Efficiency Analysis and Integrated Design of Rocket-Augmented Turbine-Based Combined Cycle Engines with Trajectory Optimization

Abstract

An integrated analysis method for a rocket-augmented turbine-based combined cycle (TBCC) engine is proposed based on the trajectory optimization method of the Gauss pseudospectral. The efficiency and energy of the vehicles with and without the rocket are analyzed. Introducing an appropriate rocket to assist the TBCC-powered vehicle will reduce the total energy consumption of drag, and increase the vehicle efficiency in the transonic and the mode transition. It results in an increase in the total efficiency despite a reduction in engine efficiency. Therefore, introducing a rocket as the auxiliary power is not only a practical solution to enable flight over a wide-speed range when the TBCC is incapable but also probably an economical scheme when the the TBCC meets the requirements of thrust. When the vehicle drag is low, the rocket works for a short time and its optimal relative thrust is small. Thus, the TBCC combined with a booster rocket will be a more simple and suitable scheme. When the vehicle drag is high, the operating time of the rocket is long and the optimal relative thrust is large. The specific impulse has a significant impact on the flight time and the total fuel consumption. Accordingly, the combination form for the rocket-based combined cycle (RBCC) engines and the turbine will be more appropriate to obtain higher economic performance.

1. Introduction

Turbine-based combined cycle (TBCC) engines are believed to be a promising means of power for wide-speed range hypersonic vehicles [1]. Due to combining turbine engines with ramjet engines or dual-mode scramjet engines, typical TBCC engines have the advantages of low fuel consumption, the high reliability of turbine engines and the high-speed cruise ability of ramjet/dual mode ramjet (DMRJ) engines [2]. However, several technical issues with TBCC engines still exist, such as “Thrust Pinch”, so the engines are incapable of providing sufficient thrust for acceleration during transonic and mode transition [3]. To resolve this problem, the “rocket-augmented TBCC” concept has been proposed as a near-term solution, where the rockets are used as auxiliary power for thrust augmentation when extra thrust is needed.

In terms of combination form, “rocket-augmented TBCC” engines are classified into two broad categories: “T+RBCC” and “R+TBCC”. For “T+RBCC” engines, the rockets are typically mounted within the high-speed ducts and integrated as ejector-ramjets (ERJs). To date, some schemes have been published, such as TriJet [3,4], TRRE [5,6], and XTER [7]. All these engines utilize the thrust and specific impulse potential of ERJs [8,9,10], but the rockets should be synergistically integrated into the thermodynamic cycle at the cost of complexity of the rocket-matching design. By comparison, the rockets in “R+TBCC” engines are installed as additional boosters, which are generally isolated from the typical TBCC engines. Many researchers have discussed these kinds of engine and vehicle as well, for example, the advanced Rocket/Dual-mode ramjet propulsion of Lapcat II [11] and the high-speed near space flight vehicle proposed by the Beijing Institute of Mechanical and Electrical Engineering [12]. Although the integration design method for the “R+TBCC” engines is relatively simple, this combination may increase extra vehicle drag and inert weight, which may raise the fuel consumption and reduce the payload weight, respectively. From the above, in order to achieve synergistic benefits from the collaboration of TBCCs and rockets excellently, the performance of the vehicle, TBCC, and rockets should be taken into account when designing the combination form of the rocket-augmented TBCC.

Generally, the rockets are introduced to meet the vehicle thrust requirement when the TBCCs are incapable of providing sufficient thrust. Naturally, due to the low specific impulse (Isp) of rockets, the economic performance of the vehicle seems to be reduced. However, in flight, the acceleration increase by the rockets reduces the flight time and results in fuel saving for the TBCC engine. Therefore, the rockets used for the thrust augmentation have a mixed effect on the total economic performance. In addition, the acceleration characteristics are related to trajectory features; for example, the trajectory of the “climb-dive” (or gravity-assist) is used to overcome the transonic thrust pinch in the acceleration phase. Thus, the tradeoff should take into consideration the trajectory features in the thrust pinch regions. The rocket-matching analysis based on trajectory optimization is necessary for the integrated design of the rocket-augmented TBCC. Trajectory optimization methods have been studied for decades [13,14,15]. The Gauss pseudospectral method as a direct method has the ability to obtain accurate estimates of the state, costate, and control for continuous time optimal control problems [16] and has been successfully implemented in trajectory optimization problems of supersonic or hypersonic vehicles [17,18]. Thus, the Gauss pseudospectral method could be directly used to evaluate the integrated design of a rocket-augmented TBCC under different rocket schemes, and the results from trajectory optimization could be used for choosing the appropriate combination form for rockets and TBCCs for vehicle systems.

The paper is organized as follows: Section 2 will give a problem statement of the rocket-augmented strategy from the aspect of efficiency. In Section 3, an integrated analysis model for a rocket-augmented TBCC-powered vehicle with the trajectory optimization method of the Gaussian pseudospectral is established. Section 4 analyzes the performance of the vehicle from the aspect of efficiency and energy, and the rocket thrust is optimized in a specified vehicle case. Based on the above understanding, the parameter study of different vehicles and rockets is discussed in Section 5, and the combination form for rockets and TBCCs is given. Finally, Section 6 summarizes this paper.

2. Problem Statement

TBCC vehicles suffer from two main thrust pinches [1], as shown in Figure 1a. One is the transonic thrust pinch, and the other is the transition thrust pinch. The thrust pinches are limited by the current performance level of vehicle and TBCC engines. The thrust pinch issues can be described as flight with small acceleration and long acceleration times, which can be measured by vehicle efficiency, ${\eta }_{veh}$ [18]:

$${\eta }_{veh}=∆E_{veh}/\left(TVcos\alpha \right)$$

(1)

$∆E_{veh}$ represents the vehicle work:

$$∆E_{veh}=\left(Tcos\alpha -D-mgsinr\right)V$$

(2)

where $T$ and $D$ denote the thrust and drag, respectively. $\alpha$ and $r$ denote angle of attack and the flight path angle, respectively. $m$ denotes the mass of the vehicle, and V denotes the flight velocity.

As shown in Figure 1b, the reference ${\eta }_{veh}$ curve could effectively reflect the acceleration efficiency characteristics corresponding to Figure 1a. It can be seen that the vehicle efficiency in the thrust pinch region is very low.

For wide-speed flight, the TBCC has the distinct advantage of utilizing oxygen in the air instead of carrying it onboard, like a rocket [19]. This results in the higher Isp of TBCC engines compared to rockets, as shown in Figure 2a [20]. Similarly, the engine efficiency ${\eta }_{enigne}$ can be formulated with the Isp [21]:

$${\eta }_{engine}=\frac{gV}{h_{PR}}·Isp$$

(3)

where $h_{PR}$ denotes the heating value of the fuel.

As shown in Figure 2b, the rockets’ engine efficiency is obviously lower than that of the air-breathing engines as well, such as turbojets, ramjets, and scramjets. However, rockets have significantly higher unit frontal area thrust (or unit weight thrust). Besides, the thrust of rockets increases gradually as the flight altitude increases, while the air-breathing engines have the opposite trend due to the reduction in air density. Therefore, a smaller rocket may generate enough thrust to accelerate the vehicle when going through the thrust pinch region, especially at high altitudes.

According to the description of Figure 1 and Figure 2, introducing a small rocket to assist the TBCC could significantly increase the vehicle efficiency, ${\eta }_{veh}$, in the transonic and mode transition. Although the engine efficiency, ${\eta }_{enigne}$, may be reduced when the rocket works, the rocket-augmented strategy is still possible to improve the local total efficiency ${\eta }_{tot}$, as shown in Figure 3. It is to be noted that ${\eta }_{tot}$ is defined as follows:

$${\eta }_{tot}={\eta }_{veh}·{\eta }_{engine}$$

(4)

This analysis shows that introducing a rocket as the auxiliary power is not only a practical solution to enable flight over a wide-speed range when the TBCC is incapable but also probably an economical scheme when the TBCC meets the requirements of thrust.

3. Integrated Analysis Method

3.1. Description of the Vehicle Model

The baseline vehicle is assumed to have a take-off weight of 19,050 kg and a wing loading of 460 kg/m². As a result of a lack of performance data for currently available TBCC-powered vehicles, the aerodynamic coefficients in high Mach (>1.5) refer to X − 43 [22], and the data for low Mach (≤1.5) are obtained from the reference [23]. It is noted that the low-Mach data in the case of X − 43 are not suitable for showing the transonic thrust pinch.

For the trajectory optimization in the conceptual design, a point mass model for motion in a vertical plane is usually quite adequate [23].

The equations of motion for this model [24] are given by:

$$\dot{h}=Vsin\gamma$$

(5)

$$\dot{V }=\frac{Tcos\alpha - D}{m} - gsin\gamma$$

(6)

$$\dot{\gamma }=\frac{Tsin\alpha +L}{mV}+\left(\frac{V}{r} - \frac{g}{V}\right)cos\gamma$$

(7)

$$\dot{m}= - \frac{T}{g·Isp}$$

(8)

The lift L and drag D are defined as:

$$L=qS{C}_L\left(\alpha ,Ma\right)$$

(9)

$$D=qS{C}_D\left(\alpha ,Ma\right)$$

(10)

where q is the flight dynamic pressure and S is the reference area of the vehicle. The lift coefficient $C_L$ and the drag coefficient $C_D$ are the interpolation functions of the attack angle $\alpha$ and the Mach number.

3.2. Description of the Engines Model

The TBCC engine model adopts the lapping combination of the Ma 4.4 turbine and the HRE scramjet from the Air Force Research Laboratory (AFRL) [22]. The ratio of the number of turbine engines to ramjet engines is 20:1, and the takeoff thrust/weight ratio is 1.0. The thrust and the Isp are obtained by interpolating the altitude and the velocity data table. To highlight the “thrust trap” feature of the mode transition, the thrust and the Isp are calculated by Equations (11) and (12) [25], respectively:

$$T\left(\lambda ,M\right)=T_{turb}{\left(\frac{M_2 - M}{M_2 - M_1}\right)}^{\lambda }+T_{ram}{\left(\frac{M - M_1}{M_2 - M_1}\right)}^{\lambda },M\in \left[M_1,M_2\right]$$

(11)

$$Isp\left(\lambda ,M\right)=\frac{T\left(\lambda ,M\right)}{\left(\frac{T_{turb}}{Is{p}_{turb}}{\left(\frac{M_2 - M}{M_2 - M_1}\right)}^{\lambda }+\frac{T_{ram}}{Is{p}_{ram}}{\left(\frac{M - M_1}{M_2 - M_1}\right)}^{\lambda }\right)},M\in \left[M_1,M_2\right]$$

(12)

where $M_1$ (4.0) and $M_2$ (4.4) are the start and the end Mach numbers of the mode transition, and the parameter $\lambda$ is the pinch coefficient and can measure the minimum thrust in mode transition. In this paper, the $T\left(\lambda ,\frac{M_1+M_2}{2}\right)$ is equal to the 2/3 of $T\left(1,\frac{M_1+M_2}{2}\right)$, in which the $\lambda$ is correspondingly given as 1.585.

The Isp of the baseline rocket is 300 s. The thrust of the baseline rocket is adjustable, and the adjustable thrust is the product of the rocket throttle, $Thr$, and the maximum of thrust, $T_{Rocket\_max}$:

$$T_{Rocket}=T_{Rocket\_max}·Thr$$

(13)

where $T_{Rocket\_max}$ is defined as the product of the vehicle takeoff weight, $m_{to}$, and the maximum relative thrust of the rocket, $R{T}_{max}$:

$$T_{Rocket\_max}=m_{to}·g·R{T}_{max}$$

(14)

In addition, introducing the rocket increases the frontal area of the vehicle and the weight of the propulsion system, which influences the performance of the vehicle. In this study, the rocket weight is estimated at 1/50 of the maximum thrust. The drag coefficient is used to estimate the performance affected by the frontal area [12].

3.3. Integrated Trajectory Optimization Method

3.3.1. Optimization Problem

According to both the vehicle model and the engine model, the flight altitude h, speed V, flight path angle $\gamma$, vehicle mass m, and attack angle $\alpha$ are set as state parameters. Meanwhile, the gradient of the attack angle $\dot{\alpha }$ and rocket throttle ratio $thr$ are set as control variables. It is noted that the attack angle $\dot{\alpha }$ is aimed at smoothing flight control. Based on the vehicle control equations, the optimization problem of the ascent trajectory is formulated with the corresponding cost function and parameter constraints.

The objective is to determine the minimum-fuel (minF) trajectory and control from takeoff to a specified speed and altitude. Besides, the minimum-time (minT) trajectory optimization problem is analyzed as a comparative case. The cost functions are formulated as

$$J_{min,fuel}=- m\left(t_f\right)$$

(15)

$$J_{min,time}=t_f$$

(16)

The boundary conditions and constraints are shown in

Table 1. Boundary conditions and constraints.

Value Type	Time	Control Parameter		State Parameter
	t/s	$\dot{\mathit{\alpha }}$(deg/s)	thr	h/km	V/(m/s)	$\mathit{\gamma }$/deg	m/kg	$\mathit{\alpha }$/deg
Initial	0	Free	Free	0	129.314	0	19,050.8	6
Terminal	Free	Free	Free	Free	1495.28	0	Free	Free
Min	0	−0.5	0	0	5	−40	22	−5
Max	1500	0.5	1	30	1794	40	19,050.8	20

.

Additionally, the operating range of dynamic pressures, q, is also restrained in different flight phases.

$$\{\begin{array}{c}10kPa\le q\le 75kPa, Ma\le M_1,Ma>M_2\\ 40kPa\le q\le 75kPa, M_1<Ma\le M_2 \end{array}$$

(17)

3.3.2. Gauss Pseudospectral Method

The optimization problem above can be converted to a continuous Bolza problem by using the Gauss pseudospectral method (GPM). The GPM constructs the Lagrange interpolation polynomials on a set of Legendre points to approximate the state variables and control variables of the system, so the continuous-optimal control problem can be transcribed into a nonlinear planning problem (NLP). Studies have shown that Karush–Kuhn–Tucker (KKT) multipliers of the NLP can be utilized to estimate the costate at both the Legendre-Gauss points and the boundary points accurately, which is due to the equivalence between the KKT conditions and the discretized first-order necessary conditions [12]. In this paper, the trajectory optimization problem is solved by the general software package GPOPS; more details of this algorithm are discussed in [26,27].

4. Efficiency Analysis and Rocket Optimization

According to the predictions in the problem statement, the engine efficiency is depressed when the rocket is introduced, yet the vehicle efficiency in thrust pinches may be improved significantly. With overall consideration, introducing a rocket is still possible to increase the total efficiency. In this section, the efficiency analysis of the trajectory with the rocket is performed on the baseline TBCC-powered vehicle by using the integrated analysis method. However, introducing a rocket increases the frontal area of the vehicle and the weight of the propulsion system. Taking the above factor into account, the rocket with an appropriate thrust will yield the optimal performance. Therefore, the rocket optimization is also studied in this section.

Figure 4 shows the trajectories of the baseline vehicle with minF and minT as the optimization objectives, and of the rocket-augmented TBCC-powered vehicle with minF as the optimization objective. Simulations are performed from Mach 0.38 to 5 with the constraints given in Table 1. For the baseline vehicle, the minF path shows that the vehicle goes through transonic and mode transition in the way of rapid climb and dive, and flies at the constant dynamic pressure of 75 kPa in most of the remaining voyage. By contrast, the minT path shows a relatively lower climb altitude during the transonic and without “climb-dive” during mode transition. It is indicated that the “climb-dive” trajectory strategy is an efficient means of fuel saving in the thrust pinch regions. It is also necessary when going through transonic in the path of minT, which has been discussed in J. Zheng et al.’s work [25].

Based on the TBCC model, the rocket with an RT_max of 10% is used to optimize the trajectory of minF, shown as the dashed line in Figure 4. Interestingly, the rocket does not work in the transonic region, but only turns on during the mode transition. Moreover, the vehicle firstly climbs to the limit boundary of the minimum dynamic pressure of 40 kPa and then takes advantage of the gravity-assist to accelerate during the mode transition. Due to the change in trajectory when the rocket works, the local performance of vehicle might be changed.

Figure 5 shows a detailed comparison of the flight time of the three trajectories above. For the TBCC-powered vehicle, the times taken to reach Mach 5 of minF and minT are 719.2 s and 614.5 s, respectively. The time difference between the minF and minT closely depends on the different trajectory strategy in the transonic. The “climb-dive” trajectory strategy for fuel saving takes a great amount of time. The total time of the rocket-augmented minF is 648.4 s, which is 9.8% less than that of the baseline minF. The time difference mainly depends on the different accelerations in the mode transition. In addition, as a result of the flight drag caused by the rocket, the trajectory of the minF with rocket takes slightly more flight time than the baseline minF to accelerate to the same Mach number.

The throttle control of the rocket is also shown in Figure 5. The rocket works at 495.8 s and lasts for 34.3 s. The corresponding Mach number ranges from 4.12 to 4.39. By contrast, the TBCC-powered vehicle without a rocket takes 119.5 s to flight over the same Mach range. In addition, the rocket throttle is allowed to vary freely between 0% and 100% in the trajectory optimization; however, the results show that the throttle mostly remains in the two states of 0% and 100%. This indicates that the adjustment of thrust is not necessary. Therefore, the fixed-thrust design could satisfy the thrust requirement and simplify the structure of the rocket.

Figure 6 shows the vehicle mass variations with the flight Mach. The total fuel consumption of the minT, the minF, and the rocket-augmented minF to reach Mach 5 are 4992.8 kg, 4568.4 kg, and 4444.2 kg, respectively. The fuel saving is 424.4 kg (8.5%) from the minT to the minF, and a further 124.2 kg (2.7%) from the minF to the rocket-augmented minF. Due to the gravity-assist, compared with the minT, the minF saves 415.3 kg of fuel in the process of transonic, contributing 97.9% of the total fuel saving. With the assistance of the rocket, an additional 201.6 kg of fuel is saved during the mode transition, which improves the economic performance of the TBCC vehicle even with the additional fuel consumption resulting from the drag of the rocket. If the weight of the rocket is considered as a part of the fuel consumption, the fuel saving is 86.1 kg (1.9%). Based on these results, it appears that the rocket-augmented TBCC is a better option to promote economic performance than the TBCC without a rocket for wide-speed range vehicles.

The TBCC-powered vehicle assisted by a rocket could improve both the flight timeliness and the economic performance, which could be explained in terms of the energy and efficiency.

From the perspective of energy, the vehicle climbs from a low-potential and low-kinetic state to a high-potential and high-kinetic state. In the process of ascending, besides increasing kinetic energy and potential energy, the thrust provides the most of the energy to overcome the drag. For instance, the drag dissipates 35.45 GJ of energy in the trajectory of the minT, while the increases in the kinetic energy and the potential energy are 15.56 GJ and 3.42 GJ, respectively. By contrast, the drag dissipates 31.31 GJ of energy in the trajectory of the minF, which is 11.7% less than that of the minT. Under the assistance of the rocket in the minF, the energy consumption of drag is reduced from 31.31 GJ to 26.58 GJ. The difference in the drag dissipation between the minF and the minT mainly lies in the transonic region, as shown in Figure 7a. The energy of the drag has a significant reduction from the minT to the minF, which results from the gravity-assist strategy. After climbing to a higher altitude, the vehicle flies with a smaller drag and lower energy consumption. Meanwhile, the thrust is reduced and the Isp of the engine remains almost constant in the process of ascending, which is one of the reasons for the fuel saving. Since the gravitational potential energy is comparable to the kinetic energy in the transonic region, the gravity-assist strategy could achieve an efficient increase in the kinetic energy from the conversion of the potential energy. However, in the mode transition, the potential energy is much less than the kinetic energy and the gravity-assist strategy is inefficient, as shown in Figure 7b. With the assistance of the rocket, the time spent in the mode transition is reduced from 164.5 s to 73.2 s, and the energy consumption of drag is reduced from 9.73 GJ to 3.47 GJ. The results indicate that the rocket-augmented scheme seems to be a better choice for the reduction of drag dissipation in mode transition.

Figure 8 shows the efficiency profiles of the two trajectories of the minF and the rocket-augmented minF. As shown in Figure 8a, the vehicle efficiency, ${\eta }_{veh}$, has a sharp decrease before Mach 0.9 due to the rapid dropping of acceleration in the phase of the climb. In the final stage of climb, the thrust is not able to accelerate the vehicle, so the value of ${\eta }_{veh}$ is negative. With the rapid dive in transonic, the ${\eta }_{veh}$ increases first and then decreases, under the combined effect of drag, thrust, and gravity. By contrast, the variation of the ${\eta }_{enigne}$ is slight, because the Isp is insensitive to altitude.

However, when the rocket works, the ${\eta }_{veh}$ is improved and the ${\eta }_{enigne}$ is reduced during the mode transition. Additionally, the ${\eta }_{tot}$ is optimized, especially in the excessively inefficient region around Ma 4.3. In order to evaluate the effectiveness of the performance optimization, the average vehicle efficiency, engine efficiency, and total efficiency are respectively defined as:

$${\overline{\eta }}_{veh }=\frac{{\displaystyle \int }\Delta E_{veh}dt}{{\displaystyle \int }TVcos\alpha dt}$$

(18)

$${\overline{\eta }}_{engine}=\frac{{\displaystyle \int }TVcos\alpha dt}{{\displaystyle \int }{h}_{PR}\dot{m_f}dt}$$

(19)

$${\overline{\eta }}_{tot}=\frac{{\displaystyle \int }\Delta E_{veh}dt}{{\displaystyle \int }{h}_{PR}\dot{m_f}dt}$$

(20)

where $\dot{m_f}$ denotes the fuel mass flow rate.

These variables in Equations (18)–(20) are defined in Section 2. The ${\overline{\eta }}_{veh}$ increases by 9.9% (from 0.415 to 0.456) after the introduction of the rocket, and the ${\overline{\eta }}_{engine}$ is reduced by 6.8%, resulting in an increase of 2.6% in the ${\overline{\eta }}_{tot}$. Therefore, introducing a suitable rocket to assist the TBCC increases the vehicle efficiency in the mode transition, which results in an increase in the total efficiency despite the reduction in engine efficiency.

It seems that the greater the thrust of the rocket, the better the vehicle efficiency. However, the increase in rocket thrust will inevitably bring increased drag and additional weight. The two factors should be considered in the thrust design for the rocket for optimal performance.

In Figure 9, when the rocket thrust is relatively small (RT_max < 6%), as the thrust of the rocket increases, the fuel consumption of the TBCC reduces rapidly. Even if the fuel consumption and the weight of the rocket are increased, the total weight of consumption is still reduced. It is noted that the total weight in Figure 9 is the sum of the fuel consumption of the TBCC, the fuel consumption of the rocket, and the weight of the rocket. As the thrust increases (RT_max > 6%), the fuel consumption of the TBCC increases slowly, which is due to the increase in the drag. In the meantime, the weight and the fuel consumption of the rocket increase almost linearly. Consequently, the total weight of consumption is at a minimum value when the relative thrust of the rocket is about 6%, which is 2.2% less than the total weight consumed by the vehicle without the rocket.

According to Equations (18) and (19), Figure 10 shows the variations in the average efficiencies with the rocket thrust. With the rocket RT_max increasing from 0% to 8%, the ${\overline{\eta }}_{veh}$ has a sharp increase (9.6%) while the ${\overline{\eta }}_{engine}$ drops by about 6%, which results in a 3% increase in the ${\overline{\eta }}_{tot}$. With the rocket RT_max further increasing (RT_max ≥ 8%), the ${\overline{\eta }}_{veh}$ shows a slightly decreasing trend and the ${\overline{\eta }}_{engine}$ keeps decreasing, so the ${\overline{\eta }}_{tot}$ starts to decrease.

With the assistance of the rocket, the time spent and the energy consumption of drag in the mode transition could be reduced significantly. Therefore, introducing an appropriate rocket to assist the TBCC-powered vehicle could increase vehicle efficiency in the mode transition, which results in an increase in total efficiency despite the reduction of engine efficiency. For a specified TBCC-powered vehicle, the thrust of the rocket has an optimal value to maximize the average total efficiency. The rocket with an overly small thrust could not improve the average vehicle efficiency sufficiently, but an overly large thrust might lead to excessive drag.

5. Parameter Study and Integrated Design

As mentioned previously, the performance of a specified TBCC-powered vehicle could be optimized by introducing the rocket. The essence of the optimization is the tradeoff between the efficiency and engine efficiency. Considering that vehicle drag and rocket Isp are respectively related to vehicle efficiency and engine efficiency, the optimization of the thrust and operating time of the rocket should involve the influence of vehicle drag and rocket Isp. The thrust, Isp, and operating time of the rocket would determine the combination form (“T+RBCC” or “R+TBCC”) of the rocket and TBCC. In this section, the study of the vehicle drag and the rocket Isp is conducted, and the combination form for the rocket and TBCC is also discussed.

Figure 11 shows the total weight consumption as a function of rocket thrust under different drag coefficients. To represent vehicles with different drag coefficients, AF_D is introduced in the present study and defined as the amplification factor of the drag coefficient of a specified vehicle to that of the baseline vehicle. Under an AF_D of 1.0 and 1.1, the TBCC engines could propel the vehicle to accelerate in the wide-speed range without a rocket. As the AF_D is increased, the TBCC thrust is unable to overcome the drag in the thrust pinch regions. Consequently, the rocket is introduced to augment the thrust. For each TBCC-powered vehicle, the thrust of the rocket has an optimal value to minimize the total weight consumption. With the increase in the AF_D, the minimum value and the optimal value of the relative thrust both increase. Although the introduction of rockets could propel the vehicle to accelerate in the wide-speed range, the total weight consumption in the process of acceleration increases with the increase in drag.

The optimal relative thrust of the rocket is presented in Figure 12 under various AF_D and Isp. In the low-AF_D cases (such as AF_D = 1, 1.1), the optimal relative thrust is insensitive to the Isp. A possible reason is that the rocket works for a short time and the consumption is relatively small. Compared to the total fuel consumption, the fuel reduction caused by the increase in the rocket Isp could be negligible. In the high-AF_D cases (such as AF_D = 1.3, 1.4), the changes in the optimal relative thrust with Isp are obvious, because the operating time of the rocket is too long to ignore the impact of Isp. The higher the Isp, the less the reduction in the engine efficiency caused by the rocket, and the more the increase in vehicle efficiency. Thus, the optimal relative thrust increases to satisfy the requirement of higher vehicle efficiency. The results suggest that it is not necessary to pay much attention to the Isp of the rocket in the case of low drag while the Isp is important in the case of high drag.

Figure 13 shows the thrust profiles of the rocket under different AF_D over the entire Mach range. To compare the thrust, the combined variable of AF_D and rocket relative thrust RT is on the horizontal axis. It is to be noted that the RT is the product of the maximum relative thrust, RT_max, and the throttle, Thr. The distance from the solid line to the dotted line represents the relative thrust of the rocket. With the increase in AF_D, the operating range of the rocket expands in transonic and mode transition, and the value of the optimal thrust increases. This could be explained in that the expansion and deepening of the thrust gaps with the increase in the AF_D call for a bigger thrust and a longer operation time of the rocket. In addition, under each AF_D, the rocket almost operates at the corresponding maximum thrust constantly in the operating range. It means the fixed-thrust design of the rocket could satisfy the thrust requirement.

Similarly, Figure 14 shows the thrust profiles of the rocket under different Isp over the entire Mach range. Under the AF_D of 1.1 but different Isp, the maximum values of the thrust are approximately equal. The higher the Isp, the less the reduction in the engine efficiency caused by the rocket. Therefore, the operating ranges of the rocket are widened. In the case of Isp = 240 s, the rocket hardly works in the transonic, while in the case of Isp = 360 s, it operates in the range of Mach 1.18 to 1.56.

No matter what AF_D and Isp are, the thrust is basically equal to the two values of 0 and maximum. This indicates that the thrust adjustment of the rocket is not necessary, and the fixed-thrust design could satisfy the thrust requirement and simplify the structure of the rocket.

Figure 15 shows the average vehicle efficiency, ${\overline{\eta }}_{veh}$, and the average engine efficiency, ${\overline{\eta }}_{engine}$, corresponding to different Isp and AF_D. With the increase in Isp, the ${\overline{\eta }}_{veh}$ increases due to the expansion of the rocket operating range, while the ${\overline{\eta }}_{engine}$ is roughly constant after including the effect of the improved engine efficiency of the rocket. When AF_D = 1.4, as the Isp increases from 240 s to 360 s, the ${\overline{\eta }}_{veh}$ is improved from 0.368 to 0.453, an increase of 23.1%. By comparison, when AF_D = 1.0, the Isp has little effect on the ${\overline{\eta }}_{engine}$ and the ${\overline{\eta }}_{veh}$, because the operating range of the rocket is very small. With the increase in AF_D, the ${\overline{\eta }}_{veh}$ decreases first and then increases. In low-AF_D cases, the TBCC provides most of the thrust for acceleration, and the rocket works within a narrow range, serving as auxiliary power. The increase in drag decreases the ${\overline{\eta }}_{veh}$. By comparison, in high-AF_D cases, the TBCC works independently in a narrow range. The rocket widens the operating range and increase the thrust of itself, and becomes a great part of the propulsion system. Due to the increase in rocket thrust, the ${\overline{\eta }}_{veh}$ increases.

Figure 16 shows the average total efficiency, ${\overline{\eta }}_{tot}$, and the total energy consumption of drag, E_Drag, corresponding to different Isp and AF_D. With the increase in Isp, the ${\overline{\eta }}_{tot}$ increases, because the ${\overline{\eta }}_{veh}$ increases, and the ${\overline{\eta }}_{engine}$ remains almost unchanged. Meanwhile, the operating range of the rocket is widened and the E_Drag is reduced. When AF_D = 1.4, as the Isp increases from 240 s to 360 s, the ${\overline{\eta }}_{tot}$ is improved from 0.066 to 0.079, namely an increase of 19.7%, and the E_Drag is decreased from 35.37 GJ to 24.71 GJ, namely a decrease of 30%.

The increase in AF_D decreases the ${\overline{\eta }}_{tot}$. Moreover, with the increase in AF_D, the E_Drag increases first and then decreases. The role played by the rocket accounts for this trend of E_Drag. When the rocket serves as an auxiliary power at low AF_D, the increase in AF_D decreases the acceleration, resulting in an increase in the E_Drag. When the rocket serves as a great part of the propulsion system at high AF_D, the increase in AF_D widens the operating range of the rocket and increases the thrust of the rocket, which results in a decrease in the E_Drag.

Figure 17 presents the total weight consumption ratio and flight time under different Isp and AF_D. With the increase in Isp, the total weight consumption and the flight time are gradually reduced, because the ${\overline{\eta }}_{tot}$ increases and the E_Drag decreases. As the AF_D increases, the total weight consumption is increased.

The effect of Isp on weight consumption and flight time varies considerably under different AF_D. Therefore, this network map can be roughly divided into two areas: low-drag area and high-drag area. In the low-drag area, the rocket works for a short time and the optimal relative thrust is small. The consumption of the rocket is relatively small compared to the total fuel consumption. It is concluded that the improvement of the Isp of the small rocket is not necessary. In this case, the small rocket could serve as an additional booster, whose structure is simple. The combination of the rocket and TBCC engine is similar to the “R+TBCC”. By contrast, in the high-drag area, the operating time of the rocket is long and the optimal relative thrust is large. The Isp has a significant impact on the flight time and the total fuel consumption. It is necessary to pay attention to the Isp of the big rocket. Introducing a big rocket as a booster might bring a large base drag. Instead, the rocket should be integrated into the TBCC engine, such as an RBCC or ERJs, which could deliver augmented thrust at an Isp performance of up to twice or triple that of a rocket through proper design. The combination of the rocket and TBCC engine is similar to the “T+RBCC”.

According to the parameter study of the vehicle drag and rocket Isp, two design recommendations for rocket-augmented TBCC engines corresponding to the low-drag area and the high-drag area are as follows. In the low-drag area, the rocket works for a short time and its optimal relative thrust is small, so optimizing the Isp of the small rocket is not necessary. The TBCC combined with a booster rocket could be a more simple and suitable scheme. By comparison, in the high-drag area, the operating time of the rocket is long and the optimal relative thrust is large. The Isp has a significant impact on the flight time and the total fuel consumption. Therefore, combining an RBCC (or ERJs) and a turbine could produce higher economic performance.

6. Conclusions

In this study, an integrated analysis method for rocket-augmented TBCC-powered vehicles was proposed based on the trajectory optimization method of the Gauss pseudospectral. The trajectories of the baseline vehicle with and without a rocket were analyzed from the aspect of efficiency and energy. The combination form for rockets and TBCC was discussed with the parameter study of vehicle drag and rocket Isp. Accordingly, the following conclusions were obtained.

(1)

Introducing an appropriate rocket to assist the TBCC-powered vehicle could reduce the total energy consumption of drag and increase the vehicle efficiency in the transonic and the mode transition. It results in an increase in total efficiency in spite of the reduction in engine efficiency. Therefore, introducing the rocket as the auxiliary power is not only a practical solution to enable flight over a wide-speed range when the TBCC is incapable but also an economical scheme when the TBCC meets the requirements of thrust.

(2)

For a specified TBCC-powered vehicle, there is an optimal thrust of the rocket to maximize the economic performance. That is to say, the rocket with an overly small thrust could not improve the vehicle efficiency sufficiently, while a rocket with an overly large thrust might lead to excessive drag. In addition, the rocket almost stays at the maximum thrust constantly when it works. It means the fixed-thrust design of the rocket could satisfy the requirement.

(3)

When the vehicle drag is low, the rocket works for a short time and its optimal relative thrust is small, thus optimizing the Isp of the small rocket is not necessary. The TBCC combined with a booster rocket will be a very simple and suitable scheme. When the vehicle drag is high, the operating time of the rocket is long and the optimal relative thrust is large. The Isp has a significant impact on the flight time and the total fuel consumption. Therefore, the combination form for an RBCC (or ERJs) and the turbine will be more appropriate to obtain higher economic performance.