Analysis of Dynamic Operating Characteristics of a Pulse Detonation Turbine Engine

Abstract

In order to obtain the dynamic operating characteristics of a pulse detonation turbine engine (PDTE), a transient model is established considering the shaft dynamics and the volumetric effect of the components in the PDTE. The accuracy of the model is verified with the experimental data from a pulse detonation prototype engine. The deviations between the calculated data and the experimental results are no more than 7.61%. The numerical results show that the operating state of the PDTE will change gradually with the increase of the fuel flow rate or the firing frequency. The quasi-steady state calculations show that there are maximum values for the rotor speed, thrust, and specific thrust when the fuel flow rate is increased from 0.0056 kg/s to 0.0129 kg/s at firing frequency of 10 Hz. The rotor speed, thrust, and specific thrust will suddenly decrease due to the over rich of fuel in the PDC with the increasing of the fuel flow rate. The specific fuel consumption has a minimize value during this process. When the firing frequency is increased from 7 Hz to 18 Hz at a fixed fuel flow rate, the performance parameters such as the thrust, specific thrust, and specific fuel consumption have a similar variation trend. The extremum values of the performance parameters are obtained at firing frequency of 12 Hz. For the dynamic operating process of the PDTE, parameters such as the rotor speed or pressure ratio of the compressor are increased in a cyclic oscillation way when the fuel flow rate or the firing frequency is changed.

1. Introduction

The pulse detonation engine (PDE) has the advantage of high thermodynamic efficiency due to the low entropy generation and pressure gain combustion process in the pulse detonation combustor (PDC) [1,2]. Relevant research showed that the thermodynamic efficiency of the PDE is 49%, which is much higher than the traditional gas turbine engine based on constant pressure combustion. Therefore, the pulse detonation turbine engine (PDTE) which utilizes the PDC to replace the constant pressure combustor in the traditional gas turbine engine is proposed to further improve the specific performance of the traditional gas turbine engine [3,4].

Due to the potential advantages of PDTE in terms of thermodynamic efficiency, many studies have been carried out to predict the theoretical performance of the engine. Goldmeer et al. [5] conducted a detailed study on PDTE performance; the results showed that the PDTE had higher thermal efficiency. Petters et al. [6] used the NPSS program to analyze the performance of the PDTE. The results indicated that the specific fuel consumption of the engine was decreased by 11% at the flight altitude of 35,000 ft and the Mach number of 0.85 after replacing the combustor of a high bypass ratio turbofan engine with the PDC. Venkat et al. [7] proposed an equivalent performance model of the PDC and then estimated the performance of the PDTE based on the model. The results stated that the efficiency of turbine components had a great impact on the thermal efficiency of the PDTE. Andrus et al. [8] evaluated the performance of a pulse detonation turbofan engine of large bypass ratio. The results showed that the specific fuel consumption of the PDTE would be reduced by 5–11% when compared with the original turbofan engine. Qiu et al. [9] established a model of detonation turbine-based combined cycle engine (DTBCC), which used the flow at the head of the PDC to drive the turbine, and the flow at the tail of the PDC was used to generate thrust. The flow into the turbine can be adjusted by regulating the valve at head of the PDC. The results showed that when the compressor pressure ratio was 2–20, the DTBCC had higher unit thrust and lower specific fuel consumption. Lu et al. [10] established an equivalent performance model of the PDTE and verified the accuracy of the model through a pulse detonation prototype engine. The calculation results showed that the performance of the ideal PDTE was much better than that of the traditional gas turbine engine. Cleopatra et al. [11] proposed a method for calculating thermal performance of the detonation engine based on the ZND model. The results showed that the specific power of the engine was higher than that of the Brayton cycle engine. Carlos et al. [12] proposed a PDC model and integrated the model within GESTPAN to evaluate the performance of three types of PDTEs: a normal PDTE, a PDTE with intercooling, and a PDTE with after-cooling. The results showed that the thermal efficiency of the normal PDTE was higher than that of the Brayton cycle engine. It demonstrated that the intercooling technology improved the cycle performance for compressor pressure ratio above 10 and contributed to an increase in specific power.

In addition to the theoretical analysis of the PDTE, many experimental studies have also been carried out to further promote the application of the PDTE. Lu et al. [13] built up a multi-tube PDC and a centrifugal compressor system. The experiment results showed that the compressor was operated closer to the surge boundary when the multi-tube PDC was fired. George et al. [14,15,16,17] designed a six-tube PDC integrated with an axial flow turbine. Interactions between the two components were experimentally investigated. The results showed that the error of using the mass averaging method to evaluate turbine efficiency was smaller than that of using time average method. When the peak pressure of the detonation gas was high, the turbine efficiency would be reduced although the specific power was higher. It was found that the thermal efficiency of detonation cycle was still higher than that of Brayton cycle under the case of serious loss of turbine efficiency. Rouser et al. [18,19,20] used a turbocharger to study the interactions between the PDC and a centrifugal turbine. The experimental results showed that the peak pressure of detonation gas decreased significantly after passing through the turbine. Li et al. [21] and Qiu et al. [22] established a pulse detonation turbine prototype engine. The experimental results showed that the transfer structure between the PDC and turbine would affect power extraction of the turbine.

The above investigations confirmed the feasibility and performance advantages of the PDTE from the theoretical and experimental perspectives. However, the above equivalent model is unable to reveal the dynamic operating characteristics of the PDTE and cannot be used in the design of the PDTE control system. In order to build up a model which can be adopted in the control system of the PDTE, a transient model is established and verified with the experimental data in the present paper. Then, the unsteady operating characteristics of the PDTE are analyzed based on the transient model. This paper will offer a useful tool for the transient performance analysis of the PDTE and the control system design of the PDTE.

2. Model of the PDTE

The model of the PDTE consists of a compressor, a transition section, multi-tube PDCs, a turbine, and a convergent nozzle with fixed outlet area. Figure 1 shows a schematic model of the PDTE. The PDTE is operated in an unsteady state at any time due to the cyclic operated PDC. However, the unsteady operating state of the PDTE can be divided into two sorts: quasi-steady state and dynamic state. The PDTE is in a quasi-steady state when the mass flow rate of fuel and firing frequency remain unchanged. For this state, the average parameters of the PDTE stay in constant values. When the mass flow rate of fuel or the firing frequency is changed, the engine is operated in a dynamic operating state.

The balance equations of the PDTE are established based on the conservation of energy and mass flow rate. The volumetric effect and the shaft dynamics are considered in the transient model. The volume of the compressor is included in the transition section. Since the volume of the turbine is small, its length and volume are considered in the modeling of the PDC. In addition, the following hypotheses are made in the transient model.

(1)

There is an ideal valve at the head of the PDC. The valve is closed when the pressure in the PDC is higher than the outlet pressure of the transition section. When the PDC is fired, the detonation products of high pressure are completely isolated by the ideal valve.

(2)

If the operating point of the compressor exceeds the surge boundary during the firing phase of the PDC, the mass flow rate of the compressor is supposed to be 0, and the air in the transition section will not flow back.

(3)

All the components in the PDTE are supposed to be adiabatic.

The transition model of each part of the PDTE is shown as follows.

2.1. Model of the Compressor

When the total pressure loss in the inlet is ignored, the compressor inlet parameters can be calculated based on the following equations.

$$p_{t2}=p_{t0}=p_0{\left(1+\frac{{\gamma }_0-1}{2}M{a}^2\right)}^{\frac{{\gamma }_0}{{\gamma }_0-1}}$$

(1)

$$T_{t2}=T_{t0}=T_0\left(1+\frac{{\gamma }_0-1}{2}M{a}^2\right)$$

(2)

Then, the compressor outlet parameters can be obtained by the following equations.

$$p_{t21}=p_{t2}{\pi }_c$$

(3)

$$T_{t21}=T_{t2}\left[1+\left({\pi }_c{}^{\frac{{\gamma }_{21}-1}{{\gamma }_{21}}}-1\right)/{\eta }_c\right]$$

(4)

$$N_c=W_2{c}_p{T}_{t2}\left[\left({\pi }_c{}^{\frac{{\gamma }_{21}-1}{{\gamma }_{21}}}-1\right)/{\eta }_c\right]$$

(5)

2.2. Model of Transition Section

The parameters in the transition section are supposed to be equally distributed in order to simplify the modeling process. The parameters in the transition section can be calculated based on the following equations.

$$\frac{d{p}_{t31}}{dt}=\rho R\frac{d{T}_{t31}}{dt}+\frac{R{T}_{t31}}{V}\left(W_3-W_{31}\right)$$

(6)

$$\frac{d{T}_{t31}}{dt}=\frac{R{T}_{t31}}{p_{t31}V}\frac{1}{c_v}\left[\left(W_3{h}_{t3}-W_{31}{h}_{t31}\right)-\frac{h_{t31}}{{\gamma }_{31}}\left(W_3-W_{31}\right)\right]$$

(7)

Since the temperature changes slowly and has little influence on the calculation of the pressure, the temperature derivative in Equation (6) can be ignored so as to simplify the calculation process in the present model.

2.3. Model of PDC

In order to simulate the dynamic operating characteristics of the engine, the unsteady operating process of the PDC should be considered. A simplified PDC model is proposed based on the research of Endo et al. [23,24]. According to their model, the parameters of the PDC during the filling process can be calculated as follows.

$$W_{31}=k\frac{p_{t31}{\delta }_{pdc}}{\sqrt{T_{t31}}}{A}_{pdc}q\left(\lambda \right)$$

(8)

$$k=\sqrt{\frac{{\gamma }_{fill}}{R}{\left(\frac{2}{{\gamma }_{fill}+1}\right)}^{\frac{{\gamma }_{fill}+1}{{\gamma }_{fill}-1}}}$$

(9)

$$q\left(\lambda \right)={\left(\frac{{\gamma }_{fill}+1}{{\gamma }_{fill}}\right)}^{\frac{1}{{\gamma }_{fill}}}\lambda {\left(1-\frac{{\gamma }_{fill}-1}{{\gamma }_{fill}+1}{\lambda }^2\right)}^{\frac{1}{{\gamma }_{fill}-1}}$$

(10)

$$c_{cr}=\sqrt{\frac{2{\gamma }_{fill}}{{\gamma }_{fill}+1}R{T}_{t31}}$$

(11)

First of all, the velocity factor λ of the PDC is calculated through Equations (8)–(10) based on the PDC inlet air flow rate W₃₁, total temperature T_t31, and total pressure p_t31. Then, the filling speed u_fill, static pressure p_fill, and static temperature T_sfill in the PDC are calculated through Equation (11) and the relationships between the total parameters and the static parameters. Finally, γ₄, M_D, and D_CJ can be calculated by the CEA program, and the transition parameters at the PDC outlet in a detonation cycle can be calculated based on the model of Endo, which are as follows:

$$p_{t4}\left(t_{\mathrm{PDC}}\right)=p_{s4}\left(t_{\mathrm{PDC}}\right){\left[1+\frac{{\gamma }_4-1}{2}{\left(\frac{u_4\left(t_{\mathrm{PDC}}\right)}{a_4\left(t_{\mathrm{PDC}}\right)}\right)}^2\right]}^{\frac{{\gamma }_4}{{\gamma }_4-1}}$$

(12)

$$T_{s4}\left(t_{\mathrm{PDC}}\right)=\frac{a_4{\left(t_{\mathrm{PDC}}\right)}^2}{{\gamma }_4{R}_4}$$

(13)

$$T_{t4}\left(t_{\mathrm{PDC}}\right)=T_{s4}\left(t_{\mathrm{PDC}}\right)\left[1+\frac{{\gamma }_4-1}{2}{\left(\frac{u_4\left(t_{\mathrm{PDC}}\right)}{a_4\left(t_{\mathrm{PDC}}\right)}\right)}^2\right]$$

(14)

$$W_4\left(t_{\mathrm{PDC}}\right)=\frac{p_{s4}\left(t_{\mathrm{PDC}}\right)}{R_4{T}_{s4}\left(t_{\mathrm{PDC}}\right)}{u}_4\left(t_{\mathrm{PDC}}\right)A_{pdc}$$

(15)

Since the change of parameters in the PDC exhaust process is in a time scale of microsecond, the transient parameters at the PDC outlet are calculated with a time step ($\mathsf{\Delta }{t}_{PDC}$) of 5 microseconds. In addition, by analyzing the experimental data of rotor speed, it is found that the cut off frequency of the rotor speed is 60 Hz when the PDTE is fired with a frequency of 30 Hz. The selection of the simulation time step is similar to the selection principle for the data acquisition system, which is generally 8–10 times the cut off frequency. Hence, the dynamic parameters of the PDTE in a detonation cycle is calculated by integrating the transient parameters over a time step of 2 ms, which are shown as follows.

$$W_4=\frac{{\displaystyle {\int }_{t-0.002}^t{W}_4\left(t_{\mathrm{PDC}}\right)d{t}_{\mathrm{PDC}}}}{0.002}$$

(16)

$$p_{t4}=\frac{{\displaystyle {\int }_{t-0.002}^t{p}_{t4}\left(t_{\mathrm{PDC}}\right)W_4\left(t_{\mathrm{PDC}}\right)d{t}_{\mathrm{PDC}}}}{{\displaystyle {\int }_{t-0.002}^t{W}_4\left(t_{\mathrm{PDC}}\right)d{t}_{\mathrm{PDC}}}}$$

(17)

$$T_{t4}=\frac{{\displaystyle {\int }_{t-0.002}^t{T}_{t4}\left(t_{\mathrm{PDC}}\right)W_4\left(t_{\mathrm{PDC}}\right)d{t}_{\mathrm{PDC}}}}{{\displaystyle {\int }_{t-0.002}^t{W}_4\left(t_{\mathrm{PDC}}\right)d{t}_{\mathrm{PDC}}}}$$

(18)

2.4. Model of Turbine

The turbine is considered to be a quasi-steady operating component under all conditions. The parameters of the turbine can be obtained by interpolation from the turbine characteristic map, then the turbine outlet parameters can be obtained as follows.

$$p_{t5}=p_{t4}/{\pi }_T$$

(19)

$$T_{t5}=T_{t4}\left[1-{\eta }_T\left(1-{\pi }_T{}^{\frac{1-{\gamma }_5}{{\gamma }_5}}\right)\right]$$

(20)

$$N_T=W_4{c}_{pg}{T}_{t4}\left[1-\left(1-{\pi }_T{}^{\frac{1-{\gamma }_5}{{\gamma }_5}}\right)\right]{\eta }_T$$

(21)

2.5. Shaft Dynamics

The acceleration of rotational speed of the rotor can be obtained based on the shaft dynamics.

$$\frac{dn}{dt}=\left(N_T{\eta }_m-N_c\right)/\left[Jn{\left(\frac{\pi }{30}\right)}^2\right]$$

(22)

2.6. Solving Algorithm of the Transition Model

In the transition model of the PDTE, the coefficient of compressor pressure ratio Z_c and turbine expansion ratio π_T are chosen as guess parameters. The choice of these two parameters is to make the following residual equations meet the given error criterion.

$$f_1=\left(W_{21}-W_3\right)/W_{21}$$

(23)

$$f_2=\left(W_5-W_7\right)/W_5$$

(24)

The Newton–Raphson (N-R) iterative algorithms are adopted to solve these equations. The flow chart of the solving process is shown in Figure 2, which can be divided into the following five steps.

Step 1: Input the necessary initial parameters such as flight altitude H, Mach number Ma, fuel mass flow rate W_f, operating frequency f, rotor speed n, pressure p_t31, and temperature T_t31 in the transition section, mass flow rate W_a31 at the transition section outlet; some parameters are listed in

Table 1. On-ground parameters used in the calculation.

Conditions	Value
Flight altitude H	0 km
Mach number Ma	0
Atmosphere temperature	288 K
Atmosphere pressure	101.325 kPa
Cross-sectional area of PDC Apdc	0.01131 m2

. Then, the parameters at each section of the PDTE can be calculated according to the above models.

Step 2: Determine whether the two residual equations are all less than 0.0001. If yes, jump to Step 4, otherwise go to Step 3.

Step 3: Adjust Z_c and π_T according to the N-R method and go back to Step 1;

Step 4: The explicit difference scheme is adopted, and n, p_t31, T_t31 at time t + Δt can be calculated as follows:

$$p_{t31}^{t+\Delta t}=p_{t31}^t+\frac{d{p}_{t31}}{dt}\Delta t$$

(25)

$$T_{t31}^{t+\Delta t}=T_{t31}^t+\frac{d{T}_{t31}}{dt}\Delta t$$

(26)

$$n^{t+\Delta t}=n^t+\frac{dn}{dt}\Delta t$$

(27)

Step 5: Output the required thermodynamics parameters of each station and go to Step 1.

3. Results and Discussion

In this section, the transition model is verified with experimental data from a pulse detonation prototype engine, in which gasoline is used for both the experiment and the transition model. Then, the quasi-steady state and dynamic operating characteristics of the PDTE at ground conditions are discussed.

3.1. Validation of the Transition Model

The pulse detonation prototype engine system is shown in Figure 3. The system consists of a turbocharger, a dual-tube PDCs, a transition section, a fuel supply system, an ignition system, and a data acquisition system. Reed valves are set in the dual-tube PDCs to isolate the reverse flow and back pressure waves. Detail of the pulse detonation prototype engine system were introduced in Ref. [22]. The pulse detonation prototype engine was operated in a self-aspiration model, which was that the turbine extracted power from the fired dual-tube PDC and drive the compressor to supply air to the PDCs through the intake transition section.

In the pulse detonation prototype engine system, a gear flowmeter is used to measure the fuel flow rate W_f supplied into the PDC. A type K thermocouple (measuring range of 0–1573.15 K, measurement accuracy of 0.75% of full scale) is installed on the transition section to measure the total temperature T_t31 in the transition section. A piezoresistive pressure sensor is installed to measure the total pressure p_t31 in the transition section (measuring range of 0–1 MPa, frequency response of 200 KHz, measurement accuracy of 0.5% of full scale). In order to monitor the operating conditions of the PDCs, two piezoelectric sensors (measuring range of 0–10 MPa, frequency response of 500 KHz, measurement accuracy of 1% of full scale) are installed on the end of the PDC. A displacement sensor (measuring range of 0–2 mm, measurement accuracy of 0.5% of full scale) is used to monitor the periodic change of the hexagonal thread on the shaft of the compressor. The rotor speed can be calculated based on the displacement sensor. All the data are collected by the DEWE3020 high-speed data acquisition instrument with a sampling rate of 200 K/s.

In order to verify the accuracy of the transition model, the unsteady operating state and the quasi-steady state of the pulse detonation prototype engine are calculated based on the experimental conditions. Then, the numerical data are compared with the experimental results. For the unsteady operating state, the comparison is carried out at 10 Hz, while the quasi-steady state comparisons are carried out for the firing frequencies of 10 Hz, 15 Hz, and 20 Hz.

Figure 4 shows a comparison of the pressure and rotor speed for the experimental and numerical results. Figure 4a shows the comparison of the pressure in the transition section. It can be seen that the experimental peak pressure is higher than the simulated peak pressure. The reason is that the reed valve in the prototype engine cannot completely isolate the reverse flow and back pressure waves in the PDCs. However, it is supposed that the back pressure waves are totally isolated in the transient model. Therefore, the experimental peak pressure is higher than the simulated peak pressure. The simulated results show that the pressure in the transition section will rise suddenly due to the closeness of the reed valve. The sudden pressure rise at the compressor outlet will make the compressor temporarily cross the surge boundary. It takes a certain time for the back pressure of the compressor to drop out of the surge line. After that, the reed valve opens and the pressure in the transition section decreases rapidly.

Figure 4b gives static pressure profiles at the outlet of the PDC. It states that the trend between the simulated static pressure and experimental static pressure is basically the same. The simulated peak values of the static pressure for different cycles are the same since the fuel and air are supposed to be the same under different cycles in the transition model. While for the real pulse detonation prototype engine, the fuel distributions in the PDC under different cycles are different due to the fuel supply or the manufacture error of the PDC. Therefore, the peak pressure of the detonation wave in the tests shows variations from cycle to cycle.

Figure 4c presents a comparison for the dynamic and average rotor speed of the PDTE. It can be concluded that the dynamic rotor speed also shows cyclic oscillations under the effect of the PDC. The rotor speed will sharply rise when the detonation gas enters into the turbine, and then the speed will decline slowly. The variation trends are basically the same for the simulation data and the experimental results. Moreover, the average rotor speed in the experiment is 23,117 r/min, while the simulated average rotor speed is 23,097 r/min. The deviation between the two is within 0.1%. These comparisons indicate that the transition model can exactly predict the dynamic operation characteristics of the PDTE.

The quasi-steady state parameters for the numerical results and experimental data are presented in

Table 2. Experiment and simulation results of PDTE at f = 10 Hz.

	Experiment	Simulation	Deviation (%)
n (RPM)	23,117	23,097	0.09
pt31 (Pa)	117,325	117,406	0.07
Tt31 (K)	307.59	308.69	0.36
Average W31 (kg/s)	0.131	0.133	1.37
Average F (N)	73.91	78.21	5.82

,

Table 3. Experiment and simulation results of PDTE at f = 15 Hz.

	Experiment	Simulation	Deviation (%)
n (RPM)	30,710	30,988	0.91
pt31 (Pa)	127,325	130,188	2.25
Tt31 (K)	322.8	321.18	0.50
Average W31 (kg/s)	0.234	0.218	6.84
Average F (N)	135.90	146.24	7.61

and

Table 4. Experiment and simulation results of PDTE at f = 20 Hz.

	Experiment	Simulation	Deviation (%)
n (RPM)	28,134	28,574	1.56
pt31 (Pa)	120,325	126,007	4.72
Tt31 (K)	311.7	317.29	1.79
Average W31 (kg/s)	0.172	0.182	5.81
Average F (N)	102.99	107.29	4.18

for the firing frequencies of 10 Hz, 15 Hz, and 20 Hz. All the experimental data presented in Table 2, Table 3 and Table 4 are the average value within 1 s, and the reliability of experimental data is evaluated by standard deviation. The standard deviation is calculated by using the average value and the experimental data, and the results show that the standard deviation is no more than 6.52%. The data in these tables state that the deviation between simulation data and experimental data are within 7.61%. The difference between the simulation data and experimental data can be explained as follows. Since the PDTE is always operated in a dynamic operating state, the characteristic maps of the compressor and turbine adopted in the transition model are based on the steady tests in a traditional gas turbine engine. This may result in errors in the simulation results. Moreover, the assumptions in the transition model will also introduce deviations in the simulation results. However, the above comparisons show that the deviation between simulation data and experimental data are small under the above operating conditions, indicating that the transition model is reliable in predicting the dynamic operating characteristics and the quasi-steady state parameters of the PDTE.

3.2. Quasi-Steady State Operating Characteristics of the PDTE

In this part, the quasi-steady state operating characteristics of the PDTE are analyzed based on the transient model. The operating frequency is 10 Hz with W_f increased from 0.0056 kg/s to 0.0129 kg/s, when the W_f is fixed, the operating frequency is increased from 7 Hz to 18 Hz.

Figure 5 shows the average performance and operating parameters of the PDTE at firing frequency of 10 Hz when the W_f is changed. As can be seen in Figure 5a, the rotor speed and thrust of the PDTE are firstly increased with the raising of the fuel flow rate W_f. When the fuel flow rate W_f increase to certain value, the rotor speed and thrust will suddenly go down and start to decrease. The specific thrust shows a similar variation trend, which can be seen in Figure 5b. For the specific fuel consumption, there is a minimum specific fuel consumption as the fuel flow rate W_f is changed. The reasons are as follows. As the fuel flow rate W_f is increased, the equivalence ratio in the PDC will increase to a value of about 1.1 under which the detonation wave has a peak pressure and temperature. Then, the equivalence ratio will further increase to the value of 1.3, as indicated in the Figure 5c. The strength of the detonation waves such as the peak pressure will be reduced due to the over rich of fuel in the PDC. Therefore, the inlet total temperature and total pressure of the turbine will firstly go up and then down as the fuel flow rate is changed from 0.0056 kg/s to 0.0129 kg/s. The mass flow rate of air at the compressor outlet will also firstly increase and then reduce due to power output of the turbine, which can be seen in Figure 5c. Hence, the rotor speed, the thrust, and the specific thrust of the PDTE will firstly increase and then decrease. In addition, the filling ratio in the PDC will exceed 1 with the further increase of W_f, resulting in the waste of fuel. The specific fuel consumption will reduce first of all with the increasing W_f due to the rapidly increased specific thrust. However, when the increasing rate of the specific thrust is lower than that of the fuel flow rate, the specific fuel consumption will raise. Therefore, there is a minimum specific fuel consumption with the increasing fuel flow rate W_f.

Figure 6 shows the average performance and operating parameters of the PDTE under different firing frequency with a fuel flow rate of 0.0094 kg/s. As stated in Figure 6a,b, the thrust and the specific thrust reach maximum values at a firing frequency of 12 Hz, while the specific fuel consumption is the lowest. When the firing frequency is lower than 12 Hz, the thrust and the specific thrust is increased with the increasing firing frequency, yet the specific fuel consumption is reduced. For the firing frequency above 12 Hz, the thrust, specific thrust, and specific fuel consumption show an opposite variation trend. The data in Figure 6c indicates that the equivalence ratio and the filling ratio are 1.02 under 12 Hz. When the firing frequency is increased from 7 Hz to 12 Hz, the filling ratio will be reduced from an over-filled state to the filling ratio of 1.02. The equivalence ratio in the PDC will also go to the stoichiometric ratio from a fuel rich state. Therefore, the strength of the detonation waves will increase, which will lead to a higher inlet total temperature and total pressure in the turbine. The rotor speed, the mass flow rate of air, the thrust, and the specific thrust will increase with the increasing firing frequency from 7 Hz to 12 Hz. When the firing frequency is increased from 12 Hz to 18 Hz, the filling ratio and the equivalence ratio will drop since the fuel flow rate remains unchanged. However, the turbine will extract more power with the increased firing frequency, which has been observed in the experimental tests [17,25,26]. The rotor speed and mass flow rate of air will further increase with the increased firing frequency. Since the fuel flow rate is constant, the increased output power of the turbine will cause a smaller exhaust velocity in the nozzle. Hence, the thrust, specific thrust, and specific fuel consumption shows an opposite variation trend when the firing frequency of the PDC is increased from 12 Hz to 18 Hz.

3.3. Dynamic Operating Characteristics of the PDTE

Figure 7 shows the dynamic operating process of the engine when the fuel flow rate of W_f is suddenly increased from 0.00936 kg/s to 0.0103 kg/s at the firing frequency of 10 Hz. The first curves in Figure 7 indicated that the rotor speed is increased in a cyclic oscillation way when the fuel flow rate is changed. It takes about 3.58 s for the rotor climbing to a quasi-steady state. The average rotor speed is increased from 23,079 r/min to 24,108 r/min, while the dynamic rotor speed shows cyclic oscillations due to the cyclic operated PDC under the quasi-steady state. The acceleration of the rotor speed is gradually decreased as the time changes. The curves in the middle of Figure 7 indicates that the total pressure at the compressor outlet shows a similar changing law with the raising of the rotor speed, indicating that the pressure ratio of the compressor will also increase oscillatory. Moreover, as presented in the lower part of Figure 7, when the fuel flow rate is suddenly increased, the rotor speed will gradually increase, which will lead to an increase of the mass flow rate of air. The fill ratio will also increase since the firing frequency is constant. Therefore, the equivalence ratio of the PDC will slowly decrease due to the gradually increased mass flow rate of air.

Figure 8 shows the dynamic operating characteristics of the PDTE when the operating frequency is increased from 10 Hz to 11 Hz at the fuel flow rate of 0.0094 kg/s. The above parameters show similar change rules as shown in Figure 8. It takes about 1.97 s for the rotor speed to increase to a quasi-steady state. The average rotor speed is increased from 23,081 r/min to 23,444 r/min with a 1 Hz increment of the firing frequency. The outlet total pressure of the compressor, the fill ratio, and the equivalence ratio show a similar dynamic trend as presented in Figure 7. The reason is similar, which is due to the increasing of mass flow rate of air caused by the increased rotor speed.

4. Conclusions

In this paper, a transient analytical model was established and verified with the experimental data to reveal the quasi-steady state and dynamic operating characteristics of the PDTE. The shaft dynamics and the volumetric effect of the compressor and transition section were considered in the transient model. The comparison of the transient model with the experimental results shows that the deviations between the calculated data and the experimental results are no more than 7.61%. The simulation results state that the operating line of the compressor will temporarily cross the surge boundary of the compressor during the firing phase of the pulse detonation combustor. However, the centrifugal compressor can still be in the normally operating state in the test. The operating state of the PDTE will change accordingly with the increasing of the fuel flow rate or the operating frequency. The rotor speed, thrust, and specific thrust will firstly increase and then go down with increasing of the fuel from 0.0056 kg/s to 0.0129 kg/s at 10 Hz. There are peak values for the rotor speed, thrust, and specific thrust when the fuel is changed, while the specific fuel consumption has a minimize value during this process. The extremum values of the performance parameters are much more obvious when the firing frequency is increased at a fixed fuel flow rate. When the firing frequency is changed from 7 Hz to 18 Hz, the thrust and specific thrust reach the maximum values at firing frequency of 12 Hz and the specific fuel consumption has a minimize value. The transient calculations show that the rotor speed and the pressure ratio of the compressor are increased in a cyclic oscillation way when the fuel flow rate or the firing frequency is suddenly increased. It takes dozens of cycles for the PDTE to go to a quasi-steady state when the fuel flow rate or the firing frequency is suddenly changed. In addition, the transient performance analysis of the PDTE under different flight conditions and the control strategy design of the PDTE will be discussed on the basis of the transition model in the future. All these studies will provide some theoretical bases to accelerate the application of the pulse detonation turbine engine.