f. (a) Different amount of heat release and (b) different exothermic regularity.
For the ramjet mode, to make the flow in the duct transition from subsonic to supersonic, dM=dx must always be positive. Therefore, ðdM=dxÞn 4 0 holds, and thus, Ψ o0 at xn . This is the second criterion for thermal throat formation. These two criteria need to be satisfied simultaneously. 2.5. Effect of rocket jet Before using Eq. (1) to solve for the Mach number distribution, it should be noted that the rocket jet is located between x ¼0 and pz. The jet exiting the nozzle expands from the center to both sides of the strut. However, its ability to expand is limited by the back pressure of the combustor and the influence of the shear layer on both sides, as a result of which its action range is limited in the center of the flow path, as shown in Fig. 6. The expansion of a fuel-rich jet during combustion is a problem that involves many complex factors. To simplify its role in one-dimensional analysis, the expansion boundary of the jet was calculated to make the one-dimensional model reflect the effect of the jet without being too complex. Assuming that the effect of the jet exists only at the expansion boundary, the existence of the jet can be
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Fig. 6. Static temperature contours of T Z 800 K in the internal flow field of RBCC combustor.
converted into a change in the cross-sectional area, namely, in the interval x ¼0 to pz, using an area distribution of AðxÞ=Ain ¼ 1 þ a1 x. For simplicity, the cross-sectional area of the duct reduces the maximum expansion boundary, and the remaining area is the actual action range of heat release from jet combustion. We then obtain the influence factor a1 to express the effective cross-sectional area of the rocket jet. Concepts from classical theory and engineering algorithms for gas jet dynamics are used to calculate the location of the expansion boundary [20,21]. The main equations [21] used in the calculation are shown below. In these equations, r and x are coordinates in the curve equation of the expansion boundary.
1 γ 1 2 γ 1 2 1 M 2 1 r¼ ln 1 þ M = 1þ Me þ ln e2 γ 1 2 2 2 M 1 ð10Þ x ¼ FðMÞ FðM 1 Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 2e 1
ð11Þ
sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 γ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 γ þ1 γ 1 2 2 M 1 U arctg M 1 F ðM Þ ¼ γ 1 γ 1 γ 1 γ þ1 ð12Þ M1 is the value of M when r ¼1, which can be determined using the following equation: (sffiffiffiffiffiffiffiffiffiffiffi ) i γ þ1 2 γ 1hπ 2 tg þ θ ðM e Þ ϑ M1 ¼ 1 þ ð13Þ γ 1 γ þ1 2 where ϑ is the expansion half-angle of the nozzle and sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ þ1 γ 1 2 arctg M e 1 arctg M2e 1 ð14Þ θðMe Þ ¼ γ 1 γ þ1 A Cþ þ program was developed to solve the above equations iteratively for the given boundary conditions of the nozzle exit: the Mach number, pressure, back pressure of the combustor, and expansion half-angle of the nozzle. Details of the calculation are omitted here for brevity. The calculated boundary curve is shown in Fig. 7. This curve is
consistent with the expansion boundary results obtained from the numerical simulation, indicating that the calculation method and expansion boundary results can be used in the analysis of the RBCC combustor. 2.6. Parameters of model All of the parameters used in the calculation of the onedimensional model have been obtained, as shown in Table 3. In Table 3, pz¼0.3, a¼ 0.6, a1 ¼1.1, d ¼ 0.848, e¼ 0.708, f¼3.37, b¼0.225, and c ¼7.61. By recording the intermediate values of M(x) as Eq. (1) is solved iteratively, the axial distribution of all other flow variables and thermo-physical properties, such as the static temperature, static pressure, and total temperature in the combustor, can be determined from the point-bypoint solution set M(x), together with the corresponding values of the prescribed functions A(x) and Tt(x), using several groups of integral relations [19], shown below. A complete one-dimensional analysis model of an RBCC combustor based on the Mach number distribution is therefore established. 2 3 2 γ1 TðxÞ T t ðxÞ4 1 þ 2 M in 5 ð15Þ ¼ T in T tin 1 þ γ 1 M 2 ðxÞ 2
pðxÞ Ain M in U ¼ pin AðxÞ MðxÞ
sffiffiffiffiffiffiffiffiffi TðxÞ T in
P t ðxÞ P ðxÞ T in T t ðxÞ γ =ðγ 1Þ ¼ P tin P in T ðxÞ T tin
ð16Þ
ð17Þ
3. Solution and verification of one-dimensional model 3.1. Solution of model Under the premise of satisfying the formation conditions, the position xn of the thermal throat can be obtained
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Fig. 7. Comparison of expansion boundary with Mach contours in jet combustion zone.
Table 3 The parameter settings for the one-dimensional model.
Interval A(x) Tt(x)
Fuel combustion zone
Jet combustion zone
pz to 1 1 þ ax dxe þ f
0 to pz 1 þ a1 x 1 þ bcx=ð1 þ ðc 1ÞxÞ
by solving Eq. (5). The fourth-order Runge–Kutta numerical method [22], which has high calculation precision, is used to solve Eq. (1) by iterating step by step from the thermal throat position to the upstream point (0, xn ) and the downstream point (xn ,1). In this calculation, the exothermic regularity represented by Eqs. (3) and (4) is employed. At position xn , the initial value is calculated using Eq. (7). If Eq. (5) has no solution, i.e., no thermal throat, for the given Mach number of the combustor entry, Eq. (1) can be solved step by step from the combustor entry toward the exit. All of the calculation processes are completed by programming, and as long as the iteration step length is sufficiently small, the precision of the calculation results can be guaranteed. After obtaining the axial distribution of M(x), along with the given functions A (x) and Tt(x), other flow parameters can be calculated using Eqs. (15)–(17). 3.2. Verification of model 3.2.1. Verification via three-dimensional numerical simulation The Mach number distribution calculated using the onedimensional model was found to be consistent with the three-dimensional numerical simulation results for Ma3 in the fuel combustion zone. After modification using the effective cross-sectional area, the results for the jet combustion zone also agreed well with the numerical simulation results, as shown in Fig. 8(a). This means that the calculation of the expansion boundary location and the analysis of its effect on the Mach number distribution were correct. Fig. 8(b)–(d) shows comparisons of the static pressure, static temperature, and total pressure distribution given by Eqs. (15)–(17) with the jet modification. The difference between the one-dimensional model results and the threedimensional numerical simulation results is slightly greater at the cavity position because the cavity not only entails a
change in the cross-sectional area but also affects the heat release of local combustion. Furthermore, the cavity does not promote combustion identically under different flow conditions and injection schemes. Therefore, it is very difficult to simplify the cavity as a “black box” that can adapt to a wide range of operation conditions in the onedimensional model. However, except for the cavity area, the overall trend and parameter values are consistent between the model and the numerical simulation. The calculation error associated with the one-dimensional model is shown in Table 4. The precision of the model used to forecast the thermal throat position and combustor performance was therefore judged to be sufficient. To calculate the error between the model and the numerical simulation, corresponding points at the same position were used to calculate the variance, the relative error, and the deviation in the thermal throat position, with the dimensionless length of the combustor taken as 1.
3.2.2. Verification via direct-connect testing To further verify the accuracy of the one-dimensional model and the reliability of the three-dimensional numerical simulation, a validation was conducted using experimental data obtained from direct-connect testing at the Science and Technology of Combustion, Internal Flow and ThermoStructure Laboratory of Northwestern Polytechnical University. The configuration of the test engine and the validation results are shown in Fig. 9. The test engine contains a nozzle IV used to simulate the Mach number of the inflow before the isolator, and the combustor configuration was the same as that shown in Fig. 2. The simulated operating conditions for the direct-connect testing were a flight height of 12 km and Ma¼ 3.0. The Mach number at the combustor entry was 1.7, and the equivalent ratios of the rocket and fuel were 0.1 and 0.7, respectively. On the whole, the pressure distributions along the RBCC combustor according to the onedimensional model and three-dimensional numerical simulation were both consistent with the test data, except for the cavity area and a short region after it. The maximum relative error of the corresponding points between the simulation results and test results was less than 8.9%, and that between the model results and test results was 10.4%. The validation results show that the one-dimensional model can be used to obtain a preliminary indication of the performance of an RBCC combustor.
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Fig. 8. Comparison of parameter distribution between 1-D model and 3-D numerical simulation. (a) Mach number, (b) static pressure, (c) static temperature, and (d) total pressure.
Table 4 Error between one-dimensional model results and three-dimensional numerical Simulation results. MSE¼ Mean Squared Error, MRE¼ Maximum Relative Error. Error in thermal Precision Static throat % pressure %
Static temperature %
Total pressure %
0.36
0.68 8.2 9.3
0.061 2.5 5.9
MSE RMSE MRE
0.69 8.3 7.4
4. Application of one-dimensional model Because the combustion flow field is uneven and the thermal throat is formed at high temperature and high speed, measurements from previous tests cannot easily be used to capture the formation process and provide an intuitive description of the thermal throat. Although a three-dimensional numerical simulation with high precision can be used to obtain the details of the whole flow field, the computational time required is too long and does not cover all possible operating conditions. Thus,
Fig. 9. Comparison of pressure distribution between 1-D model, 3-D numerical simulation, and experimental data.
performing three-dimensional numerical simulation to analyze the variation law of the thermal throat and guide the preliminary design of the combustor would be a waste of effort.
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Fig. 10. Parameter distribution for different expansion ratios. (a) Mach number and (b) static pressure.
Fig. 11. Parameter distribution for different injection positions. (a) Mach number and (b) static pressure.
In contrast, the results of this study show that by establishing a one-dimensional analysis model for an RBCC combustor to study the influence of the related parameters, the thermal throat and its effect on the combustor performance can be analyzed simultaneously.
4.1.2. Effect of injection position Fig. 11 shows the Mach number and static pressure distribution along the combustor when only the fuel injection position changes. When the injection position is upstream, the thermal throat and the corresponding highpressure region in the combustor are also located upstream, which causes the Mach number at the combustor entry to increase. In ramjet mode, on the one hand, a shock train in the isolator that depends on the amount of heat released must be established to ensure subsonic flow to the combustor. On the other hand, heat release at too forward a position will increase the inlet burden. Thus, a better fuel injection location can be determined using the onedimensional model for the ramjet mode.
4.1. Factors influencing thermal throat 4.1.1. Effect of expansion ratio Fig. 10 shows the Mach number and static pressure distribution along the combustor when the fuel injection and exothermic distribution remain the same but the expansion ratio changes. A larger expansion ratio accelerates the flow much more rapidly, causing the thermal throat to form earlier. Because the same amount of heat is released, the smaller the duct area is, the greater the increase in the static pressure is. The effect of pressure rise on the inlet needs to be considered for a small expansion ratio, which may cause an unstart of the inlet. On the other hand, the mass flow of fuel can be appropriately increased to further enhance the performance for a large expansion ratio. The amount of heat released and the duct area can be well matched through one-dimensional analysis.
4.1.3. Effect of equivalent ratio The Mach number and static pressure distribution along the combustor for various equivalent ratios were obtained using the exothermic regularity shown in Fig. 5 (a), and the results are shown in Fig. 12. The variation amplitude of the thermal throat suggests that the influence of the exothermic quantity on the thermal throat is smaller than that of the expansion ratio and exothermic position. This is because combustion with more fuel
Y. Wang et al. / Acta Astronautica 117 (2015) 130–141
Fig. 12. Parameter distribution for different equivalent ratios. (a) Mach number and (b) static pressure.
requires a longer reaction interval, which causes a delay in the thermal throat. The exothermic quantity greatly increases with an increase in the equivalence ratio, and its main effect is a rise in the static pressure in the combustor. For a fixed configuration and injection position, the maximum pressure the combustor can achieve is determined when the normal shock is formed in the isolator. The maximum pressure can be calculated from the compression ratio of normal shock, using Eq. (18) [23]. The largest equivalence ratio of the fuel can be determined from the one-dimensional model for the ramjet mode. p2 2γ γ 1 M2 ¼ p1 γ þ1 1 γ þ 1
ð18Þ
4.1.4. Effect of exothermic rate The Mach number and static pressure distribution along the combustor at different exothermic rates were obtained using the exothermic regularity shown in Fig. 5(b), and the results are shown in Fig. 13. The faster the heat release is, the earlier the thermal throat forms in the duct. According to one-dimensional theory, a subsonic flow accelerates with heating, and a supersonic flow decelerates with heating. If
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Fig. 13. Parameter distribution for different exothermic rates. (a) Mach number and (b) static pressure.
the amount of heat released at one duct position is sufficient to choke the flow but the next interval still exists, obvious heat release and the variation in the cross-sectional area cannot offset the effect of heat release, and choking will cease. This is why a thermal throat cannot form in the main exothermic zone, and therefore, the shorter the main exothermic zone is, the earlier the thermal throat forms. However, compared to the above influencing factors, the exothermic rate is more difficult to control, as it is affected by many complex factors, such as the fuel type, injection mode, mass flow rate, and inflow condition. Thus, it is very difficult to adjust the thermal throat by changing the exothermic rate of fuel in the actual operating process of an RBCC engine. 4.2. Optimization of RBCC combustor for ramjet As indicated by the above discussion, the main factors influencing the thermal throat are the expansion ratio of the duct, the injection position, and the exothermic rate of fuel. However, compared to adopting a combustor with a variable structure or changing the exothermic rate of the fuel, adjusting the injection position is obviously much
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Table 5 Optimization results for injection scheme for ramjet mode from onedimensional model. Injection scheme
a
pz
τb
xn
Thrust, N
Before After
0.7 0.7
0.3 0.4
2.5 3.0
0.78 0.97
2120 2790
easier. Thus, when an RBCC engine operates in ramjet mode, an appropriate injection position is selected with the corresponding equivalent ratio to adapt to different inflow conditions and therefore improve performance for a wide range of operating conditions by adjusting the thermal throat. From the perspective of design of a combustor for a ramjet, the thermal throat should be located as close to the end of the combustor as possible. As a result, the fuel will have a sufficient subsonic combustion interval, the static pressure can be kept high throughout the combustor region, and the influence of back pressure on the inlet can be reduced. As described in Section 4.1, downstream thermal throat formation can be achieved by increasing the fuel equivalent ratio or moving the injection position downward. The maximum pressure at the combustor entrance is the pressure value after the normal shock formed in the isolator. At the same time, the heat release in the combustor reaches a maximum. The thermal throat located at the end of the combustor and the maximum heat release can thus be realized by adjusting the injection position and equivalent ratio at the same time. The thrust performance of the engine then reaches its optimal value. This is the process of optimization. Therefore, if (a) the maximum pressure at the combustor entry reaches the compression ratio of normal shock and (b) the thermal throat is located at the end of the combustor—these being the two main optimization goals—an injection scheme for better RBCC combustor performance can be obtained using the one-dimensional model, as indicated by the optimization results shown in Table 5. The optimization object is the RBCC combustor, whose configuration encompasses both the ejector and scramjet modes, as shown in Fig. 2. The static pressure distribution in the combustor was calculated using the one-dimensional model, and the thrust was computed using the integral of the static pressure over the flow path area. The integrated thrust is the parameter used to measure the combustor performance. The thrust performance increased by 31.6% at Ma3 after optimization for the ramjet mode, indicating that the one-dimensional model can play an important role in guiding the preliminary design of an RBCC combustor. 5. Conclusions A one-dimensional analysis model for an RBCC combustor was established. This model can be used to analyze the factors that influence the thermal throat under the interaction of the exothermic distribution and the variation in the area and the effects of this interaction on the performance of the combustor. The model is based on the solution for a one-dimensional flow equation and uses
parameters for which the general laws of exothermic distribution as they apply to the characteristics of an RBCC combustor have been established. The effective crosssectional area of the rocket jet was calculated and incorporated into the model. Other related flow parameters, mainly associated with the static pressure, were solved based on the Mach number distribution to reflect the combustor performance. The results calculated using the one-dimensional model were found to be consistent with those obtained from a three-dimensional numerical simulation. The position of the thermal throat was predicted with an error of 0.36%. The maximum relative errors of the static pressure among the corresponding points were 7.4% and 9.3% for the static temperature and total pressure, respectively. The one-dimensional model and three-dimensional numerical simulation were validated using experimental data obtained in direct-connect testing. Except for the cavity region, the maximum relative error of the corresponding points between the simulation results and the test results was less than 8.9%, and that between the model results and test results was 10.4%. These results demonstrate the accuracy and precision of the one-dimensional model. The one-dimensional analysis results show that the greater the expansion ratio is, the farther ahead the injection location is, or the faster the heat release is, the earlier the thermal throat forms in the duct. Compared with the aforementioned factors, the fuel equivalence ratio has only a limited effect. An optimization study of the RBCC combustor for the ramjet mode was conducted by adjusting the thermal throat. The thrust performance improved by 31.6% at Ma3 after optimization. These results indicate the important role that the one-dimensional model can play in analyzing the thermal throat and guiding the preliminary design of an RBCC combustor. References [1] P. Roncioni, P. Natale, M. Marini, et al., Numerical simulation and performance assessment of a scramjet powered cruise vehicle at Mach 8, Aerosp. Sci. Technol. 42 (2015) 218–228. [2] Z. Zhong, Z.G. Wang, M.B. Sun, Effects of fuel cracking on combustion characteristics of a supersonic model combustor, Acta Astronaut. 110 (2015) 1–8. [3] J.M. Donohue, Dual-mode scramjet flameholding operability measurements, J. Propul. Power 30 (3) (2014) 592–603. [4] Z.P. Wang, F. Li, H.B. Gu, et al., Experiment study on the effect of combustor configuration on the performance of dual-mode combustor, Aerosp. Sci. Technol. 42 (2015) 169–175. [5] T. Sato, H. Taguchi, H. Kobayashi, T. Kojima, et al., Development study of precooled-cycle hypersonic turbojet engine for flight demonstration, Acta Astronaut. 61 (2006) 367–375. [6] J. Etele, S. Hasegawa, S. Ueda, Experiment investigation of an alternative rocket configuration for rocket-based combined cycle engines, J. Propul. Power 30 (4) (2014) 944–951. [7] M. Bulman, A. Siebenhaar, The strutjet engine: exploding the myths surrounding high speed airbreathing propulsion, AIAA Paper 952475, 1995. [8] T. Hiraiwa, K. Ito, S. Sato, S. Ueda, et al., Recent progress in scramjet/ combined cycle engines at JAXA, Kakuda space center, Acta Astronaut. 63 (2008) 565–574. [9] C.F. Ehrlich, Early studies of RBCC applications and lessons learned for today, AIAA Paper 2000-3105, 2000. [10] A.P. Kothari, J.W. Livingston, C. Tarpley, et al., Rocket based combined cycle hypersonic vehicle for orbital access, AIAA Paper 2011-2338, 2011.
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[11] T. Fujikawa, T. Tsuchiya, S. Tomioka, Multi-objective, multidisciplinary design optimization of TSTO spanes with RBCC engines, AIAA Paper 2015-0650, 2015. [12] A. Brandt, L.L. Perini, One-dimensional combustor optimization, J. Spacecr. Rockets 6 (5) (1969) 570–576. [13] T.F. O’brien, R.P. Starkey, M.J. Lewis, Quasi-one-dimensional highspeed engine model with finite-rate chemistry, J. Propul. Power 17 (6) (2001) 1366–1374. [14] L. Tian, L.H. Chen, Q. Chen, F. Li, X.Y. Chang, Quasi-one-dimensional multimodes analysis for dual-mode scramjet, J. Propul. Power 30 (6) (2014) 1559–1567. [15] W.H. Heiser, D.T. Pratt, D.H. Daley, U.B. Mehta, Hypersonic Airbreathing Propulsion, AIAA, Washington, DC, 1994, 346–350. [16] H. Ikawa, Rapid methodology for design and performance prediction of integrated scramjet/hypersonic vehicle, AlAA Paper 89-2682, 1989. [17] J.P. Drummond, Numerical investigation of the perpendicular injector flow field in a hydrogen fueled scramjet, AlAA Paper 79-1482, 1979.
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[18] H.H. Klein, P.C. Chan, A numerical investigation of scramjet combustor, AlAA Paper 89-2561, 1989. [19] H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, The Ronald Press Company, New York, 1953, 218–260. [20] R.L. Daines, C.L. Merkle, Computational analysis of mixing and jet pumping in rocket ejector engines, AlAA Paper 95-2477, 1995. [21] C.Q. Zhao, Y. Jiang, Gas Jet Dynamics, Beijing Institute of Technology Press, BeiJing, 1998, 134–143. [22] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, SpringerVerlag, New York, 1993, 434–439. [23] G. Emanuel, Gasdynamics Theory and Applications, AIAA, Washington, DC, 1986, 49–54.
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Study of thermal throat of RBCC combustor based on one-dimensional analysis Ya-jun Wang, Jiang Li, Fei Qin n, Guo-qiang He, Lei Shi Science and Technology on Combustion, Internal Flow and Thermal-Structure Laboratory, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China
a r t i c l e in f o
abstract
Article history: Received 8 March 2015 Received in revised form 14 July 2015 Accepted 31 July 2015 Available online 10 August 2015
An analysis model was developed to better understand the formation mechanism and variation law of the thermal throat in a rocket-based combined-cycle (RBCC) combustor. This analysis model is based on one-dimensional flow equations and consideration of the variation in factors such as the area, exothermic distribution, and the fuel-rich jet of the rocket. The influence law for the thermal throat under the interaction of the exothermic distribution and the variation of the area is consistent with the heat release models for a gaseous jet and liquid kerosene. The effective cross-sectional area of the jet was calculated and incorporated into the model. The results calculated using the one-dimensional model were found to be consistent with those obtained from a three-dimensional numerical simulation. The position of the thermal throat was predicted with an error of 0.36%. The maximum relative errors of the static pressure among the corresponding points were 7.4% and 9.3% for the static temperature and total pressure, respectively. The one-dimensional model and three-dimensional numerical simulation were validated using experimental data obtained in direct-connect testing. Except for the cavity region, the maximum relative error of the corresponding points between the simulation results and test results was less than 8.9%, and that between the model results and test results was 10.4%. Compared to the fuel equivalence ratio, the expansion ratio, injection location, and exothermic rate have a significant impact on the position of the thermal throat. An optimization study of the RBCC combustor for the ramjet mode was conducted by adjusting the thermal throat. The thrust performance improved by 31.6% at Ma3 after optimization. These results indicate the important role that the one-dimensional model can play in analyzing the thermal throat and guiding the preliminary design of an RBCC combustor. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Rocket-based combined-cycle One-dimensional model Thermal throat Performance optimization
1. Introduction In the development of hypersonic flight systems, several types of air-breathing engines for different application layers, such as scramjets [1,2], dual-mode scramjets [3,4], and combined-cycle engines [5,6], have long been studied.
n
Corresponding author. Tel./fax: þ 86 29 88494163. E-mail address: [email protected] (F. Qin).
http://dx.doi.org/10.1016/j.actaastro.2015.07.034 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.
A rocket-based combined-cycle (RBCC) combustor [7,8] is yet another type of advanced propulsion system. This type of system has the widest flight envelope and application scope of those mentioned, and it can satisfy the various requirements of future aircraft, such as high altitude, high speed, and flight maneuverability. RBCC combustors are considered to be among the most promising propulsion systems for reusable space transportation and hypersonic cruise vehicles [9–11]. A large amount of reference data is needed to develop a preliminary engine design, predict the
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performance of the engine, and plan various trajectories in accordance with various mission requirements. In the project demonstration stage, the use of one-dimensional analysis can greatly reduce time and cost expenditures. An RBCC engine typically has an ejector mode, a ramjet mode, a scramjet mode, and a rocket mode, all of which must operate in the same expanding duct. In the ejector mode, which is approximately from Ma 0 to 2.5, the incoming air is introduced into the duct by the highspeed jet of the rocket, which operates at a high flow rate with high combustor pressure. Extra thrust can be gained through afterburning of the fuel-rich jet or the mixing and combusting downstream fuel with the introduced air. In the ramjet mode, which is approximately from Ma 2.5 to 5.5, and scramjet mode, which is approximately from Ma 5.5 to 8, the ignition and stable combustion of fuel can be achieved with the rocket operating at a relatively low flow rate and under fuel-rich conditions. High-efficiency combustion is achieved by adjusting the injection position of the fuel depending on the incoming flow conditions. As the flow Mach increases, the inlet is shut, and the rocket operates at a high flow rate again to provide the engine the required power for speeding or climbing into orbit. To obtain satisfactory engine performance during the ejector mode at high Mach numbers as well as during the entire ramjet mode, a thermal throat formed by heat release is needed to choke, expand, and accelerate the inflow. Therefore, it is essential to investigate the formation mechanism and variation law of the thermal throat to be able to adjust and control it in an expanding duct. Traditional (quasi) one-dimensional analysis methods are mainly of two types. In the first type of method [12,13], inputs that can reflect the mass injection, chemical reaction, variation in area, and wall friction are added to the one-dimensional Euler equation, differential equations are solved iteratively from the entry to the exit in the flow direction, and numerical simulation and experimental results for various engines are obtained to improve and correct the inputs [14]. Methods of this type require one to calculate a complex chemical reaction; consider the method and interval of mass injection, combustion efficiency, and other factors; and modify the inputs so that the calculation results approximate the actual flow field of the engine. In the second type of method [15,16], a onedimensional flow equation is used, a distribution function of parameters such as the variation in area and exothermic regularity is obtained, the Mach number distribution is solved, and other related flow parameters are calculated based on the Mach number distribution. This type of method requires the distribution function of the parameters, which cannot be determined easily. However, the general functional form of the parameter distribution can be obtained based on expert knowledge, and the onedimensional analysis can be greatly simplified. In analyses focusing on only one certain parameter, such as the thermal throat in the present study, the one-dimensional analysis method is very simple and effective. Previous studies have mainly focused on supersonic combustion in a scramjet or dual-mode scramjet. However, in an RBCC engine, considering the wide operating range of subsonic combustion, it remains necessary to investigate
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the variation and adjustment law of the thermal throat. The present study was based on the one-dimensional analysis approach described above. Findings from previous studies were used to identify the general laws of exothermic distribution that describe the characteristics of an RBCC combustor. Accordingly, the generation condition and influencing factors of the thermal throat were obtained by solving the Mach number distribution based on the exothermic distribution and variation in area, and the influence of the thermal throat position on the performance of the RBCC combustor in ramjet mode was analyzed. 2. Establishment of one-dimensional model 2.1. One-dimensional flow theory Many factors influence the combustion and flow in an RBCC combustor, in which complicated physical and chemical processes occur constantly. To simplify the analysis of the thermal throat, one-dimensional theory was used in this study to describe the flow process. The differential equation for the axial variation in the Mach number is given as Eq. (1) [15]. For brevity and clarity, the effects of wall friction, drag of internal struts, cavity or fuel injection, heat convection, and mass addition due to fuel injection are not considered herein; these effects are of secondary importance compared to the strong interaction between the axial variation of the cross-sectional area A(x) and that of the total temperature Tt(x). Moreover, the present study was focused mainly on the ramjet mode, during which the low flow velocity in the combustor makes the influence of friction and other drag forces negligible. 9 !8 1 þ γ M 2 1 dT t = 1 þ γ 2 1M 2 < dM 1 dA ¼M þ : dx A dx T t dx ; 2 1 M2 ð1Þ For the expanding duct of an RBCC combustor, the axial variation of the cross-sectional area can be expressed as follows: AðxÞ ¼ 1 þax Ain
ð2Þ
where a ¼Aout/Ain 1 ; a is a function of the expansion ratio, Aout and Ain are the area of the combustor's exit and entry, respectively; and x is a dimensionless coordinate of length. For a scramjet combustor, the empirical expression of exothermic regularity obtained from the literature [15] can usually be represented in nondimensional form as follows: T t ðxÞ bcx ¼ 1þ T tin 1 þ ðc 1Þx
ð3Þ
where b¼Ttout/Ttin 1; b is a function of the total temperature rise ratio Ttb ¼Ttout/Ttin ; Ttout and Ttin are the total temperature of the combustor's exit and entry, respectively; and c is an empirical constant on the order of 1 to 10 that depends on the mode of fuel injection and fuel–air mixing. As Fig. 1 shows, the value of b determines the exothermic quantity and c, the exothermic rate; the
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noted that the range of one-dimensional analysis considered in this study is from the exit of the strut rocket nozzle (combustor entry) to the combustor exit, and the length coordinates x range from 0 to 1. 2.3. Exothermic distribution model
Fig. 1. Exothermic regularity expressed by Eq. (3) with different parameters b and c. (a) c ¼5 and (b) b ¼1.5.
greater the value of c is, the shorter the heat release distance is. 2.2. RBCC combustor configuration Fig. 2 shows the studied center strut of an RBCC combustor. The strut contains two expanding ducts, II and III. The rocket engine is embedded in the center strut in isolator I. Two pylons for fuel injection and a cavity for the flame holder are located in expanding duct II. When the RBCC engine operates in ramjet mode, the strut rocket operates at a relatively low flow rate and in fuel-rich conditions, and its high-temperature jet ensures the reliable ignition and stable combustion of fuel. Thus, compared to a traditional ramjet or scramjet engine, the studied RBCC combustor contains the combustion zone of the fuel-rich jet of the rocket, except for the zone of fuel combustion. The cut-off point between these two zones is the fuel injection location (pz). Because the fuel-rich jet mainly contains the gaseous component CO and the fuel is liquid kerosene, there are obviously many differences in the exothermic regularity between these two fuels. Therefore, it is necessary to study the exothermic distribution between the partitions in the RBCC combustor. It should be
With data obtained from a three-dimensional numerical simulation of a typical point Ma3, 12 km in the ramjet mode, Eq. (3) was used to fit the data for the total temperature distribution in the combustor. The numerical simulation was carried out using the software FLUENT. Three-dimensional unsteady Reynolds-averaged Navier– Stokes (N–S) equations were chosen as the governing equations, and the shear stress transport (SST) k–ω model for viscous flow was used. This model has a short calculation period and high calculation accuracy for the turbulence in free shear flow and for flows in which separation occurs. A simplified three-step kinetic model, shown in Table 1 for JP-10 kerosene, for which C10H16 was used as a surrogate, was used to calculate the chemical reaction rate. The Lagrange two-phase flow method was used for liquid fuel injection, and the Taylor analogy breakup (TAB) model was used for droplet breakup. The data fit the model well in the combustion zone of the fuel-rich jet but badly in the fuel combustion zone, as shown in Fig. 3. The main reason for this is that the exothermic regularity expressed by Eq. (3) has a faster rate, allowing it to complete the main heat release within a short distance, as shown in Fig. 1, which is closer to the exothermic regularity of a gaseous fuel. This is consistent with the use of gaseous hydrogen fuel in early studies on ramjet and scramjet engines [17,18]. In comparison, because of the longer ignition delay time, mixing, and cracking interval, the exothermic rate and reaction interval of liquid kerosene are much greater. Apparently, it is not appropriate to express the exothermic regularity of kerosene using Eq. (3). Thus, the total temperature distribution of fuel combustion was fitted again. The results show that the exothermic distribution of kerosene is closer to that described by Eq. (4), shown below. T t ðxÞ ¼ dxe þ f T tin
ð4Þ
Table 2 shows the precision of the exothermic distribution model fitted in different zones by Eqs. (3) and (4). Using Eqs. (3) and (4) to express the exothermic regularity of rocket jet and kerosene fuel, respectively, conforms to the characteristics of the flow field in an RBCC combustor. To verify the exothermic regularity of kerosene, the total temperature data obtained from the numerical simulation for other typical points (Ma4, 17 km and Ma5, 21 km) in ramjet mode were fitted using Eq. (4). The results shown in Fig. 4 and Table 2 indicate that Eq. (4) can be used to characterize the exothermic regularity of kerosene in an RBCC combustor. According to the mathematical characterization of Eq. (4), the total amount of heat release is determined by the parameters d and f, and the exothermic rate is determined by the parameters d and e. In a preliminary analysis, for simplicity, parameter d was assumed to be unchanging,
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Fig. 2. Configuration of RBCC combustor with center strut.
Table 1 Three-step kinetic model for JP-10 kerosene. Reaction
A (cm3/(mol s))
B
Ea (J/(kg mol))
C10H16 þ5O2 ¼10COþ 8H2 2COþO2 ¼2CO2 2H2 þ O2 ¼ 2H2O
23500 3.48E8 3.0E17
1 2 1
1.633E8 8.42E7 0
Table 2 Fitted precision of different exothermic distribution models in fuel and jet combustion zones according to Eqs. (3) and (4). SSE¼ Sum of Squared Errors, RMSE¼ Root Mean Squared Error. Precision
SSE RMSE
Fuel combustion zone
Jet combustion zone
Eq. (3)
Eq. (3)
Eq. (4)
0.0025% 0.25%
0.013% 0.66%
10.65% 8.42%
Eq. (4) Ma3
Ma4
Ma5
1.25% 2.99%
0.95% 2.6%
0.12% 0.93%
and parameters e and f were changed simultaneously. A similar characterization for expressing different exothermic quantities, as shown in Fig. 1(a), was obtained, as shown in Fig. 5(a). The total amount of heat release was assumed to be unchanging, and parameters d, e, and f were changed simultaneously. A similar characterization for expressing different exothermic rates, as determined mainly by the parameter e, as shown in Fig. 1(b), was obtained, as shown in Fig. 5(b). 2.4. Generation condition of thermal throat Previously, one-dimensional analysis of the Mach number distribution could be conducted only when Tt(x) was clear in the RBCC combustor. This is mathematically represented in Eq. (1) by the requirement that as M goes to 1 at the thermal throat, the singularity due to the (1 M2) term in the denominator must be offset by the polynomial in brackets going to 0 at exactly the same location. Therefore, the position of the thermal throat can be obtained from Eq. (5), as shown below, where the asterisk * denotes the thermal throat, at which M¼1. 1 þ γ M 2 1 dT t 1 dA þ ¼0 ð5Þ A dx n T t dx n 2
Fig. 3. Curves fitted with the data of total temperature distribution at Ma3, 12 km. (a) Fuel combustion zone and (b) jet combustion zone.
Apparently, the thermal throat in an RBCC combustor is located between x¼ pz and 1. The value of xn can be determined from Eq. (5) for a given pair of functions A(x) and Tt(x) when the equation has a solution between x¼pz and 1; when there is no such solution, it means that there is no thermal throat.
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Fig. 4. Curves fitted with the data of total temperature distribution in fuel combustion zone. (a) Ma4, 17 km and (b) Ma5, 21 km.
The expressions A(x) and Tt(x) can be substituted into Eq. (5) to obtain the following expression: γ þ1 γ þ1 e 1 e 1 xe þ dex f ðxÞ ¼ ad af 0 2 2 pz r x r 1 ð6Þ As is known from the characteristics of the fitted curve for Tt(x), do0, eo0, and f40; therefore, f(pz)4 0 and f(1) r0 are necessary requirements to ensure that Eq. (6) has a root between x ¼pz and 1. This is the first criterion for thermal throat formation. From Eq. (1), ðdM=dxÞn ¼ 0=0 is algebraically indeterminate at xn . Thus, L’Hôpital’s rule is used to evaluate Eq. (1) at this critical point [19]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dM 1 ¼ Ω 7 Ω2 4Ψ ð7Þ dx n 4 where
"
2
d ðlnAÞ Ψ ≡ ð1 þ γ Þ dx2
Ω
γ ðγ þ 1Þ 1 dT t 2
T t dx
# ⁎
" # 2 ð1 þγ Þ2 d ðlnT t Þ þ 2 dx2
ð8Þ ⁎
ð9Þ n
Fig. 5. Exothermic regularity expressed by Eq. (4) with different parameters d, e, and f. (a) Different amount of heat release and (b) different exothermic regularity.
For the ramjet mode, to make the flow in the duct transition from subsonic to supersonic, dM=dx must always be positive. Therefore, ðdM=dxÞn 4 0 holds, and thus, Ψ o0 at xn . This is the second criterion for thermal throat formation. These two criteria need to be satisfied simultaneously. 2.5. Effect of rocket jet Before using Eq. (1) to solve for the Mach number distribution, it should be noted that the rocket jet is located between x ¼0 and pz. The jet exiting the nozzle expands from the center to both sides of the strut. However, its ability to expand is limited by the back pressure of the combustor and the influence of the shear layer on both sides, as a result of which its action range is limited in the center of the flow path, as shown in Fig. 6. The expansion of a fuel-rich jet during combustion is a problem that involves many complex factors. To simplify its role in one-dimensional analysis, the expansion boundary of the jet was calculated to make the one-dimensional model reflect the effect of the jet without being too complex. Assuming that the effect of the jet exists only at the expansion boundary, the existence of the jet can be
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135
Fig. 6. Static temperature contours of T Z 800 K in the internal flow field of RBCC combustor.
converted into a change in the cross-sectional area, namely, in the interval x ¼0 to pz, using an area distribution of AðxÞ=Ain ¼ 1 þ a1 x. For simplicity, the cross-sectional area of the duct reduces the maximum expansion boundary, and the remaining area is the actual action range of heat release from jet combustion. We then obtain the influence factor a1 to express the effective cross-sectional area of the rocket jet. Concepts from classical theory and engineering algorithms for gas jet dynamics are used to calculate the location of the expansion boundary [20,21]. The main equations [21] used in the calculation are shown below. In these equations, r and x are coordinates in the curve equation of the expansion boundary.
1 γ 1 2 γ 1 2 1 M 2 1 r¼ ln 1 þ M = 1þ Me þ ln e2 γ 1 2 2 2 M 1 ð10Þ x ¼ FðMÞ FðM 1 Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 2e 1
ð11Þ
sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 γ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 γ þ1 γ 1 2 2 M 1 U arctg M 1 F ðM Þ ¼ γ 1 γ 1 γ 1 γ þ1 ð12Þ M1 is the value of M when r ¼1, which can be determined using the following equation: (sffiffiffiffiffiffiffiffiffiffiffi ) i γ þ1 2 γ 1hπ 2 tg þ θ ðM e Þ ϑ M1 ¼ 1 þ ð13Þ γ 1 γ þ1 2 where ϑ is the expansion half-angle of the nozzle and sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ þ1 γ 1 2 arctg M e 1 arctg M2e 1 ð14Þ θðMe Þ ¼ γ 1 γ þ1 A Cþ þ program was developed to solve the above equations iteratively for the given boundary conditions of the nozzle exit: the Mach number, pressure, back pressure of the combustor, and expansion half-angle of the nozzle. Details of the calculation are omitted here for brevity. The calculated boundary curve is shown in Fig. 7. This curve is
consistent with the expansion boundary results obtained from the numerical simulation, indicating that the calculation method and expansion boundary results can be used in the analysis of the RBCC combustor. 2.6. Parameters of model All of the parameters used in the calculation of the onedimensional model have been obtained, as shown in Table 3. In Table 3, pz¼0.3, a¼ 0.6, a1 ¼1.1, d ¼ 0.848, e¼ 0.708, f¼3.37, b¼0.225, and c ¼7.61. By recording the intermediate values of M(x) as Eq. (1) is solved iteratively, the axial distribution of all other flow variables and thermo-physical properties, such as the static temperature, static pressure, and total temperature in the combustor, can be determined from the point-bypoint solution set M(x), together with the corresponding values of the prescribed functions A(x) and Tt(x), using several groups of integral relations [19], shown below. A complete one-dimensional analysis model of an RBCC combustor based on the Mach number distribution is therefore established. 2 3 2 γ1 TðxÞ T t ðxÞ4 1 þ 2 M in 5 ð15Þ ¼ T in T tin 1 þ γ 1 M 2 ðxÞ 2
pðxÞ Ain M in U ¼ pin AðxÞ MðxÞ
sffiffiffiffiffiffiffiffiffi TðxÞ T in
P t ðxÞ P ðxÞ T in T t ðxÞ γ =ðγ 1Þ ¼ P tin P in T ðxÞ T tin
ð16Þ
ð17Þ
3. Solution and verification of one-dimensional model 3.1. Solution of model Under the premise of satisfying the formation conditions, the position xn of the thermal throat can be obtained
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Fig. 7. Comparison of expansion boundary with Mach contours in jet combustion zone.
Table 3 The parameter settings for the one-dimensional model.
Interval A(x) Tt(x)
Fuel combustion zone
Jet combustion zone
pz to 1 1 þ ax dxe þ f
0 to pz 1 þ a1 x 1 þ bcx=ð1 þ ðc 1ÞxÞ
by solving Eq. (5). The fourth-order Runge–Kutta numerical method [22], which has high calculation precision, is used to solve Eq. (1) by iterating step by step from the thermal throat position to the upstream point (0, xn ) and the downstream point (xn ,1). In this calculation, the exothermic regularity represented by Eqs. (3) and (4) is employed. At position xn , the initial value is calculated using Eq. (7). If Eq. (5) has no solution, i.e., no thermal throat, for the given Mach number of the combustor entry, Eq. (1) can be solved step by step from the combustor entry toward the exit. All of the calculation processes are completed by programming, and as long as the iteration step length is sufficiently small, the precision of the calculation results can be guaranteed. After obtaining the axial distribution of M(x), along with the given functions A (x) and Tt(x), other flow parameters can be calculated using Eqs. (15)–(17). 3.2. Verification of model 3.2.1. Verification via three-dimensional numerical simulation The Mach number distribution calculated using the onedimensional model was found to be consistent with the three-dimensional numerical simulation results for Ma3 in the fuel combustion zone. After modification using the effective cross-sectional area, the results for the jet combustion zone also agreed well with the numerical simulation results, as shown in Fig. 8(a). This means that the calculation of the expansion boundary location and the analysis of its effect on the Mach number distribution were correct. Fig. 8(b)–(d) shows comparisons of the static pressure, static temperature, and total pressure distribution given by Eqs. (15)–(17) with the jet modification. The difference between the one-dimensional model results and the threedimensional numerical simulation results is slightly greater at the cavity position because the cavity not only entails a
change in the cross-sectional area but also affects the heat release of local combustion. Furthermore, the cavity does not promote combustion identically under different flow conditions and injection schemes. Therefore, it is very difficult to simplify the cavity as a “black box” that can adapt to a wide range of operation conditions in the onedimensional model. However, except for the cavity area, the overall trend and parameter values are consistent between the model and the numerical simulation. The calculation error associated with the one-dimensional model is shown in Table 4. The precision of the model used to forecast the thermal throat position and combustor performance was therefore judged to be sufficient. To calculate the error between the model and the numerical simulation, corresponding points at the same position were used to calculate the variance, the relative error, and the deviation in the thermal throat position, with the dimensionless length of the combustor taken as 1.
3.2.2. Verification via direct-connect testing To further verify the accuracy of the one-dimensional model and the reliability of the three-dimensional numerical simulation, a validation was conducted using experimental data obtained from direct-connect testing at the Science and Technology of Combustion, Internal Flow and ThermoStructure Laboratory of Northwestern Polytechnical University. The configuration of the test engine and the validation results are shown in Fig. 9. The test engine contains a nozzle IV used to simulate the Mach number of the inflow before the isolator, and the combustor configuration was the same as that shown in Fig. 2. The simulated operating conditions for the direct-connect testing were a flight height of 12 km and Ma¼ 3.0. The Mach number at the combustor entry was 1.7, and the equivalent ratios of the rocket and fuel were 0.1 and 0.7, respectively. On the whole, the pressure distributions along the RBCC combustor according to the onedimensional model and three-dimensional numerical simulation were both consistent with the test data, except for the cavity area and a short region after it. The maximum relative error of the corresponding points between the simulation results and test results was less than 8.9%, and that between the model results and test results was 10.4%. The validation results show that the one-dimensional model can be used to obtain a preliminary indication of the performance of an RBCC combustor.
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Fig. 8. Comparison of parameter distribution between 1-D model and 3-D numerical simulation. (a) Mach number, (b) static pressure, (c) static temperature, and (d) total pressure.
Table 4 Error between one-dimensional model results and three-dimensional numerical Simulation results. MSE¼ Mean Squared Error, MRE¼ Maximum Relative Error. Error in thermal Precision Static throat % pressure %
Static temperature %
Total pressure %
0.36
0.68 8.2 9.3
0.061 2.5 5.9
MSE RMSE MRE
0.69 8.3 7.4
4. Application of one-dimensional model Because the combustion flow field is uneven and the thermal throat is formed at high temperature and high speed, measurements from previous tests cannot easily be used to capture the formation process and provide an intuitive description of the thermal throat. Although a three-dimensional numerical simulation with high precision can be used to obtain the details of the whole flow field, the computational time required is too long and does not cover all possible operating conditions. Thus,
Fig. 9. Comparison of pressure distribution between 1-D model, 3-D numerical simulation, and experimental data.
performing three-dimensional numerical simulation to analyze the variation law of the thermal throat and guide the preliminary design of the combustor would be a waste of effort.
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Fig. 10. Parameter distribution for different expansion ratios. (a) Mach number and (b) static pressure.
Fig. 11. Parameter distribution for different injection positions. (a) Mach number and (b) static pressure.
In contrast, the results of this study show that by establishing a one-dimensional analysis model for an RBCC combustor to study the influence of the related parameters, the thermal throat and its effect on the combustor performance can be analyzed simultaneously.
4.1.2. Effect of injection position Fig. 11 shows the Mach number and static pressure distribution along the combustor when only the fuel injection position changes. When the injection position is upstream, the thermal throat and the corresponding highpressure region in the combustor are also located upstream, which causes the Mach number at the combustor entry to increase. In ramjet mode, on the one hand, a shock train in the isolator that depends on the amount of heat released must be established to ensure subsonic flow to the combustor. On the other hand, heat release at too forward a position will increase the inlet burden. Thus, a better fuel injection location can be determined using the onedimensional model for the ramjet mode.
4.1. Factors influencing thermal throat 4.1.1. Effect of expansion ratio Fig. 10 shows the Mach number and static pressure distribution along the combustor when the fuel injection and exothermic distribution remain the same but the expansion ratio changes. A larger expansion ratio accelerates the flow much more rapidly, causing the thermal throat to form earlier. Because the same amount of heat is released, the smaller the duct area is, the greater the increase in the static pressure is. The effect of pressure rise on the inlet needs to be considered for a small expansion ratio, which may cause an unstart of the inlet. On the other hand, the mass flow of fuel can be appropriately increased to further enhance the performance for a large expansion ratio. The amount of heat released and the duct area can be well matched through one-dimensional analysis.
4.1.3. Effect of equivalent ratio The Mach number and static pressure distribution along the combustor for various equivalent ratios were obtained using the exothermic regularity shown in Fig. 5 (a), and the results are shown in Fig. 12. The variation amplitude of the thermal throat suggests that the influence of the exothermic quantity on the thermal throat is smaller than that of the expansion ratio and exothermic position. This is because combustion with more fuel
Y. Wang et al. / Acta Astronautica 117 (2015) 130–141
Fig. 12. Parameter distribution for different equivalent ratios. (a) Mach number and (b) static pressure.
requires a longer reaction interval, which causes a delay in the thermal throat. The exothermic quantity greatly increases with an increase in the equivalence ratio, and its main effect is a rise in the static pressure in the combustor. For a fixed configuration and injection position, the maximum pressure the combustor can achieve is determined when the normal shock is formed in the isolator. The maximum pressure can be calculated from the compression ratio of normal shock, using Eq. (18) [23]. The largest equivalence ratio of the fuel can be determined from the one-dimensional model for the ramjet mode. p2 2γ γ 1 M2 ¼ p1 γ þ1 1 γ þ 1
ð18Þ
4.1.4. Effect of exothermic rate The Mach number and static pressure distribution along the combustor at different exothermic rates were obtained using the exothermic regularity shown in Fig. 5(b), and the results are shown in Fig. 13. The faster the heat release is, the earlier the thermal throat forms in the duct. According to one-dimensional theory, a subsonic flow accelerates with heating, and a supersonic flow decelerates with heating. If
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Fig. 13. Parameter distribution for different exothermic rates. (a) Mach number and (b) static pressure.
the amount of heat released at one duct position is sufficient to choke the flow but the next interval still exists, obvious heat release and the variation in the cross-sectional area cannot offset the effect of heat release, and choking will cease. This is why a thermal throat cannot form in the main exothermic zone, and therefore, the shorter the main exothermic zone is, the earlier the thermal throat forms. However, compared to the above influencing factors, the exothermic rate is more difficult to control, as it is affected by many complex factors, such as the fuel type, injection mode, mass flow rate, and inflow condition. Thus, it is very difficult to adjust the thermal throat by changing the exothermic rate of fuel in the actual operating process of an RBCC engine. 4.2. Optimization of RBCC combustor for ramjet As indicated by the above discussion, the main factors influencing the thermal throat are the expansion ratio of the duct, the injection position, and the exothermic rate of fuel. However, compared to adopting a combustor with a variable structure or changing the exothermic rate of the fuel, adjusting the injection position is obviously much
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Table 5 Optimization results for injection scheme for ramjet mode from onedimensional model. Injection scheme
a
pz
τb
xn
Thrust, N
Before After
0.7 0.7
0.3 0.4
2.5 3.0
0.78 0.97
2120 2790
easier. Thus, when an RBCC engine operates in ramjet mode, an appropriate injection position is selected with the corresponding equivalent ratio to adapt to different inflow conditions and therefore improve performance for a wide range of operating conditions by adjusting the thermal throat. From the perspective of design of a combustor for a ramjet, the thermal throat should be located as close to the end of the combustor as possible. As a result, the fuel will have a sufficient subsonic combustion interval, the static pressure can be kept high throughout the combustor region, and the influence of back pressure on the inlet can be reduced. As described in Section 4.1, downstream thermal throat formation can be achieved by increasing the fuel equivalent ratio or moving the injection position downward. The maximum pressure at the combustor entrance is the pressure value after the normal shock formed in the isolator. At the same time, the heat release in the combustor reaches a maximum. The thermal throat located at the end of the combustor and the maximum heat release can thus be realized by adjusting the injection position and equivalent ratio at the same time. The thrust performance of the engine then reaches its optimal value. This is the process of optimization. Therefore, if (a) the maximum pressure at the combustor entry reaches the compression ratio of normal shock and (b) the thermal throat is located at the end of the combustor—these being the two main optimization goals—an injection scheme for better RBCC combustor performance can be obtained using the one-dimensional model, as indicated by the optimization results shown in Table 5. The optimization object is the RBCC combustor, whose configuration encompasses both the ejector and scramjet modes, as shown in Fig. 2. The static pressure distribution in the combustor was calculated using the one-dimensional model, and the thrust was computed using the integral of the static pressure over the flow path area. The integrated thrust is the parameter used to measure the combustor performance. The thrust performance increased by 31.6% at Ma3 after optimization for the ramjet mode, indicating that the one-dimensional model can play an important role in guiding the preliminary design of an RBCC combustor. 5. Conclusions A one-dimensional analysis model for an RBCC combustor was established. This model can be used to analyze the factors that influence the thermal throat under the interaction of the exothermic distribution and the variation in the area and the effects of this interaction on the performance of the combustor. The model is based on the solution for a one-dimensional flow equation and uses
parameters for which the general laws of exothermic distribution as they apply to the characteristics of an RBCC combustor have been established. The effective crosssectional area of the rocket jet was calculated and incorporated into the model. Other related flow parameters, mainly associated with the static pressure, were solved based on the Mach number distribution to reflect the combustor performance. The results calculated using the one-dimensional model were found to be consistent with those obtained from a three-dimensional numerical simulation. The position of the thermal throat was predicted with an error of 0.36%. The maximum relative errors of the static pressure among the corresponding points were 7.4% and 9.3% for the static temperature and total pressure, respectively. The one-dimensional model and three-dimensional numerical simulation were validated using experimental data obtained in direct-connect testing. Except for the cavity region, the maximum relative error of the corresponding points between the simulation results and the test results was less than 8.9%, and that between the model results and test results was 10.4%. These results demonstrate the accuracy and precision of the one-dimensional model. The one-dimensional analysis results show that the greater the expansion ratio is, the farther ahead the injection location is, or the faster the heat release is, the earlier the thermal throat forms in the duct. Compared with the aforementioned factors, the fuel equivalence ratio has only a limited effect. An optimization study of the RBCC combustor for the ramjet mode was conducted by adjusting the thermal throat. The thrust performance improved by 31.6% at Ma3 after optimization. These results indicate the important role that the one-dimensional model can play in analyzing the thermal throat and guiding the preliminary design of an RBCC combustor. References [1] P. Roncioni, P. Natale, M. Marini, et al., Numerical simulation and performance assessment of a scramjet powered cruise vehicle at Mach 8, Aerosp. Sci. Technol. 42 (2015) 218–228. [2] Z. Zhong, Z.G. Wang, M.B. Sun, Effects of fuel cracking on combustion characteristics of a supersonic model combustor, Acta Astronaut. 110 (2015) 1–8. [3] J.M. Donohue, Dual-mode scramjet flameholding operability measurements, J. Propul. Power 30 (3) (2014) 592–603. [4] Z.P. Wang, F. Li, H.B. Gu, et al., Experiment study on the effect of combustor configuration on the performance of dual-mode combustor, Aerosp. Sci. Technol. 42 (2015) 169–175. [5] T. Sato, H. Taguchi, H. Kobayashi, T. Kojima, et al., Development study of precooled-cycle hypersonic turbojet engine for flight demonstration, Acta Astronaut. 61 (2006) 367–375. [6] J. Etele, S. Hasegawa, S. Ueda, Experiment investigation of an alternative rocket configuration for rocket-based combined cycle engines, J. Propul. Power 30 (4) (2014) 944–951. [7] M. Bulman, A. Siebenhaar, The strutjet engine: exploding the myths surrounding high speed airbreathing propulsion, AIAA Paper 952475, 1995. [8] T. Hiraiwa, K. Ito, S. Sato, S. Ueda, et al., Recent progress in scramjet/ combined cycle engines at JAXA, Kakuda space center, Acta Astronaut. 63 (2008) 565–574. [9] C.F. Ehrlich, Early studies of RBCC applications and lessons learned for today, AIAA Paper 2000-3105, 2000. [10] A.P. Kothari, J.W. Livingston, C. Tarpley, et al., Rocket based combined cycle hypersonic vehicle for orbital access, AIAA Paper 2011-2338, 2011.
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