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Numerical investigation of flow particle paths and thermodynamic performance of continuously rotating detonation engines
Combustion and Flame 159 (2012) 3632–3645
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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Numerical investigation of flow particle paths and thermodynamic performance of continuously rotating detonation engines Zhou Rui, Wang Jian-Ping ⇑ State Key Laboratory of Turbulence and Complex System, Department of Mechanics and Aerospace Engineering, Center for Combustion Propulsion and Power, College of Engineering, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 22 July 2011 Received in revised form 8 April 2012 Accepted 11 July 2012 Available online 11 August 2012 Keywords: RDE Particle path Thermodynamic performance p–v Diagram T–s Diagram
a b s t r a c t Based on the two-dimensional numerical simulation of continuously rotating detonations in an annular chamber, the paths of flow particles burned by three different processes are tracked and analyzed in detail. The detonation wave, the deflagration wave, the oblique shock wave, and the contact surface have a small influence on the paths of flow particles. The fluctuation of paths in the azimuthal direction is less than 12% of the circumference of the combustion chamber. The path will deflect when the flow particle encounters the detonation wave or the oblique shock wave, and it will not deflect when encounters the deflagration wave or the contact surface. About 23.6% fuel is burned by deflagration, and the left is burned by rotating detonation wave. The thermodynamic performance of continuously rotating detonations is then discussed. The p–v and T–s diagrams obtained by numerical simulation are qualitatively consistent with the ideal ZND model. The average thermal efficiency of the detonation combustion in 2D RDE is 31%, and its average net mechanical work is 1.3 MJ/kg. The thermal efficiency of the entire RDE is 26.4%, and its net mechanical work is 30% of the ideal ZND model. The superior performance of continuously rotating detonations is determined. Crown Copyright Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. All rights reserved.
1. Introduction Detonation combustion has an inherently higher thermodynamic efficiency than constant-pressure combustion [1–4]. In recent years, the continuously rotating detonation engine (RDE) based on detonation combustion has been extensively studied [5,6]. In the RDE, fuel is continuously injected at a high velocity, which greatly increases the average flow rate of the fuel. Therefore, the RDE has high propulsion performance. Voitsekhoviskii [7] initially proposed the concept of continuously rotating detonation and achieved a brief continuous detonation of premixed acetylene and ethylene in a circular tube. Bykovskii et al. [8] performed experiments using various fuels and achieved steady long-duration continuous detonation wave propagation. Davidenko [9] numerically simulated continuous spin detonation using a H2/O2 two-dimensional (2D) model. Hishida et al. [10] computationally modeled the detonation cellular structure at the head of 2D rotating detonations. Shao et al. [11–13] comprehensively studied three-dimensional (3D) numerical simulations in continuously rotating detonations. They obtained multicycles of continuously rotating detonations and discussed several
⇑ Corresponding author.
key issues, including the fuel injection limit, self-ignition, thrust performance, and nozzle effects. So far, there has been little research, by both numerical simulations and experiments, on the thermodynamic performance of continuously rotating detonations. Heiser and Pratt [14] have analyzed the thermodynamic cycle of pulse detonation engines, but only studied progress of the ideal cycle, which cannot be achieved in practice. In addition, most of the previous numerical simulation studies have focused on the overall description of the flow-field within a continuously rotating detonation combustion chamber. The complicated flow-field of continuously rotating detonations not only consists of a detonation wave, a deflagration wave, and an oblique shock wave, but also of a contact surface. However, the paths of flow particles and changes in the variables of the flow particles have not been studied. The process from premixed gas injection to combustion product ejection has not been investigated in any detail. It is natural to think that the thermodynamic properties can be obtained by tracking flow particles from injection into the combustion chamber to ejection from the exit. In this study, a new method for analyzing the continuously rotating detonation engines flow-field and describing its thermodynamic properties is proposed. The paths of flow particles are tracked, and the effects of the detonation wave, the deflagration wave, the oblique shock wave, and the contact surface on the paths
E-mail address: [email protected] (J.-P. Wang). 0010-2180/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. All rights reserved. http://dx.doi.org/10.1016/j.combustflame.2012.07.007
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Nomenclature p R
q T v e q
c Ds
pressure gas constant density temperature specific volume total energy heat released per unit mass of reactants ratio of specific heats entropy increment
and flow parameters are investigated. The corresponding p–v and T–s diagrams can then be used to calculate the net mechanical work and thermal efficiency. The results from numerical simulations are qualitatively and quantitatively compared with ideal ZND model.
E1, E2 Cv Cp k1, k2 r, h, z uh uz
activation energies specific heat at constant volume specific heat at constant pressure rate constants cylindrical coordinates flow velocity in circumferential direction flow velocity in axial direction progress variable of induction reaction progress variable of exothermic reaction
a b
duction, and mass diffusion are ignored in this study. Euler equations are
@U @F @G þ þ ¼S @t @h @z
ð1Þ
where 2. Physical modeling and numerical method
Z
2.1. Physical modeling Y
2.2. Numerical method The flow-field is governed by the 2D conservative Euler equations in cylindrical coordinates and the Korobeinikov et al. [17] two-step chemical reaction model. Viscosity, thermal con-
X
2
1b
1a
(a)
0.06
1a
0.05
p (atm) 26 24 22 20 18 16 14 12 10 8 6 4 2
0.04
r⋅θ (m)
The combustion chamber of the RDE is a coaxial cavity with a toroidal section. A detonation wave propagates azimuthally in the annulus while a combustible mixture is injected from the head-end, and then, the burnt gas spurts out of the downstream exit. Assuming that the depth along the radial direction is much smaller than the diameter and axial length, and then, the flow-field can be approximated as a 2D cylindrical chamber without thickness [10,13], as shown in Fig. 1a. At the head-end, there are a large number of Laval micro-nozzles to axially inject premixed stoichiometric hydrogen/oxygen gas into the combustion chamber. The mass flux of the incoming fuel is controlled by the relationship between stagnation and flow pressures at the head-end with a set exit to throat area ratio. The flow-field is initially filled with the premixed stoichiometric hydrogen and oxygen mixture at 1 atm and 300 K, except from the head wall region where a 1D C-J detonation wave is artificially placed. It is placed along the axial direction for a short distance, in order to initiate 2D detonation. Figure 1b shows the initial pressure distribution by extending the thin chamber to a plane. In the inlet nozzles at the head-end, the ratio of the exit to throat area is set to 3.7 and the stagnation pressure is set to 3 MPa, which results in a critical pressure at the nozzle exit of 1.6 MPa. If the pressure at the head-end is larger than the stagnation pressure, the fuel cannot be injected. Therefore, a rigid wall condition is used at the head-end. If the pressure at the head-end is smaller than the inflow pressure, the inflow condition is given by Laval nozzle theory [15,16]. At the downstream exit, we use non-reflecting boundary conditions, with a relaxation factor of 0.05. The radius of the combustion chamber is 1 cm and the length is 5 cm. The number of grid points is 400 in the azimuthal direction and 500 in axial direction, resulting in Dz = 0.1 mm and D(r h) = 0.157 mm. The grid dependency has been validated in Ref. [12]. The upper and lower sides are connected by periodic boundary conditions.
2
0.03
0.02
1b 0.01
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Z (m)
(b) Fig. 1. (a) Initial pressure distributions of the annulus chamber without thickness and (b) the plane extended from (a).
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R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645 (T/300)/K
Contact Surface
0.04
2 0.03 0.02
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Induced Shock Wave 0
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ρ (kg/m3)
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3
β
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2 0.03
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Fresh Gas
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Induced Shock Wave
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1
Z (m)
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2 0.03
Z (m)
0.04
0
0.04
2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6
1
50 45 40 35 30 25 20 15 10 5
Axial Direction
Detonation Front
0.05
0.01
p (atm)
Oblique Shock Wave
3
3
0.05
Azimuthal Direction
Deflagration 0.01
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r⋅θ (m)
Detonation Front
0.05
r⋅θ (m)
9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
Oblique Shock Wave
3
r⋅θ (m)
0.06
0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2
1
0
0
0.01
0.02
0.03
Z (m)
Z (m)
(b)
(d)
0.04
0.05
0.06
Fig. 2. Flow distribution and initial positions of the just-injected fuel with a stable propagating detonation wave. (a) Temperature contours; (b) density contours; (c) pressure contours and streamlines; (d) contours of the progress variable for chemical reaction.
Partical path3 0.06
3
ΙΙ
0.05
r⋅θ(m)
Ι
p (atm)
0.055
0.05
Partical path 2
0.04
50 45 40 35 30 25 20 15 10 5
r⋅θ(m)
0.06
0.03
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0
0.005
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0.015
Z(m)
0
Partical path 1 0
0.01
0.02
0.03
0.04
Z(m) Fig. 3. Local enlargement of the pressure distribution near the detonation wave front.
Fig. 4. Initial paths of the three particles.
0.05
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d 1
c b
qa
0.8
2
0.7
0.5
2
g
0.4
quz a 0
0.01
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0.05
e¼
Fig. 5. Path of particle 1.
3
50
Pressure (atm)
b d
p
0
10
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40
1500
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1000
d b e
g
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(b)
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UZ Uθ
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a c
a
f
b
Time (μs)
600
g
25
Time (μs)
1600
2000
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30
0
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2500
P T
e
5 a a
0
ð4Þ
d
10
g
0.5 a
1 1 þ bqq þ qu2h þ qu2z 2 2
35
15
f
e
UZ (m/s)
Density (kg/m3 )
2
ð3Þ
c1
40
1
qxa
c c
45
c
1.5
ð2Þ
p ¼ qRT
Z(m)
2.5
3
0
0
10
20
30
40
50
Time (μs)
(c) Fig. 6. Change in the flow parameters of particle 1. (a) Density; (b) pressure and temperature; (c) axial and azimuthal velocity.
T(K)
e
3
7 7 6 6 7 6 6 0 7 7 7 6 6 7 7 6 6 7 6 6 0 7 7 S¼6 7 G¼6 7 7 6 6 6 ðe þ pÞuz 7 6 0 7 7 7 6 6 6 qu b 7 6 qx 7 5 4 4 z b5
f
0.6
quz quh uz qu2z þ p
quh a
Uθ (m/s)
a 0.9
3
2
q quh 6 qu 7 6 qu2 þ p 7 7 6 h7 6 h 7 7 6 6 6 quz 7 6 quh uz 7 1 7 7 6 U¼6 6 e 7 F ¼ r 6 ðe þ pÞu 7 h7 7 6 6 7 7 6 6 4 qb 5 4 quh b 5
1.1
r⋅θ(cm)
3
2
R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
0.06
0.06
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r⋅θ (m)
r⋅θ (m)
3636
0.03
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a 0
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Z(m)
(a)
(b) 0.06
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e 0
0
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f 0.03
Z(m)
(e)
0.04
0.05
0.06
0
0
0.01
0.02
0.03
Z(m)
(f)
Fig. 7. Temperature distribution of flow particle 1 around the points of Fig. 5.
R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
0.06 T/300 (K) 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
0.05
r⋅θ(m)
0.04
0.03
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0.01
g 0
0
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0.02
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Z(m)
(g) Fig. 7 (continued)
4.2
g
e
4
r⋅θ(cm)
f 3.8
3.6
a
b
3.4
c d
0
0.01
0.02
0.03
0.04
0.05
Z(m) Fig. 8. Path of particle 2.
xa ¼
da ¼ k1 q expðE1 =RTÞ dt
db xb ¼ ¼ dt
(
ð5Þ
h i 2 k2 p2 b2 exp E ð1 bÞ2 exp E2RTþq ða ¼ 0Þ RT 0 ð1 > a > 0Þ
3637
cycles. The fuel injected into the combustion chamber can only be burned by two types of processes: deflagration and detonation. Figure 2 shows the distributions of temperature, density, pressure with streamlines, and the progress variable b. In Fig. 2a, the following can be identified: the detonation front; the attached oblique shock wave; the contact surface between the current cycle’s chemical reaction products and the previous cycle’s products; deflagration occurring at the contact surface of the fresh gas mixture with the high-temperature burned gas; and the induced shock wave near the head-end. Furthermore, in the close-up of the pressure distribution near the detonation front (Fig. 3), it can be seen that the detonation front is composed of two parts. Part I burns off the fresh gas mixture before the induced shock wave, while part II burns off the gas mixture after the induced shock wave. As a consequence, detonation combustion is divided into two cases in this study. Figure 3 shows the local enlargement of the pressure distribution near the detonation wave front. The increase in pressure at detonation front II is larger than at detonation front I, leading to different paths and flow variables when a particle encounters each one. One flow particle is injected into the head-end and its path, and flow variables are tracked for each case. Generally, the particle paths are different from the streamlines in unsteady flows. It is necessary to calculate the particle paths step-by-step by solving the governing equations. The particle’s velocity can be calculated by area interpolation of the flow velocities at the four closest grid points. The time step is very small in the numerical simulations, so that the distance moved by each particle in a time step is less than the grid size. The initial positions of the three typical particles that were followed are shown in Fig. 2. At this instant, the flow particles have just been injected into the head-end of the combustion chamber. After detailed analysis, particle 1 is burned by detonation wave II, particle 2 is burned by the deflagration wave, and particle 3 is burned by detonation wave I. More details will be discussed in the following paragraphs. Figure 4 shows the paths of the three particles from injection into the combustion chamber to exhaustion. The horizontal line is the boundary of cycle. The particle paths have small fluctuations in the azimuthal direction, which are less than 12% of the circumference, independent of their initial positions. The average time of the three flow particles in the combustion chamber is 58.4 ls. The period of detonation propagating in the azimuthal direction is 26.2 ls, which indicates that detonation propagates almost two cycles when the particles are moving through the chamber. The particle paths differ depending on whether the flow particles encounter the contact surface, the detonation, deflagration, or oblique shock waves, and where they encounter them. Thus, the changes in the flow variables are also different. In the following section, we analyze particle 1 in detail, with the other two only discussed if their behavior is different to particle 1.
ð6Þ The parameters in the equations are set to be the same as Refs. [17,18]. The spatial terms are discretized with a 5-step WENO scheme [19,20], and the temporal terms are discretized with the 4-step Runge–Kutta method [21,22]. 3. Results and discussions 3.1. Trajectory analysis After initiation of the detonation wave by the extended 1D C-J detonation, the wave becomes stable after propagating for eight
3.1.1. Particle 1 Figure 5 shows the path of particle 1 enlarged about eight times in the azimuthal direction. We use a–g to mark seven key points in the path of particle 1 and discuss the properties of the particle in the phases between these points. Figure 6 shows the variation of particle 1’s density, pressure, temperature, axial velocity, and azimuthal velocity with time. Figure 7 shows the temperature distributions and the locations of particle 1 in the flow-field when particle 1 moves via the points in Fig. 5. The path of particle 1 and the change in conditions of the flow parameters are as follows.
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2 P T
25
f
e
Pressure (atm)
20
b 1
g
f 0.5 a
c
c
g 2000
d
1500
15
10
f
b
e d
c
b
T(K)
1.5
Density (kg/m3)
2500
g 1000
e d 500
5 a 0
a
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c
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e
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U Z (m/s)
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a
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-500 -1000
500
-1500 0
0
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40
50
Time (μs)
(c) Fig. 9. Changes in flow parameters of particle 2. (a) Density; (b) pressure and temperature; (c) axial and azimuthal velocities.
(1) a to b: First, particle 1 is injected into the head wall of the combustion chamber, as shown by point a in Figs. 5 and 7a. In the streamline distribution (Fig. 2c), particle 1 first moves to the right and then down with the flow, until it reaches point b where it is just in front of detonation wave II (Fig. 7b). (2) b to c: In this phase, the particle encounters detonation wave II. It only moves a very small distance, but its path is deflected, as shown in Figs. 5 and 7b and c. At the instant, it encounters the detonation wave, its density, pressure, temperature, and azimuthal velocity increase rapidly, while its axial velocity decreases. As shown in Fig. 6, the maximum pressure and temperature are 50 atm and 2700 K, respectively. (3) c to d: Particle 1 follows detonation wave II until the azimuthal velocity decrease to zero (point d in Figs. 5 and 7d), where its path is deflected again. In this phase, the pressure, density, and azimuthal velocity of particle 1 decrease rapidly. The temperature also decreases, but it remains high.
The axial velocity fluctuates with small amplitude (Fig. 6). This phase confirms the expansion phenomenon after the detonation wave. (4) d to e: Initially, the density, pressure, temperature and azimuthal velocity of particle 1 all decrease, while the axial velocity increases. The thermodynamic parameters tend to be stabilized in this phase, and particle 1 moves downstream. After the detonation wave has propagated almost one cycle, the particle encounters the oblique shock wave coupled with the detonation wave, where its path deflects again, as shown in Fig. 5 and near point e in Fig. 7e. At this moment, its density, pressure, temperature, azimuthal and axial velocities all increase. The variations in the trends of the flow parameters, apart from the axial velocity, are similar when the particle encounters the oblique shock wave and when it encounters the detonation wave. Only the degree of the change is different, there is a greater change when encountering the detonation wave than the oblique shock wave. The paths are deflected in both cases.
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0.06
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b 0.03
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Z(m)
Z(m)
(e)
(f)
Fig. 10. Temperature distributions of particle 2 around the points labeled in Fig. 8.
0.04
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0.06
T/300 (K) 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
0.05
g
r⋅θ(m)
0.04
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0
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Z(m)
(g) Fig. 10 (continued)
6.4
d
r⋅θ(cm)
0
b
6.2
e
c
6
a
0
0.01
0.02
0.03
0.04
0.05
Z(m) Fig. 11. Path of particle 3.
(5) e to f: After passing through the oblique shock wave, the density, pressure, temperature, azimuthal, and axial velocities of particle 1 decrease and tend to stabilize. The particle follows the oblique shock wave, as shown at point f in Figs. 5 and 7f, until the azimuthal velocity becomes negative and then the path deflects again. (6) f to e: The particle continues moving downstream until it is exhausted out of the combustion chamber, as shown at point g in Figs. 5 and 7g. In summary, particle 1 encounters detonation wave II once, the oblique shock wave once, and does not encounter the contact surface or the deflagration wave. 3.1.2. Particles 2 and 3 In contrast to particle 1, flow particle 2 is burned by the deflagration wave. Figures 8–10 show the path of particle 2, changes
in the flow parameters, and the temperature distributions around the key points labeled in Fig. 8. First, flow particle 2 is injected into point at the head wall and its axial velocity decreases rapidly due to the induced shock wave (Fig. 10a). Particle 2 encounters the deflagration wave at point b and passes through it to point c. The progress variable of exothermic reaction has a value of 0.9 at point b and 0.2 at point c. This suggests that the process occurring between these points is deflagration combustion. It is observed that the particle path is not affected during the process of deflagration combustion. Figure 9 shows that the temperature increases to 2000 K, the pressure slightly decreases in this phase. Particle 2 continues moving to point d, where it encounters the oblique shock wave and its path deflects (point d in Fig. 10d). All of the flow variables shown in Fig. 9 increase due to the shock wave. The particle moves to point e and encounters the contact surface, as shown at point e in Fig. 10e. The strength of contact with the surface is so weak that the flow parameters are almost unaffected and the path does not change. The particle moves to point f, where it encounters the oblique shock wave again (Fig. 10f). Finally, the particle exits the combustion chamber at point g. In summary, flow particle 2 encounters the deflagration wave once, the oblique shock wave twice, the contact surface once, and it does not encounter the detonation wave. Figure 11 shows the path of particle 3, and the horizontal line is the boundary of cycle. Particle 3 is injected into the combustion chamber and is instantly burned by detonation wave I. Its pressure, temperature, density, and azimuthal velocity increase rapidly, while its axial velocity decreases, as shown in Fig. 12. The maximum pressure is 22 atm, and the maximum temperature is 2700 K. Although the maximum pressure is only 44% of the maximum pressure of particle 1, which is burned by detonation wave II, the maximum temperature increases to the same level. After detonation wave I passes through particle 3, the particle travels closely to the detonation wave. At point b, the azimuthal velocity decreases to zero, and the path deflects. It keeps moving to point c and encounters the oblique shock wave, as shown in Fig. 13c. The subsequent phases are the same as for particle 1. The variation trends of the flow parameters are similar when the particle encounters detonation wave I and detonation wave II, but are different in magnitude. In summary, flow particle 3 encounters the detonation wave I once, the oblique shock wave once, and does not encounter the contact surface or the deflagration wave. Figure 14 shows the changes in pressure of the three flow particles with time. The pressure of particle 1, burned by detonation wave II, reaches about 50 atm, while the other two particles only reach about 20 atm. In other words, the fuel burned by detonation wave II contributes most to the performance of RDEs. The detailed analysis of the movement of the three typical flow particles shows that the detonation wave and the oblique shock wave do have an impact on the path of particles. This is due to deflection of the paths, and the variations of the flow parameters are more significant than the fluctuations of the paths. The pressures, temperatures, and densities of all 3 particles increased when they encountered the detonation and oblique shock waves. However, the paths do not deflect when the particles encounter the contact surface and the deflagration wave. 3.2. Analysis and comparison of thermodynamic performance Pressure as a function of specific volume is defined by the p–v diagram, which represents the mechanical work of the thermodynamic processes. Temperature as a function of the entropy increment is defined by the T–s diagram, which represents the heat release of the thermodynamic processes. In this paper, the p–v
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1.5 3000 P T
1
d e
0.5
a
d
2000
15
e 1500
c 10
T (K)
c b
2500
c
b
Pressure (atm)
Density (kg/m3 )
20
b d
1000
e 5
0
0
10
20
30
40
0
50
500
a a 0
10
20
30
Time (μs)
Time (μs)
(a)
(b)
2500
40
50
0
1000 UZ Uθ
2000
500
U Z (m/s)
d
b
a
1500 a
0
e c
1000
d e
-500
Uθ (m/s)
c
b 500
0
-1000
0
10
20
30
40
50
-1500
Time (μs)
(c) Fig. 12. Changes of the flow parameters of particle 3. (a) Density; (b) pressure and temperature; (c) axial and azimuthal velocities.
and T–s diagrams of flow particle 1 were obtained. The relaxation length is set at the exit to ensure that the p–v and T–s curves are closed, in which the values of the physical parameters in the combustion flow-field decrease to the ambient values at infinity. Figure 15 shows the p–v and T–s diagrams of flow particle 1 from 2D numerical simulation. The points marked in Fig. 15 correspond to the same points in Fig. 5. At point e, the particle encounters the oblique shock wave, causing fluctuation of the p–v and T–s curves and a slight increase in Ds. However, the general trends of the p–v and T–s curves do not change. Therefore, the accompanying oblique shock wave causes only a small loss in performance of the continuously rotating detonation engine. The black dashed-dotted line in Fig. 16 shows the ideal ZND model [23–25]. The ideal ZND model has four phases. (1) The fuel with initial state 1 is injected into the combustion chamber and is compressed by the leading shock wave. It increases to the Von Neumann Spike along the Hugoniot line of the shock wave in the p–v diagram (point 2 in Fig. 16). Both the temperature and entropy
increase in this phase. (2) The fuel undergoes an intense chemical reaction and releases heat. The combustion products begin to expand, causing a decrease in pressure and density and an increase in temperature and entropy. It then reaches point C-J (point 3) from the Von Neumann Spike along the Rayleigh line in the p–v diagram. (3) The detonation products expand until the pressure reaches the initial pressure (point 4), according to the isentropic law. (4) The curve is forced to close isobarically, and the cycle is complete. In order to systematically compare the results of the numerical simulation with the ideal ZND model, the choice of parameters in the ideal ZND model are the same as those in the numerical simulation. A premixed stoichiometric hydrogen/oxygen mixture is injected into the combustion chamber with an initial pressure of 1 atm, an initial temperature of 300 K, a specific heat ratio of c = 1.4, the gas constant is R = 692.9 J/(kg K), the heat released per unit mass of reactants of q = 4 106 J/kg. The formula for calculating the entropy is:
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0.05
0.04
0.04
r⋅θ(m)
0.05
0.03
0.02
0.01
0.01
0
0.01
0.02
0.03
0.04
0.05
0
0.06
0.03
(b)
0.04
0.05
0.06
0.04
0.05
0.06
0.06
0.04
r⋅θ(m)
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0.01
0.02
(a)
0.05
0
0.01
Z(m)
0.05
0
0
Z(m)
c
0.06
r⋅θ(m)
0.03
0.02
0
b
0.06
a
0.02
0.03
0.04
0.05
0
0.06
d 0
0.01
0.02
Z(m)
0.03
Z(m)
(c)
(d) e
0.06
T/300 (K) 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
0.05
0.04
r⋅θ(m)
r⋅θ(m)
0.06
0.03
0.02
0.01
0
0
0.01
0.02
0.03
0.04
0.05
0.06
Z(m)
(e) Fig. 13. Temperature distributions of particle 2 around the points labeled in Fig. 11.
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1
50
50
c
Pressure (atm)
Pressure (atm)
40
Particle1 Particle2 Particle3
40
30
20
2
3
1
2
30
20
d e
10
10
b
f a
0
0
0
10
20
30
40
50
0.5
60
1
1.5
2
v (m3/kg)
Time (μs) Fig. 14. Change in pressure for the three flow particles.
(a)
3000
c dp dv þ Cp p v
In addition, one-dimensional detonation was numerically simulated, where the numerical method and the reaction model were the same as the 2D numerical simulation. Figure 16 shows the comparison of the p–v and T–s curves for the 2D RDE, the 1D detonation, and the ideal ZND model. The results of the 1D and 2D numerical simulations are qualitatively consistent with the ideal ZND model. The compression process of the leading shock wave in the 2D numerical simulation (state 1 to state 2 in Fig. 16) coincides well with the ideal ZND model, although they differ for the combustion process (state 2 to state 3). The maximum pressure obtained by 2D numerical simulation is much higher than by 1D simulation and the Von Neumann Spike of the ideal ZND model, which is caused by the complex structure of 2D detonation. The maximum temperatures obtained by 1D and 2D numerical simulations are similar, but are lower than that of ideal ZND model. The maximum entropy increment by 2D numerical simulation is less than by 1D numerical simulation and the ideal ZND model. The T–s diagram shows that the expansion process is not entirely isentropic in the numerical simulation caused by the non-adiabatic movement process. That is because the length of the combustion chamber is limited and there is no nozzle connected at the exit, resulting in an incomplete expansion process in 2D p–v diagram. Through tracking large number of flow particles at the head end, we know that the fuel injected between the point B and the point C will be burned by deflagration wave, and the fuel injected between the point A and the point B will be burned by detonation wave, as shown in Fig. 17. Therefore, about 23.6% fuel is burned by deflagration, and the left is burned by detonation. This number will be different because of different combustion chamber size. The net mechanical work and thermal efficiency of the cycle can be calculated by using the p–v and T–s curves. The thermal efficiency of the particle 1 in 2D numerical simulation is 34.5%, and its net mechanical work is 1.5 MJ/kg. In order to obtain the entire performance of RDE, we calculate the average net mechanical work and thermal efficiency of the large number of particles burned by the detonation wave (injected between point A and point B in Fig. 17) and the deflagration wave (injected between point B and point C in Fig. 17), as shown in Table 1. Table 1 shows the net
d
2000
e
f T(K)
ds ¼ C v
2500
1500
1000
b
500
a 0
0
1000
2000
3000
Δs (J/(kg*K))
(b) Fig. 15. p–v and T–s diagrams of flow particle 1. (a) p–v diagram; (b) T–s diagram.
mechanical work and the thermal efficiency of the three methods. The thermal efficiencies of the ideal ZND model, 1D detonation, and 2D RDE are 51.1%, 39.7%, and 26.4%, respectively. The net mechanical work of the 1D straight-tube detonation and 2D RDE are 52.9% and 30% of the ideal ZND model. The average thermal efficiency of the detonation combustion in 2D RDE is 31%, and its average net mechanical work is 1.3 MJ/kg. Increasing the proportion of the detonation combustion can improve the performance of RDE. If a nozzle is attached at the exit in the 2D numerical simulation, the net mechanical work and the thermal efficiency will be closer to the ideal values. 4. Conclusions (1) In continuously rotating detonations, the detonation wave, the deflagration wave, the oblique shock wave, and the contact surface have a small influence on the paths of flow particles. Flow particles are injected into the combustion chamber, burned by the detonation wave or deflagration
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R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
50
40
0.05
35
2(Von Neumann)
30
r⋅θ(m)
Pressure (atm)
0.06
2D Numerical Simulation 1D Numerical Simulation Ideal ZND
45
25 20
3(C-J point)
15
0.04
C
1.1cm Deflagration: 23.6% Detonation : 76.4%
B
0.03
4.66cm
0.02
10
h=0.8cm 0.01
5
1
4
0
0
2
4
6
8
h A
10
0
0
0.01
v (m3/kg)
0.02
0.03
0.04
0.05
0.06
Z(m)
(a)
Fig. 17. Deflagration influence in RDE.
3500 3000
(3) When a flow particle encounters the deflagration wave or the contact surface, its path does not deflect. (4) The p–v and T–s diagrams obtained by numerical simulation are qualitatively consistent with the ideal ZND model. The accompanying oblique shock wave causes only a small loss in performance of the continuously rotating detonation engine performance. About 23.6% fuel is burned by deflagration, and the left is burned by detonation in this study. (5) The thermal efficiency of the entire 2D RDE is 26.4%, and its net mechanical work of a cycle is 30% of the ideal ZND model. The average thermal efficiency of detonation combustion in 2D RDE is 31%, and its average net mechanical work is 1.3 MJ/kg. Further investigation should be undertaken with a nozzle connected to the combustion chamber.
3
2D Numerical Simulation 1D Numerical Simulation Ideal ZND
T(K)
2500 2000
2 1500
4 1000 500
1 0
0
1000
2000
3000
4000
Δs (J/(kg*K))
Acknowledgment The present work is supported by the Aeronautical Science Foundation of China under Grant No. 2008ZH71006.
(b) Fig. 16. Comparison of the T–s and p–v diagrams from 1D numerical simulation, 2D numerical simulations, and the ideal ZND model. (a) p–v diagram; (b) T–s diagram.
Table 1 Net mechanical work and thermal efficiency. Combustion type
Net mechanical work (MJ/kg)
Thermal efficiency (%)
Ideal ZND model 1D detonation 2D RDE
3.6 1.9 1.1
51.1 39.7 26.4
wave, and then rapidly ejected almost along the axial direction. The fluctuation in the azimuthal direction is less than 12% of the circumference of the combustion chamber. (2) When a flow particle encounters the detonation wave or the oblique shock wave, its path is deflected. At this instant, its pressure, temperature, and density increase rapidly. After the detonation wave or oblique shock wave has traveled through the flow particle, the flow particle follows the detonation or oblique shock wave for a while. Its path then deflects again until the azimuthal velocity decreases to zero.
References [1] K. Kailasanath, AIAA J. 38 (2000) 1698–1708. [2] G.D. Roy, S.M. Frolov, A.A. Borisov, D.W. Netzer, Prog. Energy Combust. Sci. 30 (2004) 545–672. [3] S.R. Turns, An Introduction to Combustion: Concept and Application, McGrawHill, Singapore, 2000. [4] F. Ma, J.-Y. Choi, V. Yang, J. Propul. Power 22 (2006) 1188–1203. [5] T.-H. Yi, J. Lou, C. Turangan, J. Propul. Power 27 (2011) 171–181. [6] J. Kindracki, P. Wolanski, Z. Gut, Shock Waves 21 (2011) 75–84. [7] B.V. Voitsekhovskii, Sov. J. Appl. Mech. Tech. Phys. 3 (1960) 157–164. [8] F.A. Bykovskii, S.A. Zhdan, E.F. Vedernikov, J. Propul. Power 22 (2006) 1204– 1216. [9] D.M. Davidenko, Numerical simulation of H2/O2 continuous spin detonation with a detailed chemical mechanism, 21st ICDERS (CDROM), 2007. [10] M. Hishida, T. Fujiwara, P. Wolanski, Shock Waves 19 (2009) 1–10. [11] Y.T. Shao, J.P. Wang, Chin. Phys. Lett. 27 (2010) 034705. [12] Y.T. Shao, M. Liu, J.P. Wang, Combust. Sci. Technol. 182 (2010) 1586–1597. [13] Y.T. Shao, J.P. Wang, Chin. J. Aeronaut. 23 (2010) 647–652. [14] W.H. Heiser, D.T. Pratt, J. Propul. Power 18 (2002) 68–76. [15] B.G. Tong, X.Y. Kong, G.H. Deng, Gas Dynamics, Higher Education Press, Beijing, 1990. [16] T. Yamada, A.K. Hayashi, E. Yamada, N. Tsuboi, V.E. Tangirala, T. Fujiwara, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, AIAA, Orlando, 2010. [17] V.P. Korobeinikov, V.A. Levin, V.V. Markov, G.G. Chernyi, Astronaut. Acta 17 (1972) 529–537.
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S. Taki, T. Fujiwara, AIAA J. 16 (1978) 73–77. X.D. Liu, S. Osher, T. Chan, J. Comput. Phys. 115 (1994) 200–212. G.S. Jiang, C.W. Shu, J. Comput. Phys. 126 (1996) 202–228. C.W. Shu, S. Osher, J. Comput. Phys. 83 (1989) 32–78. A.P. Graf, U. Riedel, Numerical simulation of supersonic reactive flows using explicit Runge–Kutta methods, AIAA Paper 00–0438, 2000.
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[23] K.K. Kuo, Principles of Combustion, John Wiley & Sons, New York, 1986. [24] K. Kailasanath, Applications of detonations to propulsion: a review, AIAA Paper 99–1067, 1999. [25] R.A. Strehlow, Combustion Fundamentals, McGraw-Hill, New York, 1984. pp. 127–138, 302–307.
Contents lists available at SciVerse ScienceDirect
Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Numerical investigation of flow particle paths and thermodynamic performance of continuously rotating detonation engines Zhou Rui, Wang Jian-Ping ⇑ State Key Laboratory of Turbulence and Complex System, Department of Mechanics and Aerospace Engineering, Center for Combustion Propulsion and Power, College of Engineering, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 22 July 2011 Received in revised form 8 April 2012 Accepted 11 July 2012 Available online 11 August 2012 Keywords: RDE Particle path Thermodynamic performance p–v Diagram T–s Diagram
a b s t r a c t Based on the two-dimensional numerical simulation of continuously rotating detonations in an annular chamber, the paths of flow particles burned by three different processes are tracked and analyzed in detail. The detonation wave, the deflagration wave, the oblique shock wave, and the contact surface have a small influence on the paths of flow particles. The fluctuation of paths in the azimuthal direction is less than 12% of the circumference of the combustion chamber. The path will deflect when the flow particle encounters the detonation wave or the oblique shock wave, and it will not deflect when encounters the deflagration wave or the contact surface. About 23.6% fuel is burned by deflagration, and the left is burned by rotating detonation wave. The thermodynamic performance of continuously rotating detonations is then discussed. The p–v and T–s diagrams obtained by numerical simulation are qualitatively consistent with the ideal ZND model. The average thermal efficiency of the detonation combustion in 2D RDE is 31%, and its average net mechanical work is 1.3 MJ/kg. The thermal efficiency of the entire RDE is 26.4%, and its net mechanical work is 30% of the ideal ZND model. The superior performance of continuously rotating detonations is determined. Crown Copyright Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. All rights reserved.
1. Introduction Detonation combustion has an inherently higher thermodynamic efficiency than constant-pressure combustion [1–4]. In recent years, the continuously rotating detonation engine (RDE) based on detonation combustion has been extensively studied [5,6]. In the RDE, fuel is continuously injected at a high velocity, which greatly increases the average flow rate of the fuel. Therefore, the RDE has high propulsion performance. Voitsekhoviskii [7] initially proposed the concept of continuously rotating detonation and achieved a brief continuous detonation of premixed acetylene and ethylene in a circular tube. Bykovskii et al. [8] performed experiments using various fuels and achieved steady long-duration continuous detonation wave propagation. Davidenko [9] numerically simulated continuous spin detonation using a H2/O2 two-dimensional (2D) model. Hishida et al. [10] computationally modeled the detonation cellular structure at the head of 2D rotating detonations. Shao et al. [11–13] comprehensively studied three-dimensional (3D) numerical simulations in continuously rotating detonations. They obtained multicycles of continuously rotating detonations and discussed several
⇑ Corresponding author.
key issues, including the fuel injection limit, self-ignition, thrust performance, and nozzle effects. So far, there has been little research, by both numerical simulations and experiments, on the thermodynamic performance of continuously rotating detonations. Heiser and Pratt [14] have analyzed the thermodynamic cycle of pulse detonation engines, but only studied progress of the ideal cycle, which cannot be achieved in practice. In addition, most of the previous numerical simulation studies have focused on the overall description of the flow-field within a continuously rotating detonation combustion chamber. The complicated flow-field of continuously rotating detonations not only consists of a detonation wave, a deflagration wave, and an oblique shock wave, but also of a contact surface. However, the paths of flow particles and changes in the variables of the flow particles have not been studied. The process from premixed gas injection to combustion product ejection has not been investigated in any detail. It is natural to think that the thermodynamic properties can be obtained by tracking flow particles from injection into the combustion chamber to ejection from the exit. In this study, a new method for analyzing the continuously rotating detonation engines flow-field and describing its thermodynamic properties is proposed. The paths of flow particles are tracked, and the effects of the detonation wave, the deflagration wave, the oblique shock wave, and the contact surface on the paths
E-mail address: [email protected] (J.-P. Wang). 0010-2180/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. All rights reserved. http://dx.doi.org/10.1016/j.combustflame.2012.07.007
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Nomenclature p R
q T v e q
c Ds
pressure gas constant density temperature specific volume total energy heat released per unit mass of reactants ratio of specific heats entropy increment
and flow parameters are investigated. The corresponding p–v and T–s diagrams can then be used to calculate the net mechanical work and thermal efficiency. The results from numerical simulations are qualitatively and quantitatively compared with ideal ZND model.
E1, E2 Cv Cp k1, k2 r, h, z uh uz
activation energies specific heat at constant volume specific heat at constant pressure rate constants cylindrical coordinates flow velocity in circumferential direction flow velocity in axial direction progress variable of induction reaction progress variable of exothermic reaction
a b
duction, and mass diffusion are ignored in this study. Euler equations are
@U @F @G þ þ ¼S @t @h @z
ð1Þ
where 2. Physical modeling and numerical method
Z
2.1. Physical modeling Y
2.2. Numerical method The flow-field is governed by the 2D conservative Euler equations in cylindrical coordinates and the Korobeinikov et al. [17] two-step chemical reaction model. Viscosity, thermal con-
X
2
1b
1a
(a)
0.06
1a
0.05
p (atm) 26 24 22 20 18 16 14 12 10 8 6 4 2
0.04
r⋅θ (m)
The combustion chamber of the RDE is a coaxial cavity with a toroidal section. A detonation wave propagates azimuthally in the annulus while a combustible mixture is injected from the head-end, and then, the burnt gas spurts out of the downstream exit. Assuming that the depth along the radial direction is much smaller than the diameter and axial length, and then, the flow-field can be approximated as a 2D cylindrical chamber without thickness [10,13], as shown in Fig. 1a. At the head-end, there are a large number of Laval micro-nozzles to axially inject premixed stoichiometric hydrogen/oxygen gas into the combustion chamber. The mass flux of the incoming fuel is controlled by the relationship between stagnation and flow pressures at the head-end with a set exit to throat area ratio. The flow-field is initially filled with the premixed stoichiometric hydrogen and oxygen mixture at 1 atm and 300 K, except from the head wall region where a 1D C-J detonation wave is artificially placed. It is placed along the axial direction for a short distance, in order to initiate 2D detonation. Figure 1b shows the initial pressure distribution by extending the thin chamber to a plane. In the inlet nozzles at the head-end, the ratio of the exit to throat area is set to 3.7 and the stagnation pressure is set to 3 MPa, which results in a critical pressure at the nozzle exit of 1.6 MPa. If the pressure at the head-end is larger than the stagnation pressure, the fuel cannot be injected. Therefore, a rigid wall condition is used at the head-end. If the pressure at the head-end is smaller than the inflow pressure, the inflow condition is given by Laval nozzle theory [15,16]. At the downstream exit, we use non-reflecting boundary conditions, with a relaxation factor of 0.05. The radius of the combustion chamber is 1 cm and the length is 5 cm. The number of grid points is 400 in the azimuthal direction and 500 in axial direction, resulting in Dz = 0.1 mm and D(r h) = 0.157 mm. The grid dependency has been validated in Ref. [12]. The upper and lower sides are connected by periodic boundary conditions.
2
0.03
0.02
1b 0.01
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Z (m)
(b) Fig. 1. (a) Initial pressure distributions of the annulus chamber without thickness and (b) the plane extended from (a).
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R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645 (T/300)/K
Contact Surface
0.04
2 0.03 0.02
1
0
0.06
Induced Shock Wave 0
0.01
0.02
0.03
0.04
0.05
r⋅θ (m)
0.02
0.01
0
0.06
0
0.01
0.02
0.03
(a)
(c) 0.06
ρ (kg/m3)
2
0.02
0.02
0.03
0.04
0.05
0.06
0.04
0.05
0.06
3
β
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2 0.03
0.02
Fresh Gas
0.01
Induced Shock Wave
0.01
1
Z (m)
0.03
0
2 0.03
Z (m)
0.04
0
0.04
2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6
1
50 45 40 35 30 25 20 15 10 5
Axial Direction
Detonation Front
0.05
0.01
p (atm)
Oblique Shock Wave
3
3
0.05
Azimuthal Direction
Deflagration 0.01
0.06
r⋅θ (m)
Detonation Front
0.05
r⋅θ (m)
9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
Oblique Shock Wave
3
r⋅θ (m)
0.06
0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2
1
0
0
0.01
0.02
0.03
Z (m)
Z (m)
(b)
(d)
0.04
0.05
0.06
Fig. 2. Flow distribution and initial positions of the just-injected fuel with a stable propagating detonation wave. (a) Temperature contours; (b) density contours; (c) pressure contours and streamlines; (d) contours of the progress variable for chemical reaction.
Partical path3 0.06
3
ΙΙ
0.05
r⋅θ(m)
Ι
p (atm)
0.055
0.05
Partical path 2
0.04
50 45 40 35 30 25 20 15 10 5
r⋅θ(m)
0.06
0.03
0.02
0.01
0
0.005
0.01
0.015
Z(m)
0
Partical path 1 0
0.01
0.02
0.03
0.04
Z(m) Fig. 3. Local enlargement of the pressure distribution near the detonation wave front.
Fig. 4. Initial paths of the three particles.
0.05
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d 1
c b
qa
0.8
2
0.7
0.5
2
g
0.4
quz a 0
0.01
0.02
0.03
0.04
0.05
e¼
Fig. 5. Path of particle 1.
3
50
Pressure (atm)
b d
p
0
10
20
30
40
1500
20
1000
d b e
g
0
10
20
30
(a)
(b)
1400 1200
e
0
600
f
1000
g 800
400 200
b
e d
0
f g
400
c
-200
d 200
50
800
UZ Uθ
b
40
500
1000
a c
a
f
b
Time (μs)
600
g
25
Time (μs)
1600
2000
f
30
0
50
2500
P T
e
5 a a
0
ð4Þ
d
10
g
0.5 a
1 1 þ bqq þ qu2h þ qu2z 2 2
35
15
f
e
UZ (m/s)
Density (kg/m3 )
2
ð3Þ
c1
40
1
qxa
c c
45
c
1.5
ð2Þ
p ¼ qRT
Z(m)
2.5
3
0
0
10
20
30
40
50
Time (μs)
(c) Fig. 6. Change in the flow parameters of particle 1. (a) Density; (b) pressure and temperature; (c) axial and azimuthal velocity.
T(K)
e
3
7 7 6 6 7 6 6 0 7 7 7 6 6 7 7 6 6 7 6 6 0 7 7 S¼6 7 G¼6 7 7 6 6 6 ðe þ pÞuz 7 6 0 7 7 7 6 6 6 qu b 7 6 qx 7 5 4 4 z b5
f
0.6
quz quh uz qu2z þ p
quh a
Uθ (m/s)
a 0.9
3
2
q quh 6 qu 7 6 qu2 þ p 7 7 6 h7 6 h 7 7 6 6 6 quz 7 6 quh uz 7 1 7 7 6 U¼6 6 e 7 F ¼ r 6 ðe þ pÞu 7 h7 7 6 6 7 7 6 6 4 qb 5 4 quh b 5
1.1
r⋅θ(cm)
3
2
R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
0.06
0.06
0.05
0.05
0.04
0.04
r⋅θ (m)
r⋅θ (m)
3636
0.03
0.02
0.02
0.01
a 0
0.01
0.02
0.03
0.04
0.05
0
0.06
0
0.01
Z(m)
(a)
(b) 0.06
0.05
0.05
0.04
0.04
0.03
0.03
0.04
0.05
0.06
0.03
0.04
0.05
0.06
0.04
0.05
0.06
0.03
0.02
0.02
c
0.01
0
d
0.01
0.01
0.02
0.03
0.04
0.05
0
0.06
0
0.01
Z(m)
0.02
Z(m)
(c)
(d)
0.06
0.06
0.05
0.05
0.04
0.04
r⋅θ (m)
r⋅θ (m)
0.02
Z(m)
0.06
0
b
0.01
r⋅θ (m)
r⋅θ (m)
0
0.03
0.03
0.02
0.03
0.02
0.01
0.01
e 0
0
0.01
0.02
f 0.03
Z(m)
(e)
0.04
0.05
0.06
0
0
0.01
0.02
0.03
Z(m)
(f)
Fig. 7. Temperature distribution of flow particle 1 around the points of Fig. 5.
R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
0.06 T/300 (K) 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
0.05
r⋅θ(m)
0.04
0.03
0.02
0.01
g 0
0
0.01
0.02
0.03
0.04
0.05
0.06
Z(m)
(g) Fig. 7 (continued)
4.2
g
e
4
r⋅θ(cm)
f 3.8
3.6
a
b
3.4
c d
0
0.01
0.02
0.03
0.04
0.05
Z(m) Fig. 8. Path of particle 2.
xa ¼
da ¼ k1 q expðE1 =RTÞ dt
db xb ¼ ¼ dt
(
ð5Þ
h i 2 k2 p2 b2 exp E ð1 bÞ2 exp E2RTþq ða ¼ 0Þ RT 0 ð1 > a > 0Þ
3637
cycles. The fuel injected into the combustion chamber can only be burned by two types of processes: deflagration and detonation. Figure 2 shows the distributions of temperature, density, pressure with streamlines, and the progress variable b. In Fig. 2a, the following can be identified: the detonation front; the attached oblique shock wave; the contact surface between the current cycle’s chemical reaction products and the previous cycle’s products; deflagration occurring at the contact surface of the fresh gas mixture with the high-temperature burned gas; and the induced shock wave near the head-end. Furthermore, in the close-up of the pressure distribution near the detonation front (Fig. 3), it can be seen that the detonation front is composed of two parts. Part I burns off the fresh gas mixture before the induced shock wave, while part II burns off the gas mixture after the induced shock wave. As a consequence, detonation combustion is divided into two cases in this study. Figure 3 shows the local enlargement of the pressure distribution near the detonation wave front. The increase in pressure at detonation front II is larger than at detonation front I, leading to different paths and flow variables when a particle encounters each one. One flow particle is injected into the head-end and its path, and flow variables are tracked for each case. Generally, the particle paths are different from the streamlines in unsteady flows. It is necessary to calculate the particle paths step-by-step by solving the governing equations. The particle’s velocity can be calculated by area interpolation of the flow velocities at the four closest grid points. The time step is very small in the numerical simulations, so that the distance moved by each particle in a time step is less than the grid size. The initial positions of the three typical particles that were followed are shown in Fig. 2. At this instant, the flow particles have just been injected into the head-end of the combustion chamber. After detailed analysis, particle 1 is burned by detonation wave II, particle 2 is burned by the deflagration wave, and particle 3 is burned by detonation wave I. More details will be discussed in the following paragraphs. Figure 4 shows the paths of the three particles from injection into the combustion chamber to exhaustion. The horizontal line is the boundary of cycle. The particle paths have small fluctuations in the azimuthal direction, which are less than 12% of the circumference, independent of their initial positions. The average time of the three flow particles in the combustion chamber is 58.4 ls. The period of detonation propagating in the azimuthal direction is 26.2 ls, which indicates that detonation propagates almost two cycles when the particles are moving through the chamber. The particle paths differ depending on whether the flow particles encounter the contact surface, the detonation, deflagration, or oblique shock waves, and where they encounter them. Thus, the changes in the flow variables are also different. In the following section, we analyze particle 1 in detail, with the other two only discussed if their behavior is different to particle 1.
ð6Þ The parameters in the equations are set to be the same as Refs. [17,18]. The spatial terms are discretized with a 5-step WENO scheme [19,20], and the temporal terms are discretized with the 4-step Runge–Kutta method [21,22]. 3. Results and discussions 3.1. Trajectory analysis After initiation of the detonation wave by the extended 1D C-J detonation, the wave becomes stable after propagating for eight
3.1.1. Particle 1 Figure 5 shows the path of particle 1 enlarged about eight times in the azimuthal direction. We use a–g to mark seven key points in the path of particle 1 and discuss the properties of the particle in the phases between these points. Figure 6 shows the variation of particle 1’s density, pressure, temperature, axial velocity, and azimuthal velocity with time. Figure 7 shows the temperature distributions and the locations of particle 1 in the flow-field when particle 1 moves via the points in Fig. 5. The path of particle 1 and the change in conditions of the flow parameters are as follows.
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R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
2 P T
25
f
e
Pressure (atm)
20
b 1
g
f 0.5 a
c
c
g 2000
d
1500
15
10
f
b
e d
c
b
T(K)
1.5
Density (kg/m3)
2500
g 1000
e d 500
5 a 0
a
0
10
20
30
40
0
50
0
10
20
30
40
Time (μs)
Time (μs)
(a)
(b) 2500
50
0
1500 UZ Uθ
2000
1000
a
f b
c
e
d
0
f
1000
b
c
g
e
d
g
Uθ (m/s)
U Z (m/s)
500
a
1500
-500 -1000
500
-1500 0
0
10
20
30
40
50
Time (μs)
(c) Fig. 9. Changes in flow parameters of particle 2. (a) Density; (b) pressure and temperature; (c) axial and azimuthal velocities.
(1) a to b: First, particle 1 is injected into the head wall of the combustion chamber, as shown by point a in Figs. 5 and 7a. In the streamline distribution (Fig. 2c), particle 1 first moves to the right and then down with the flow, until it reaches point b where it is just in front of detonation wave II (Fig. 7b). (2) b to c: In this phase, the particle encounters detonation wave II. It only moves a very small distance, but its path is deflected, as shown in Figs. 5 and 7b and c. At the instant, it encounters the detonation wave, its density, pressure, temperature, and azimuthal velocity increase rapidly, while its axial velocity decreases. As shown in Fig. 6, the maximum pressure and temperature are 50 atm and 2700 K, respectively. (3) c to d: Particle 1 follows detonation wave II until the azimuthal velocity decrease to zero (point d in Figs. 5 and 7d), where its path is deflected again. In this phase, the pressure, density, and azimuthal velocity of particle 1 decrease rapidly. The temperature also decreases, but it remains high.
The axial velocity fluctuates with small amplitude (Fig. 6). This phase confirms the expansion phenomenon after the detonation wave. (4) d to e: Initially, the density, pressure, temperature and azimuthal velocity of particle 1 all decrease, while the axial velocity increases. The thermodynamic parameters tend to be stabilized in this phase, and particle 1 moves downstream. After the detonation wave has propagated almost one cycle, the particle encounters the oblique shock wave coupled with the detonation wave, where its path deflects again, as shown in Fig. 5 and near point e in Fig. 7e. At this moment, its density, pressure, temperature, azimuthal and axial velocities all increase. The variations in the trends of the flow parameters, apart from the axial velocity, are similar when the particle encounters the oblique shock wave and when it encounters the detonation wave. Only the degree of the change is different, there is a greater change when encountering the detonation wave than the oblique shock wave. The paths are deflected in both cases.
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R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
0.06
0.06
0.05
0.05
0.04
b 0.03
0.02
0.02
0.01
0.01
0
r⋅θ (m)
r⋅θ (m)
a 0.03
0
0.01
0.02
0.03
0.04
0.05
0
0.06
0.05
0.04
0.04
c 0.03
0.01
0.01
0.03
0.04
0.05
0
0.06
0
0.01
(c) 0.06
0.05
0.05
e r⋅θ (m)
r⋅θ (m)
0.04
0.03
0.03
0.04
0.05
0.06
0.05
0.06
0.03
0.02
0.01
0.01
0.03
0.04
0.05
0.06
f
0.04
0.02
0.02
0.02
(d)
0.06
0.01
0.06
Z(m)
Z(m)
0
0.05
0.03
0.02
0.02
0.04
d
0.02
0
0.03
(b)
0.05
0.01
0.02
(a) 0.06
0
0.01
Z(m)
0.06
0
0
Z(m)
r⋅θ (m)
r⋅θ(m)
0.04
0
0
0.01
0.02
0.03
Z(m)
Z(m)
(e)
(f)
Fig. 10. Temperature distributions of particle 2 around the points labeled in Fig. 8.
0.04
3640
R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
0.06
T/300 (K) 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
0.05
g
r⋅θ(m)
0.04
0.03
0.02
0.01
0
0
0.01
0.02
0.03
0.04
0.05
0.06
Z(m)
(g) Fig. 10 (continued)
6.4
d
r⋅θ(cm)
0
b
6.2
e
c
6
a
0
0.01
0.02
0.03
0.04
0.05
Z(m) Fig. 11. Path of particle 3.
(5) e to f: After passing through the oblique shock wave, the density, pressure, temperature, azimuthal, and axial velocities of particle 1 decrease and tend to stabilize. The particle follows the oblique shock wave, as shown at point f in Figs. 5 and 7f, until the azimuthal velocity becomes negative and then the path deflects again. (6) f to e: The particle continues moving downstream until it is exhausted out of the combustion chamber, as shown at point g in Figs. 5 and 7g. In summary, particle 1 encounters detonation wave II once, the oblique shock wave once, and does not encounter the contact surface or the deflagration wave. 3.1.2. Particles 2 and 3 In contrast to particle 1, flow particle 2 is burned by the deflagration wave. Figures 8–10 show the path of particle 2, changes
in the flow parameters, and the temperature distributions around the key points labeled in Fig. 8. First, flow particle 2 is injected into point at the head wall and its axial velocity decreases rapidly due to the induced shock wave (Fig. 10a). Particle 2 encounters the deflagration wave at point b and passes through it to point c. The progress variable of exothermic reaction has a value of 0.9 at point b and 0.2 at point c. This suggests that the process occurring between these points is deflagration combustion. It is observed that the particle path is not affected during the process of deflagration combustion. Figure 9 shows that the temperature increases to 2000 K, the pressure slightly decreases in this phase. Particle 2 continues moving to point d, where it encounters the oblique shock wave and its path deflects (point d in Fig. 10d). All of the flow variables shown in Fig. 9 increase due to the shock wave. The particle moves to point e and encounters the contact surface, as shown at point e in Fig. 10e. The strength of contact with the surface is so weak that the flow parameters are almost unaffected and the path does not change. The particle moves to point f, where it encounters the oblique shock wave again (Fig. 10f). Finally, the particle exits the combustion chamber at point g. In summary, flow particle 2 encounters the deflagration wave once, the oblique shock wave twice, the contact surface once, and it does not encounter the detonation wave. Figure 11 shows the path of particle 3, and the horizontal line is the boundary of cycle. Particle 3 is injected into the combustion chamber and is instantly burned by detonation wave I. Its pressure, temperature, density, and azimuthal velocity increase rapidly, while its axial velocity decreases, as shown in Fig. 12. The maximum pressure is 22 atm, and the maximum temperature is 2700 K. Although the maximum pressure is only 44% of the maximum pressure of particle 1, which is burned by detonation wave II, the maximum temperature increases to the same level. After detonation wave I passes through particle 3, the particle travels closely to the detonation wave. At point b, the azimuthal velocity decreases to zero, and the path deflects. It keeps moving to point c and encounters the oblique shock wave, as shown in Fig. 13c. The subsequent phases are the same as for particle 1. The variation trends of the flow parameters are similar when the particle encounters detonation wave I and detonation wave II, but are different in magnitude. In summary, flow particle 3 encounters the detonation wave I once, the oblique shock wave once, and does not encounter the contact surface or the deflagration wave. Figure 14 shows the changes in pressure of the three flow particles with time. The pressure of particle 1, burned by detonation wave II, reaches about 50 atm, while the other two particles only reach about 20 atm. In other words, the fuel burned by detonation wave II contributes most to the performance of RDEs. The detailed analysis of the movement of the three typical flow particles shows that the detonation wave and the oblique shock wave do have an impact on the path of particles. This is due to deflection of the paths, and the variations of the flow parameters are more significant than the fluctuations of the paths. The pressures, temperatures, and densities of all 3 particles increased when they encountered the detonation and oblique shock waves. However, the paths do not deflect when the particles encounter the contact surface and the deflagration wave. 3.2. Analysis and comparison of thermodynamic performance Pressure as a function of specific volume is defined by the p–v diagram, which represents the mechanical work of the thermodynamic processes. Temperature as a function of the entropy increment is defined by the T–s diagram, which represents the heat release of the thermodynamic processes. In this paper, the p–v
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R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
1.5 3000 P T
1
d e
0.5
a
d
2000
15
e 1500
c 10
T (K)
c b
2500
c
b
Pressure (atm)
Density (kg/m3 )
20
b d
1000
e 5
0
0
10
20
30
40
0
50
500
a a 0
10
20
30
Time (μs)
Time (μs)
(a)
(b)
2500
40
50
0
1000 UZ Uθ
2000
500
U Z (m/s)
d
b
a
1500 a
0
e c
1000
d e
-500
Uθ (m/s)
c
b 500
0
-1000
0
10
20
30
40
50
-1500
Time (μs)
(c) Fig. 12. Changes of the flow parameters of particle 3. (a) Density; (b) pressure and temperature; (c) axial and azimuthal velocities.
and T–s diagrams of flow particle 1 were obtained. The relaxation length is set at the exit to ensure that the p–v and T–s curves are closed, in which the values of the physical parameters in the combustion flow-field decrease to the ambient values at infinity. Figure 15 shows the p–v and T–s diagrams of flow particle 1 from 2D numerical simulation. The points marked in Fig. 15 correspond to the same points in Fig. 5. At point e, the particle encounters the oblique shock wave, causing fluctuation of the p–v and T–s curves and a slight increase in Ds. However, the general trends of the p–v and T–s curves do not change. Therefore, the accompanying oblique shock wave causes only a small loss in performance of the continuously rotating detonation engine. The black dashed-dotted line in Fig. 16 shows the ideal ZND model [23–25]. The ideal ZND model has four phases. (1) The fuel with initial state 1 is injected into the combustion chamber and is compressed by the leading shock wave. It increases to the Von Neumann Spike along the Hugoniot line of the shock wave in the p–v diagram (point 2 in Fig. 16). Both the temperature and entropy
increase in this phase. (2) The fuel undergoes an intense chemical reaction and releases heat. The combustion products begin to expand, causing a decrease in pressure and density and an increase in temperature and entropy. It then reaches point C-J (point 3) from the Von Neumann Spike along the Rayleigh line in the p–v diagram. (3) The detonation products expand until the pressure reaches the initial pressure (point 4), according to the isentropic law. (4) The curve is forced to close isobarically, and the cycle is complete. In order to systematically compare the results of the numerical simulation with the ideal ZND model, the choice of parameters in the ideal ZND model are the same as those in the numerical simulation. A premixed stoichiometric hydrogen/oxygen mixture is injected into the combustion chamber with an initial pressure of 1 atm, an initial temperature of 300 K, a specific heat ratio of c = 1.4, the gas constant is R = 692.9 J/(kg K), the heat released per unit mass of reactants of q = 4 106 J/kg. The formula for calculating the entropy is:
3642
R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
0.05
0.04
0.04
r⋅θ(m)
0.05
0.03
0.02
0.01
0.01
0
0.01
0.02
0.03
0.04
0.05
0
0.06
0.03
(b)
0.04
0.05
0.06
0.04
0.05
0.06
0.06
0.04
r⋅θ(m)
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0.01
0.02
(a)
0.05
0
0.01
Z(m)
0.05
0
0
Z(m)
c
0.06
r⋅θ(m)
0.03
0.02
0
b
0.06
a
0.02
0.03
0.04
0.05
0
0.06
d 0
0.01
0.02
Z(m)
0.03
Z(m)
(c)
(d) e
0.06
T/300 (K) 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
0.05
0.04
r⋅θ(m)
r⋅θ(m)
0.06
0.03
0.02
0.01
0
0
0.01
0.02
0.03
0.04
0.05
0.06
Z(m)
(e) Fig. 13. Temperature distributions of particle 2 around the points labeled in Fig. 11.
3643
R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
1
50
50
c
Pressure (atm)
Pressure (atm)
40
Particle1 Particle2 Particle3
40
30
20
2
3
1
2
30
20
d e
10
10
b
f a
0
0
0
10
20
30
40
50
0.5
60
1
1.5
2
v (m3/kg)
Time (μs) Fig. 14. Change in pressure for the three flow particles.
(a)
3000
c dp dv þ Cp p v
In addition, one-dimensional detonation was numerically simulated, where the numerical method and the reaction model were the same as the 2D numerical simulation. Figure 16 shows the comparison of the p–v and T–s curves for the 2D RDE, the 1D detonation, and the ideal ZND model. The results of the 1D and 2D numerical simulations are qualitatively consistent with the ideal ZND model. The compression process of the leading shock wave in the 2D numerical simulation (state 1 to state 2 in Fig. 16) coincides well with the ideal ZND model, although they differ for the combustion process (state 2 to state 3). The maximum pressure obtained by 2D numerical simulation is much higher than by 1D simulation and the Von Neumann Spike of the ideal ZND model, which is caused by the complex structure of 2D detonation. The maximum temperatures obtained by 1D and 2D numerical simulations are similar, but are lower than that of ideal ZND model. The maximum entropy increment by 2D numerical simulation is less than by 1D numerical simulation and the ideal ZND model. The T–s diagram shows that the expansion process is not entirely isentropic in the numerical simulation caused by the non-adiabatic movement process. That is because the length of the combustion chamber is limited and there is no nozzle connected at the exit, resulting in an incomplete expansion process in 2D p–v diagram. Through tracking large number of flow particles at the head end, we know that the fuel injected between the point B and the point C will be burned by deflagration wave, and the fuel injected between the point A and the point B will be burned by detonation wave, as shown in Fig. 17. Therefore, about 23.6% fuel is burned by deflagration, and the left is burned by detonation. This number will be different because of different combustion chamber size. The net mechanical work and thermal efficiency of the cycle can be calculated by using the p–v and T–s curves. The thermal efficiency of the particle 1 in 2D numerical simulation is 34.5%, and its net mechanical work is 1.5 MJ/kg. In order to obtain the entire performance of RDE, we calculate the average net mechanical work and thermal efficiency of the large number of particles burned by the detonation wave (injected between point A and point B in Fig. 17) and the deflagration wave (injected between point B and point C in Fig. 17), as shown in Table 1. Table 1 shows the net
d
2000
e
f T(K)
ds ¼ C v
2500
1500
1000
b
500
a 0
0
1000
2000
3000
Δs (J/(kg*K))
(b) Fig. 15. p–v and T–s diagrams of flow particle 1. (a) p–v diagram; (b) T–s diagram.
mechanical work and the thermal efficiency of the three methods. The thermal efficiencies of the ideal ZND model, 1D detonation, and 2D RDE are 51.1%, 39.7%, and 26.4%, respectively. The net mechanical work of the 1D straight-tube detonation and 2D RDE are 52.9% and 30% of the ideal ZND model. The average thermal efficiency of the detonation combustion in 2D RDE is 31%, and its average net mechanical work is 1.3 MJ/kg. Increasing the proportion of the detonation combustion can improve the performance of RDE. If a nozzle is attached at the exit in the 2D numerical simulation, the net mechanical work and the thermal efficiency will be closer to the ideal values. 4. Conclusions (1) In continuously rotating detonations, the detonation wave, the deflagration wave, the oblique shock wave, and the contact surface have a small influence on the paths of flow particles. Flow particles are injected into the combustion chamber, burned by the detonation wave or deflagration
3644
R. Zhou, J.-P. Wang / Combustion and Flame 159 (2012) 3632–3645
50
40
0.05
35
2(Von Neumann)
30
r⋅θ(m)
Pressure (atm)
0.06
2D Numerical Simulation 1D Numerical Simulation Ideal ZND
45
25 20
3(C-J point)
15
0.04
C
1.1cm Deflagration: 23.6% Detonation : 76.4%
B
0.03
4.66cm
0.02
10
h=0.8cm 0.01
5
1
4
0
0
2
4
6
8
h A
10
0
0
0.01
v (m3/kg)
0.02
0.03
0.04
0.05
0.06
Z(m)
(a)
Fig. 17. Deflagration influence in RDE.
3500 3000
(3) When a flow particle encounters the deflagration wave or the contact surface, its path does not deflect. (4) The p–v and T–s diagrams obtained by numerical simulation are qualitatively consistent with the ideal ZND model. The accompanying oblique shock wave causes only a small loss in performance of the continuously rotating detonation engine performance. About 23.6% fuel is burned by deflagration, and the left is burned by detonation in this study. (5) The thermal efficiency of the entire 2D RDE is 26.4%, and its net mechanical work of a cycle is 30% of the ideal ZND model. The average thermal efficiency of detonation combustion in 2D RDE is 31%, and its average net mechanical work is 1.3 MJ/kg. Further investigation should be undertaken with a nozzle connected to the combustion chamber.
3
2D Numerical Simulation 1D Numerical Simulation Ideal ZND
T(K)
2500 2000
2 1500
4 1000 500
1 0
0
1000
2000
3000
4000
Δs (J/(kg*K))
Acknowledgment The present work is supported by the Aeronautical Science Foundation of China under Grant No. 2008ZH71006.
(b) Fig. 16. Comparison of the T–s and p–v diagrams from 1D numerical simulation, 2D numerical simulations, and the ideal ZND model. (a) p–v diagram; (b) T–s diagram.
Table 1 Net mechanical work and thermal efficiency. Combustion type
Net mechanical work (MJ/kg)
Thermal efficiency (%)
Ideal ZND model 1D detonation 2D RDE
3.6 1.9 1.1
51.1 39.7 26.4
wave, and then rapidly ejected almost along the axial direction. The fluctuation in the azimuthal direction is less than 12% of the circumference of the combustion chamber. (2) When a flow particle encounters the detonation wave or the oblique shock wave, its path is deflected. At this instant, its pressure, temperature, and density increase rapidly. After the detonation wave or oblique shock wave has traveled through the flow particle, the flow particle follows the detonation or oblique shock wave for a while. Its path then deflects again until the azimuthal velocity decreases to zero.
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