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Investigation on the detonation propagation limit criterion for methane-oxygen mixtures in tubes with different scales
Fuel 239 (2019) 617–622
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Full Length Article
Investigation on the detonation propagation limit criterion for methaneoxygen mixtures in tubes with different scales ⁎
T
⁎
Bo Zhanga, , Hong Liua, , Bingjian Yanb a b
Shanghai Jiao Tong University, School of Aeronautics and Astronautics, Shanghai 200240, China East China University of Science and Technology, School of Resources and Environmental Engineering, Shanghai 200237, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Detonation limits Instability Boundary conditions
The detonation propagation limits of CH4-O2 mixtures (φ = 0.25–2) in a macro-channel (D = 36 mm) and in micro-channels with heights of 2 mm, 4.5 mm and 7 mm were studied. The critical pressure (pc) below which detonation cannot maintain steady propagation was measured experimentally for each mixture in different channels. At the condition of pc, the scaling between detonation cell size (λ) and tube dimension (DH) were also analyzed to explore the detonation failure mechanism in channels of macro- and micro-scale. The experimental results show that the detonation propagation limit criterion λ = πD (D is the inner diameter) holds in a macrochannel for the mixture with larger value of instability; however, the detonation instability is reduced as the mixture tends to fuel-rich condition, where this typical relation breaks down. The results indicate that the propagation limit criterion in macro-channel depends less on the boundary conditions of the tube and more on the detonation instability. In the micro-channels, because of the losses from the walls and mass divergence into the boundary layers, detonation propagates in a thinner channel with larger velocity deficit (VD), thereby increasing both the reaction zone length and the detonation cell size; therefore, λ = πD is unsuitable as a criterion of the detonation propagation limit, which suggests that for detonation propagating in the micro-channels, the propagation mechanism is mainly governed by the boundary conditions.
1. Introduction
unstable propagation behaviors appear, for instance, spinning detonation, galloping detonation, etc., as abundantly reported in the relevant literature. When the detonation is near the limits, the difference that distinguishes the failure and successful propagation is ambiguous [11,12]. Therefore, a criterion to estimate the onset of detonation limits is required to further explore its propagation and failure mechanism. One qualitative criterion to describe the limits is the appearance of the single head spinning structure in the tube, e.g., Lee [1,13] proposed λ = πD as a criterion for near-limit condition of detonations propagate in round tubes, where λ represents the cell size, D is the inner diameter of a tube. This criterion corresponds to the first appearance of a single headed spinning structure. When the detonation structure is singleheaded, the vertical dimension of the foil is the circumference of the tube, i.e., πD. Dupré et al. [14] investigated the near-limit detonation limits through changing the composition of H2-air mixtures in different diameter tubes. For the cell size less than value of πD, the unstable propagation behavior of detonation was observed and the velocity was found to suffered significant fluctuation. Recently, the criterion of λ = πD for detonation limits was confirmed by Gao et al. [15], Fischer et al. [16] and Yoshida et al. [17].
A detonation is a chemical reaction transport process accompanied by high-speed energy release [1]. On one hand, detonation is destructive in nature; unintentional detonation would cause casualties and make industrial facilities suffered heavy losses, thereby presenting great security risks especially in chemical and mining industries. On the other hand, because detonation has excellent thermal propulsion performances, detonation performed in a controlled manner could be applied in the field of advanced aerospace propulsion devices, for example, pulse detonation engines [2], rotating detonation engines [3], and oblique detonation propulsion technology [4–10]. Either to suppress destructive detonation or to improve safety during controlled propagation, a comprehensive understanding of the conditions that the detonation propagation can be sustained is of great importance for industrial processes and detonation applications. When the detonation process is well within its limits, the detonation velocity is steady, and it is therefore considered as a self-sustained detonation. In contrast, as the detonation limits are approached, the velocity fluctuates and has a significant deficit, and a variety of
⁎
Corresponding authors. E-mail addresses: [email protected] (B. Zhang), [email protected] (H. Liu).
https://doi.org/10.1016/j.fuel.2018.11.062 Received 27 July 2018; Received in revised form 31 October 2018; Accepted 11 November 2018 0016-2361/ © 2018 Elsevier Ltd. All rights reserved.
Fuel 239 (2019) 617–622
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Nomenclature CJ VCJ p0 Pc DH ΔI RT TOA χ V ρ0
VD ZND φ λ w D DDT AC LCell δ* μe x
Chapman-Jouguet CJ detonation velocity Initial Pressure Critical Pressure Tube dimension Induction Zone Length Round tube time-of-arrival Stability parameter Detonation velocity Initial density
Velocity deficit Zel’dovich-von Neumann-Döring Equivalence ratio Width of cell size Height of channel Inner diameter of the round tube Deflagration-to-Detonation Transition Annular channel Detonation cell length Boundary layer displacement thickness Viscosity Reaction zone thickness
The driver section was filled with equi-molar C2H2-O2, which was one of the most sensitive mixtures and readily to directly form a detonation. The test mixture was filled into the tube through different inlet. The length of guide and test section was 1.15 m and 1.35 m, respectively. The length of the round tube was 2.5 m. Similar as previous studies [15,21,22], in which optical sensors were applied to calculate the velocity of a deflagration or detonation. In this work, we also employed this technique to measure the velocity of combustion wave in a short distance. To validate the reliability of optical sensors, we compared the results with those used shock pins, it can be found that the signals from the shock pin and the optical fiber located at the same cross section appeared simultaneously for a successful detonation, whereas, for the deflagration case, the reaction zone is decoupled from precursor shock, and therefore, the TOA from the optical fiber was slower than that from the shock pin; the details can be found in Ref. [23]. Typical TOA signals from the optical detectors for steady and unsteady velocities are given in Fig. 2 and 3. The x-axis is the distance apart from the first optical sensor. The average velocity of the wave is computed by taking the slope of the curve. Fig. 2 illustrates that in the 7-mm annular channel section and at p0 = 14 kPa, the velocity has a slight deficit but is still steady, and the velocity is of 84.2% VCJ. As the p0 is somewhat lowered to 13 kPa, although the detonation velocity in the guide section is still 96.4% VCJ, the detonation velocity in the annulus deviates from the CJ value (given in Fig. 3), and the detonation completely fails at the end of the annular section. It is noteworthy that VCJ is the theoretical Chapman-Jouguet (CJ) detonation velocity, it is independent of boundary conditions but depends merely on the thermodynamic properties of the combustible mixture. In this study, VCJ is computed by CEA [24], which is a thermodynamic equilibrium program. After inputting the thermodynamic parameters of the mixture (e.g., initial pressure, composition, initial temperature, etc.), the value of VCJ is then given by CEA. In this study, methane-oxygen mixtures of different compositions (φ = 0.25, 0.33, 0.4, 0.5, 0.8, 1, 1.33, 1.6 and 2) were considered as test mixtures. The equivalence ratio values covered fuel-lean to fuelrich conditions that have different degrees of detonation instability. The mixtures were prepared by partial pressure. OMEGA PX409 (0–150 PSI) was used to measure the initial pressure of test mixture. To make sure the homogeneity of the mixture by molecular diffusion, the mixtures
Previous studies have suggested that λ = πD is an applicable criterion for determining the propagation limits for most gaseous combustible mixture detonations in macro-channels. However, recently, by investigating the detonation propagation limits of stable mixtures (with very regular cellular pattern, e.g., C2H2-2.5O2-70%Ar, C2H2-2.5O285%Ar) in ducts, Zhang et al. [18] claimed the relation λ = πD breaks down and the ratio of λ/D is much lower than π at near detonation-limit condition, which differs from the general knowledge of detonation limit criterion. It is known the detonation instability of a stable mixture is much lower than that of unstable mixtures, for the latter mixtures, they usually have very irregular cellular pattern. The above result indicated the limit criterion λ = πD may depends on the property of the mixture. Because most of the previous studies were performed in unstable mixtures at stoichiometric condition, to confirm whether the instability affects the detonation limit criterion λ = πD, a scaling analysis of detonation limits for the mixture that comprises a broad range of equivalence ratios from lean to rich (i.e., the instability is varied accordingly) is required. In addition, because the criterion of λ = πD is based on the results in channels with macro-scale, for the detonation propagation in micro-tube (e.g., thin annular channels), where the detonation is strongly dependent on the boundary conditions of the channel, and it subjects to significant loss (e.g., energy and momentum) from the walls, it is reasonable to speculate that the failure mechanism should be different from that in macro-channels; however, under such condition, the scaling between the tube dimension and the detonation structure length scale near the detonation-limit remains unclear. In this work, a systematic investigation is carried out to analysis the features of near-limit behavior of detonation in macro- and microchannels using typical unstable mixture (CH4-O2) with various equivalence ratios (φ = 0.25–2). The purpose of this work is twofold: 1) the instability of CH4-O2 mixture is varied by changing the equivalence ratio to explore the influence of instability of the mixture on the detonation limit criterion in macro-channel, and 2) detonation limits in thin annular channels (micro-scale) are also studied to explore the failure mechanism under high loss conditions. 2. Experimental details Three annular channels (ACs) and one round tube (RT) were used in the present investigation, for the round tube, the inner diameter D is 36 mm, and the heights of the annulus are 2 mm, 4.5 mm and 7 mm. The height of the annular channel was varied by changing the length of fins. Wu et al. [19] as well as Han et al. [20] suggested the geometry of O (10 mm) and larger is considered as macro-scale and that the geometry less than O (10 mm) mm is considered as micro-scale. A schematic of the apparatus is given by Fig. 1. The tube in Fig. 1 was divided into three sections. For the driver section, the length was 1.2 m, and the diameter was 68 mm; a spark seated at the beginning of the driver section, which was used to supply initiation energy for a detonation.
Fig. 1. Experimental apparatus. 618
Fuel 239 (2019) 617–622
B. Zhang et al. 0.5
1.5x10
-0.5
Voltage / V
-3
Previous researchers have suggested the detonation VD was caused by three aspects: First, the wall effect results in heat and momentum losses [29,30]; Second, the divergence of mass into the boundary layers, this effect from the annular channel results in the streamlines in the chemical reaction zone and then augments the VD [23,31,32]. Third, the quenched layer adjacent to the tube wall decreases the reaction rate, and the chemical energy that supports the propagation detonation is also reduced, which eventually accounts for the VD [33]. Fig. 5 illustrates the variation of the experimentally measured pc of the detonation limits versus the equivalence ratio for CH4-O2 mixture in different channels. The experimental data indicate that the value of pc together with its φ is in a “U” behavior, the minimum of the pc locates at approximately φ = 1. This behavior indicates the mixture near the condition of stoichiometric is the most sensitive for detonation; when at φ < 1 or φ > 1 condition, the critical pressure for detonation limits is also increased for the same type of channel.
0.0
-3
2.0x10
-1.0 -1.5 -2.0 -2.5 -3.0
TOA / s
-3.5
0.0
5.0x10-4
1.0x10-3
1.5x10-3
TOA / s -3
1.0x10
-4
5.0x10
2
1
3
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
distance / m Fig. 2. Typical steady detonation (CH4-2.5O2, p0 = 14 kPa, 7 mm AC, VCJ = 2198.3 m/s) 1: driver section. 2: guide section, V1 = 2147.6 m/s, 97.7% VCJ. 3: annular channel section, V2 = 1851.9 m/s, 84.2% VCJ.
3.2. Scaling analysis of detonation propagation limit The scaling between tube dimension and the detonation structure is performed to examine the propagation limit criterion in different channels. It has been suggested by previous researchers [11,34] that limits are greatly affected by the dimension of tube and the instability of detonation, based on this consideration, the scale of the channel (D or w) is compared with detonation cell size λ, the latter is a characterization length scale of detonation structure. First, the hydraulic diameter (DH) is introduced to scale with the results from CT and ACs based on the same standard criterion. DH was used in our former investigations [18,35] to construct the theoretical model of detonation limit. Using DH, the data from RT and ACs can be compared according to the same standard. For the ACs, DH is determined by:
no steady velocity
2.5x10-3 0.0
TOA / s
1.5x10-3
Voltage / V
-0.5
2.0x10-3
-1.0 -1.5 -2.0 -2.5 -3.0 -3.5
1.0x10-3
0.0
5.0x10-4
1.0x10-3
1.5x10-3
2.0x10-3
TOA / s
5.0x10-4 0.0
2
1 0.0
0.5
1.0
3 1.5
2.0
2.5
DH = 4 × 3.0
π (D 2 − d 2) /{π (D + d )} = 2w 4
(1)
where, D represents the inner diameter of steel tube, and d refers to the outer diameter of inserted tube. Second, an appropriate method is required to calculate λ with reasonable accuracy because CH4-O2 is a typical unstable mixture; the transverse waves in CH4-O2 detonation are very evident and cellular structure characterized by high irregularity. Fig. 6 shows a detonation cellular pattern of CH4-4O2 at p0 = 40 kPa in D = 36 mm RT. It can be seen the cellular structure is particularly irregular; therefore, the direct measurements of its cell size may fluctuate by a factor of two. On the other hand, λ can be calculated from a theoretical model within a reasonable accuracy, the theoretical model is established based on ZND model together with chemical kinetic mechanism. In this study, Konnov mechanism (0.4) [36] is applied for calculating the λ. A similar calculation was performed in our recent work for determining the detonation cells of CH4-H2-O2 mixtures [37]. Similar to our previous studies [37–40], the Ng [41,42] model is employed to predict λ. In the model, λ is computed from the correlation: λ = AΔI (A is a coefficient, ΔI represents induction zone length in ZND structure); the details can be found in Refs. [37,40,41]. The cell sizes both from experiment and theoretical model (dash lines) are shown in Fig. 7. The experimental data of λ are from Zhang et al. [43] and the Caltech detonation database [44]. In general, the theoretical results agree well with the experimental data if the
distance / m Fig. 3. Typical unsteady detonation (CH4-2.5O2, p0 = 13 kPa, 7 mm AC, VCJ = 2195.3 m/s) 1: driver section. 2: guide section, V1 = 2116.9 m/s, 96.4% VCJ. 3: annular channel section, no steady velocity.
were kept stationary for at least 24 h [15,22,25–28]. Before each shot, the tube was evacuated to at least 0.1 kPa, this pressure was monitored by OMEGA HHP242 (0–30 PSI). It should be noted that, the uncertainties must be considered in the experiment, which mainly stem from the pressure sensors, and hence, the accuracy and the maximum error of pressure sensors were shown in Table 1. 3. Results and discussions 3.1. Pc and VD of the detonation limit Fig. 4 illustrates the normalized velocity (average velocity of detonation or deflagration/CJ detonation velocity) versus the p0 for a CH42.5O2 (φ = 0.8) mixture in the round tube (RT) and annular channels (ACs) with different heights. The solid symbols represent steady velocity, and the “X” symbol represents unsteady velocity that is slightly below the critical pressure (pc). With the decreasing of the p0, the limit is eventually achieved, and the detonation velocity is found to progressively go down and away from the CJ value. Below the critical pressure, the steady velocity is no longer possible in 36 mm RT, 7 mm AC, 4.5 mm AC, and 2 mm AC; the critical pressures at which the corresponding velocity deficit (VD) occur are 10 kPa and 6.4% VCJ, 14 kPa and 15.8% VCJ, 15 kPa and 22.4% VCJ, 19 kPa and 28.9% VCJ. Both the critical pressure and the VD increase as the channel height is narrowed.
Table 1 Parameters of pressure sensors.
619
Model
Accuracy
Maximum Error/kPa
PX409-150A5V HHP242-030A
± 0.08% FS ± 0.10% FS
± 0.83 ± 0.20
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200
1.0
Steady (36mm RT) Unsteady (36mm RT) Steady (7mm AC) Unsteady(7mm AC) Steady (4.5mm AC) Unsteady(4.5mm AC) Steady (2mm AC) Unsteady(2mm AC)
0.6 0.4 0.2
Cell size / mm
V/VCJ
0.8
150
100
50
CH4-2.5O2
0
0.0 0
10
20
30
40
Zhang et al. ( 1.33) Zhang et al. ( 1) Caltech database ( 1) Zhang et al. ( 0.5)
50
0
5
Initial Pressure / kPa
10
15
20
25
30
35
40
45
50
Initial Pressure / kPa Fig. 7. λ (experimental and theoretical) at different initial pressures (dash lines are the theoretical prediction curves).
Fig. 4. Velocity versus initial pressure in RT and ACs.
60
40
36 mm 7 mm 4.5 mm 2 mm
RT AC AC AC
CH4-O2 CH4-O2 C2H2-2.5O2-70% Ar, Zhang et al. C2H2-2.5O2-85% Ar, Zhang et al.
5 4
30
/D
Initial Pressure / kPa
50
20
3
=
d
2 10
1 0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
0
Equivalence Ratio / Fig. 5. pc of the detonation limits versus the equivalence ratio in different channels.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Fig. 8. λ/D versus equivalence ratio in macro-scale tube.
measurement error is considered. To minimize the error of the analysis and because of the uncertainness of the measurement, the theoretically predicted cell size is used in the following discussion. Due to λ/DH represents the propagation ability of a detonation in channels, and hence, it can be considered as an applicable parameter of detonation limits, and for a round tube, DH = D. Fig. 8 first gives the λ/ D of RT for CH4-O2 at various equivalence ratios under the detonation limit condition. It can be seen that the propagation limit criterion of λ = πD is suitable in the macro-scale tube (D = 36 mm) for most conditions (i.e., φ = 0.25–1.4, red color) because, near the detonation propagation limits, the single-headed spin structure of the detonation can be observed, it is noted spin detonation is not the regular mode detonation, which occurs only at the condition near the limits; therefore, the Mylar foil utilized to register the detonation structure covers the entire internal face of the round tube of circumference of πD. However, the mixture tends to the fuel-rich condition as more methane is added; Fig. 8 shows that for the ratio between λ/D smaller than π, the average values of the ratio are 1.42 and 1.85 for CH4-1.25O2 (φ = 1.6) and CH4-O2 (φ = 2), respectively. In other words, the propagation limit criterion λ = πD is not valid for the CH4-O2 mixture at fuel-rich condition. In the study of Zhang et al. [18], they investigated the
detonation propagation limit in C2H2-2.5O2 with high concentration Ar dilution (70% and 85%) in the same tube and observed that λ/D is equal to 1.12 and 2.04, respectively (also shown in Fig. 8 for comparison); those values are also much smaller than π, which indicates the relation λ = πD breaks down for those stable mixtures. The irregular characteristic of detonation cellular structure depends on the degree of detonation instability, and for the mixture (e.g., CH4O2) presents more irregular structure usually has a larger value of χ, and its corresponding instability is higher [45].
χ = εI
ΔI σ̇ = εI ΔI max ′ ΔR uCJ
(2)
′ represent the thermicity at where, εI is activation energy, σ̇max and uCJ maximum value and CJ velocity for particle [41]. The parameter χ as a function of equivalence ratio at different p0 is given in Fig. 9. The value of χ is found to decrease with the increase of φ. Note that Zhang et al. [27] reported the value of χ is approximately 4 for 70%Ar diluted mixture at 50 kPa, and χ should be smaller for 85% Ar diluted mixture. As suggested by our previous work [40], mixtures with smaller value of χ tend to be more insensitive to form a detonation. Fig. 6. Typical detonation cells in CH4-4O2 mixture (p0 = 40 kPa, the foil is 70 cm in length and 10 cm in width).
620
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110 100 90
Recently, Gao et al. [32] have examined the influence of tube diameter on the propagation behavior in explosive mixtures, their investigations indicated the boundary layer reaching to the same order of magnitude of the diameter scale when the inner diameter of tube is small, on this condition, the boundary effect becoming more significant than that in the macro-scale tube. In this study, the displacement thickness of boundary layer δ* was also calculated to analysis the boundary effect on detonation propagation. δ* was defined by Gooderum [47] based on the data in shock tube experiments:
10 kPa 30 kPa 50 kPa
80 70 60
δ ∗ = 0.22x 0.8 (
50 40
μe 0.2 ) ρ0 V
(2)
In which, V is the velocity of detonation, μe is the viscosity, x refers to the reaction zone thickness, and ρ0 represents the initial density. x is computed from the empirical relationship of λ, the latter is calculated from prediction model (Fig. 7). Lee [1] pointed out that, the cell length of detonation (LCell) approximately equals to the x, this is because LCell characterizes a pulsation period of detonation propagation. Gao et al. [32] also employed x = LCell to theoretically calculate the velocity deficit, they found the calculation based on the above assumption agreed well with experiment. Furthermore, the LCell is approximate 1.5λ, i.e., LCell ∼ 1.5λ, which was verified by Edwards et al. [48]. Therefore, x ∼ 1.5 λ in Eq. (2). Fig. 11 illustrates the δ*/DH as a function of p0 for CH4-2O2 mixture for an example. Fig. 11 indicates that with the reducing of p0 and the height of the channel, the value of δ* accordingly goes up. It is the enlarged displacement thickness of boundary layer in the smaller scale tube that causes more severe mass divergence into the boundary layers and furthermore the enlargement of velocity deficits, which eventually results in a longer reaction zone length and a larger cell size.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Fig. 9. The parameter χ versus equivalence ratio.
Therefore, the scaling results of Fig. 8 and the stability parameter calculation shown in Fig. 9 indicate that in the macro-channel, the detonation propagation limit criterion λ = πD is valid in the mixture with larger value of χ, but as the value of stability parameter gradually decreases, λ = πD subsequently becomes unsuitable. The results suggest the failure and propagation mechanism of the unstable and stable detonation are distinct because of the instability property of the mixture. For most explosive mixtures, detonations in those mixtures are unstable with higher instability, the boundary layer effects have minor effect on the detonation propagation since the instability is the primary control mechanism of its propagation; thus, the successful propagation of detonations in unstable mixtures originates from local areas in the failure wave, at those places, the front of detonation was sustained by the amplified detonation instabilities, however, whenever the detonation instabilities are restrained and local explosions can not further develop in the failure wave, then the detonation would eventually quench [1,46]. However, as the instability is reduced by changing the properties, e.g., increase the equivalence ratio (shown in Fig. 9) or increase in the amount of argon dilution [27], the mixture tends to more stable, and the instability lost its dominate role in detonation propagation, with the failure of the detonation related to the excessive curvature of the detonation front. Fig. 10 shows the scaling of λ/DH for different channels. It can be seen that in the micro-annular channels, the detonation propagation limits criterion λ = πD breaks down in most cases, and the proportionality coefficient between λ and DH is larger in the smaller channels, indicating the detonation cell size is significantly enlarged in the smaller channel near the limit condition. As detonation propagates in the thin annular channel, the propagation mechanism becomes strongly dependent on boundary conditions (e.g., channel scale or geometry). In the smaller channels, the velocity deficit is also increased, primarily because of heat and momentum losses and mass divergence into the boundary layers. Because heat and momentum losses associated with viscous and thermal boundary layers, Han et al. [20] have confirmed that viscous wall friction plays an important role in self-sustained propagation in micro-scale ducts, and viscous drag and heat transfer are with a linear relationship to the annulus’s circumference multiplied by the length of the reaction zone; therefore, the self-sustained detonation with higher velocity deficit has a longer reaction zone length and bigger λ near the detonation limits, which explains the results in Fig. 10. It is well established that the classical CJ theory independent of the initial and boundary conditions, but depends merely on the mixture’s thermodynamic properties. However, in the experiment, it was suggested that detonation propagation is greatly affected by the combination effect from boundary conditions and initial conditions [1,31].
4. Conclusions In this work, the critical pressure near the limits of detonation propagation of CH4-O2 was experimentally measured in macro- and micro-channels (i.e., D = 36 mm round tube; w = 2, 4.5 and 7 mm annular channels), and the scaling between detonation cell size λ and tube dimension (DH) was also analyzed to explore the failure mechanism of detonation in channels. The present experimental results indicated that in a macro-channel (e.g., D = 36 mm RT), the detonation propagation limit criterion λ = πD holds for the mixture with larger value of instability, the instability is reduced by adding more fuel, where this typical relation breaks down; therefore, the propagation limit criterion depends on the detonation instability in a macrochannel. In contrast, in micro-channels, λ = πD is unsuitable as a limit 25
0.25 0.333 0.4 0.5 0.8 1 1.33 1.6 2
20
DH
15 10 5 0
= 0 1 2 3 4 5 6 7
34
35
36
37
d 38
channel gap or tube diameter Fig. 10. The result of λ/DH for the mixtures with varying φ in different channels. 621
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[17]
/DH , %
10
[18] [19]
[20]
1
AC (2 mm) AC (4.5 mm) AC (7 mm) RT (36 mm)
[21] [22]
0.1 0
5
10
15
20
25
30
35
40
45
[23]
50
Initial Pressure / kPa
[24]
Fig. 11. δ /DH versus p0 for detonation in RT and ACs (CH4-2O2). *
[25]
criterion for detonation because the propagation mechanism is strongly dependent on the boundary conditions. The above conclusions are indispensable supplements of typical detonation limit criterion λ = πD, which explains the detonation failure mechanism both in macro-/ micro-channels, and unstable/stable mixtures.
[26]
[27]
[28]
Acknowledgments [29]
This work is supported by the National Natural Science Foundation of China – China (Grant Nos.: 11772199, 91741114 and 11325209), and the project of State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology (Grant No.: KFJJ17-15M).
[30] [31] [32]
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Full Length Article
Investigation on the detonation propagation limit criterion for methaneoxygen mixtures in tubes with different scales ⁎
T
⁎
Bo Zhanga, , Hong Liua, , Bingjian Yanb a b
Shanghai Jiao Tong University, School of Aeronautics and Astronautics, Shanghai 200240, China East China University of Science and Technology, School of Resources and Environmental Engineering, Shanghai 200237, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Detonation limits Instability Boundary conditions
The detonation propagation limits of CH4-O2 mixtures (φ = 0.25–2) in a macro-channel (D = 36 mm) and in micro-channels with heights of 2 mm, 4.5 mm and 7 mm were studied. The critical pressure (pc) below which detonation cannot maintain steady propagation was measured experimentally for each mixture in different channels. At the condition of pc, the scaling between detonation cell size (λ) and tube dimension (DH) were also analyzed to explore the detonation failure mechanism in channels of macro- and micro-scale. The experimental results show that the detonation propagation limit criterion λ = πD (D is the inner diameter) holds in a macrochannel for the mixture with larger value of instability; however, the detonation instability is reduced as the mixture tends to fuel-rich condition, where this typical relation breaks down. The results indicate that the propagation limit criterion in macro-channel depends less on the boundary conditions of the tube and more on the detonation instability. In the micro-channels, because of the losses from the walls and mass divergence into the boundary layers, detonation propagates in a thinner channel with larger velocity deficit (VD), thereby increasing both the reaction zone length and the detonation cell size; therefore, λ = πD is unsuitable as a criterion of the detonation propagation limit, which suggests that for detonation propagating in the micro-channels, the propagation mechanism is mainly governed by the boundary conditions.
1. Introduction
unstable propagation behaviors appear, for instance, spinning detonation, galloping detonation, etc., as abundantly reported in the relevant literature. When the detonation is near the limits, the difference that distinguishes the failure and successful propagation is ambiguous [11,12]. Therefore, a criterion to estimate the onset of detonation limits is required to further explore its propagation and failure mechanism. One qualitative criterion to describe the limits is the appearance of the single head spinning structure in the tube, e.g., Lee [1,13] proposed λ = πD as a criterion for near-limit condition of detonations propagate in round tubes, where λ represents the cell size, D is the inner diameter of a tube. This criterion corresponds to the first appearance of a single headed spinning structure. When the detonation structure is singleheaded, the vertical dimension of the foil is the circumference of the tube, i.e., πD. Dupré et al. [14] investigated the near-limit detonation limits through changing the composition of H2-air mixtures in different diameter tubes. For the cell size less than value of πD, the unstable propagation behavior of detonation was observed and the velocity was found to suffered significant fluctuation. Recently, the criterion of λ = πD for detonation limits was confirmed by Gao et al. [15], Fischer et al. [16] and Yoshida et al. [17].
A detonation is a chemical reaction transport process accompanied by high-speed energy release [1]. On one hand, detonation is destructive in nature; unintentional detonation would cause casualties and make industrial facilities suffered heavy losses, thereby presenting great security risks especially in chemical and mining industries. On the other hand, because detonation has excellent thermal propulsion performances, detonation performed in a controlled manner could be applied in the field of advanced aerospace propulsion devices, for example, pulse detonation engines [2], rotating detonation engines [3], and oblique detonation propulsion technology [4–10]. Either to suppress destructive detonation or to improve safety during controlled propagation, a comprehensive understanding of the conditions that the detonation propagation can be sustained is of great importance for industrial processes and detonation applications. When the detonation process is well within its limits, the detonation velocity is steady, and it is therefore considered as a self-sustained detonation. In contrast, as the detonation limits are approached, the velocity fluctuates and has a significant deficit, and a variety of
⁎
Corresponding authors. E-mail addresses: [email protected] (B. Zhang), [email protected] (H. Liu).
https://doi.org/10.1016/j.fuel.2018.11.062 Received 27 July 2018; Received in revised form 31 October 2018; Accepted 11 November 2018 0016-2361/ © 2018 Elsevier Ltd. All rights reserved.
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Nomenclature CJ VCJ p0 Pc DH ΔI RT TOA χ V ρ0
VD ZND φ λ w D DDT AC LCell δ* μe x
Chapman-Jouguet CJ detonation velocity Initial Pressure Critical Pressure Tube dimension Induction Zone Length Round tube time-of-arrival Stability parameter Detonation velocity Initial density
Velocity deficit Zel’dovich-von Neumann-Döring Equivalence ratio Width of cell size Height of channel Inner diameter of the round tube Deflagration-to-Detonation Transition Annular channel Detonation cell length Boundary layer displacement thickness Viscosity Reaction zone thickness
The driver section was filled with equi-molar C2H2-O2, which was one of the most sensitive mixtures and readily to directly form a detonation. The test mixture was filled into the tube through different inlet. The length of guide and test section was 1.15 m and 1.35 m, respectively. The length of the round tube was 2.5 m. Similar as previous studies [15,21,22], in which optical sensors were applied to calculate the velocity of a deflagration or detonation. In this work, we also employed this technique to measure the velocity of combustion wave in a short distance. To validate the reliability of optical sensors, we compared the results with those used shock pins, it can be found that the signals from the shock pin and the optical fiber located at the same cross section appeared simultaneously for a successful detonation, whereas, for the deflagration case, the reaction zone is decoupled from precursor shock, and therefore, the TOA from the optical fiber was slower than that from the shock pin; the details can be found in Ref. [23]. Typical TOA signals from the optical detectors for steady and unsteady velocities are given in Fig. 2 and 3. The x-axis is the distance apart from the first optical sensor. The average velocity of the wave is computed by taking the slope of the curve. Fig. 2 illustrates that in the 7-mm annular channel section and at p0 = 14 kPa, the velocity has a slight deficit but is still steady, and the velocity is of 84.2% VCJ. As the p0 is somewhat lowered to 13 kPa, although the detonation velocity in the guide section is still 96.4% VCJ, the detonation velocity in the annulus deviates from the CJ value (given in Fig. 3), and the detonation completely fails at the end of the annular section. It is noteworthy that VCJ is the theoretical Chapman-Jouguet (CJ) detonation velocity, it is independent of boundary conditions but depends merely on the thermodynamic properties of the combustible mixture. In this study, VCJ is computed by CEA [24], which is a thermodynamic equilibrium program. After inputting the thermodynamic parameters of the mixture (e.g., initial pressure, composition, initial temperature, etc.), the value of VCJ is then given by CEA. In this study, methane-oxygen mixtures of different compositions (φ = 0.25, 0.33, 0.4, 0.5, 0.8, 1, 1.33, 1.6 and 2) were considered as test mixtures. The equivalence ratio values covered fuel-lean to fuelrich conditions that have different degrees of detonation instability. The mixtures were prepared by partial pressure. OMEGA PX409 (0–150 PSI) was used to measure the initial pressure of test mixture. To make sure the homogeneity of the mixture by molecular diffusion, the mixtures
Previous studies have suggested that λ = πD is an applicable criterion for determining the propagation limits for most gaseous combustible mixture detonations in macro-channels. However, recently, by investigating the detonation propagation limits of stable mixtures (with very regular cellular pattern, e.g., C2H2-2.5O2-70%Ar, C2H2-2.5O285%Ar) in ducts, Zhang et al. [18] claimed the relation λ = πD breaks down and the ratio of λ/D is much lower than π at near detonation-limit condition, which differs from the general knowledge of detonation limit criterion. It is known the detonation instability of a stable mixture is much lower than that of unstable mixtures, for the latter mixtures, they usually have very irregular cellular pattern. The above result indicated the limit criterion λ = πD may depends on the property of the mixture. Because most of the previous studies were performed in unstable mixtures at stoichiometric condition, to confirm whether the instability affects the detonation limit criterion λ = πD, a scaling analysis of detonation limits for the mixture that comprises a broad range of equivalence ratios from lean to rich (i.e., the instability is varied accordingly) is required. In addition, because the criterion of λ = πD is based on the results in channels with macro-scale, for the detonation propagation in micro-tube (e.g., thin annular channels), where the detonation is strongly dependent on the boundary conditions of the channel, and it subjects to significant loss (e.g., energy and momentum) from the walls, it is reasonable to speculate that the failure mechanism should be different from that in macro-channels; however, under such condition, the scaling between the tube dimension and the detonation structure length scale near the detonation-limit remains unclear. In this work, a systematic investigation is carried out to analysis the features of near-limit behavior of detonation in macro- and microchannels using typical unstable mixture (CH4-O2) with various equivalence ratios (φ = 0.25–2). The purpose of this work is twofold: 1) the instability of CH4-O2 mixture is varied by changing the equivalence ratio to explore the influence of instability of the mixture on the detonation limit criterion in macro-channel, and 2) detonation limits in thin annular channels (micro-scale) are also studied to explore the failure mechanism under high loss conditions. 2. Experimental details Three annular channels (ACs) and one round tube (RT) were used in the present investigation, for the round tube, the inner diameter D is 36 mm, and the heights of the annulus are 2 mm, 4.5 mm and 7 mm. The height of the annular channel was varied by changing the length of fins. Wu et al. [19] as well as Han et al. [20] suggested the geometry of O (10 mm) and larger is considered as macro-scale and that the geometry less than O (10 mm) mm is considered as micro-scale. A schematic of the apparatus is given by Fig. 1. The tube in Fig. 1 was divided into three sections. For the driver section, the length was 1.2 m, and the diameter was 68 mm; a spark seated at the beginning of the driver section, which was used to supply initiation energy for a detonation.
Fig. 1. Experimental apparatus. 618
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1.5x10
-0.5
Voltage / V
-3
Previous researchers have suggested the detonation VD was caused by three aspects: First, the wall effect results in heat and momentum losses [29,30]; Second, the divergence of mass into the boundary layers, this effect from the annular channel results in the streamlines in the chemical reaction zone and then augments the VD [23,31,32]. Third, the quenched layer adjacent to the tube wall decreases the reaction rate, and the chemical energy that supports the propagation detonation is also reduced, which eventually accounts for the VD [33]. Fig. 5 illustrates the variation of the experimentally measured pc of the detonation limits versus the equivalence ratio for CH4-O2 mixture in different channels. The experimental data indicate that the value of pc together with its φ is in a “U” behavior, the minimum of the pc locates at approximately φ = 1. This behavior indicates the mixture near the condition of stoichiometric is the most sensitive for detonation; when at φ < 1 or φ > 1 condition, the critical pressure for detonation limits is also increased for the same type of channel.
0.0
-3
2.0x10
-1.0 -1.5 -2.0 -2.5 -3.0
TOA / s
-3.5
0.0
5.0x10-4
1.0x10-3
1.5x10-3
TOA / s -3
1.0x10
-4
5.0x10
2
1
3
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
distance / m Fig. 2. Typical steady detonation (CH4-2.5O2, p0 = 14 kPa, 7 mm AC, VCJ = 2198.3 m/s) 1: driver section. 2: guide section, V1 = 2147.6 m/s, 97.7% VCJ. 3: annular channel section, V2 = 1851.9 m/s, 84.2% VCJ.
3.2. Scaling analysis of detonation propagation limit The scaling between tube dimension and the detonation structure is performed to examine the propagation limit criterion in different channels. It has been suggested by previous researchers [11,34] that limits are greatly affected by the dimension of tube and the instability of detonation, based on this consideration, the scale of the channel (D or w) is compared with detonation cell size λ, the latter is a characterization length scale of detonation structure. First, the hydraulic diameter (DH) is introduced to scale with the results from CT and ACs based on the same standard criterion. DH was used in our former investigations [18,35] to construct the theoretical model of detonation limit. Using DH, the data from RT and ACs can be compared according to the same standard. For the ACs, DH is determined by:
no steady velocity
2.5x10-3 0.0
TOA / s
1.5x10-3
Voltage / V
-0.5
2.0x10-3
-1.0 -1.5 -2.0 -2.5 -3.0 -3.5
1.0x10-3
0.0
5.0x10-4
1.0x10-3
1.5x10-3
2.0x10-3
TOA / s
5.0x10-4 0.0
2
1 0.0
0.5
1.0
3 1.5
2.0
2.5
DH = 4 × 3.0
π (D 2 − d 2) /{π (D + d )} = 2w 4
(1)
where, D represents the inner diameter of steel tube, and d refers to the outer diameter of inserted tube. Second, an appropriate method is required to calculate λ with reasonable accuracy because CH4-O2 is a typical unstable mixture; the transverse waves in CH4-O2 detonation are very evident and cellular structure characterized by high irregularity. Fig. 6 shows a detonation cellular pattern of CH4-4O2 at p0 = 40 kPa in D = 36 mm RT. It can be seen the cellular structure is particularly irregular; therefore, the direct measurements of its cell size may fluctuate by a factor of two. On the other hand, λ can be calculated from a theoretical model within a reasonable accuracy, the theoretical model is established based on ZND model together with chemical kinetic mechanism. In this study, Konnov mechanism (0.4) [36] is applied for calculating the λ. A similar calculation was performed in our recent work for determining the detonation cells of CH4-H2-O2 mixtures [37]. Similar to our previous studies [37–40], the Ng [41,42] model is employed to predict λ. In the model, λ is computed from the correlation: λ = AΔI (A is a coefficient, ΔI represents induction zone length in ZND structure); the details can be found in Refs. [37,40,41]. The cell sizes both from experiment and theoretical model (dash lines) are shown in Fig. 7. The experimental data of λ are from Zhang et al. [43] and the Caltech detonation database [44]. In general, the theoretical results agree well with the experimental data if the
distance / m Fig. 3. Typical unsteady detonation (CH4-2.5O2, p0 = 13 kPa, 7 mm AC, VCJ = 2195.3 m/s) 1: driver section. 2: guide section, V1 = 2116.9 m/s, 96.4% VCJ. 3: annular channel section, no steady velocity.
were kept stationary for at least 24 h [15,22,25–28]. Before each shot, the tube was evacuated to at least 0.1 kPa, this pressure was monitored by OMEGA HHP242 (0–30 PSI). It should be noted that, the uncertainties must be considered in the experiment, which mainly stem from the pressure sensors, and hence, the accuracy and the maximum error of pressure sensors were shown in Table 1. 3. Results and discussions 3.1. Pc and VD of the detonation limit Fig. 4 illustrates the normalized velocity (average velocity of detonation or deflagration/CJ detonation velocity) versus the p0 for a CH42.5O2 (φ = 0.8) mixture in the round tube (RT) and annular channels (ACs) with different heights. The solid symbols represent steady velocity, and the “X” symbol represents unsteady velocity that is slightly below the critical pressure (pc). With the decreasing of the p0, the limit is eventually achieved, and the detonation velocity is found to progressively go down and away from the CJ value. Below the critical pressure, the steady velocity is no longer possible in 36 mm RT, 7 mm AC, 4.5 mm AC, and 2 mm AC; the critical pressures at which the corresponding velocity deficit (VD) occur are 10 kPa and 6.4% VCJ, 14 kPa and 15.8% VCJ, 15 kPa and 22.4% VCJ, 19 kPa and 28.9% VCJ. Both the critical pressure and the VD increase as the channel height is narrowed.
Table 1 Parameters of pressure sensors.
619
Model
Accuracy
Maximum Error/kPa
PX409-150A5V HHP242-030A
± 0.08% FS ± 0.10% FS
± 0.83 ± 0.20
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200
1.0
Steady (36mm RT) Unsteady (36mm RT) Steady (7mm AC) Unsteady(7mm AC) Steady (4.5mm AC) Unsteady(4.5mm AC) Steady (2mm AC) Unsteady(2mm AC)
0.6 0.4 0.2
Cell size / mm
V/VCJ
0.8
150
100
50
CH4-2.5O2
0
0.0 0
10
20
30
40
Zhang et al. ( 1.33) Zhang et al. ( 1) Caltech database ( 1) Zhang et al. ( 0.5)
50
0
5
Initial Pressure / kPa
10
15
20
25
30
35
40
45
50
Initial Pressure / kPa Fig. 7. λ (experimental and theoretical) at different initial pressures (dash lines are the theoretical prediction curves).
Fig. 4. Velocity versus initial pressure in RT and ACs.
60
40
36 mm 7 mm 4.5 mm 2 mm
RT AC AC AC
CH4-O2 CH4-O2 C2H2-2.5O2-70% Ar, Zhang et al. C2H2-2.5O2-85% Ar, Zhang et al.
5 4
30
/D
Initial Pressure / kPa
50
20
3
=
d
2 10
1 0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
0
Equivalence Ratio / Fig. 5. pc of the detonation limits versus the equivalence ratio in different channels.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Fig. 8. λ/D versus equivalence ratio in macro-scale tube.
measurement error is considered. To minimize the error of the analysis and because of the uncertainness of the measurement, the theoretically predicted cell size is used in the following discussion. Due to λ/DH represents the propagation ability of a detonation in channels, and hence, it can be considered as an applicable parameter of detonation limits, and for a round tube, DH = D. Fig. 8 first gives the λ/ D of RT for CH4-O2 at various equivalence ratios under the detonation limit condition. It can be seen that the propagation limit criterion of λ = πD is suitable in the macro-scale tube (D = 36 mm) for most conditions (i.e., φ = 0.25–1.4, red color) because, near the detonation propagation limits, the single-headed spin structure of the detonation can be observed, it is noted spin detonation is not the regular mode detonation, which occurs only at the condition near the limits; therefore, the Mylar foil utilized to register the detonation structure covers the entire internal face of the round tube of circumference of πD. However, the mixture tends to the fuel-rich condition as more methane is added; Fig. 8 shows that for the ratio between λ/D smaller than π, the average values of the ratio are 1.42 and 1.85 for CH4-1.25O2 (φ = 1.6) and CH4-O2 (φ = 2), respectively. In other words, the propagation limit criterion λ = πD is not valid for the CH4-O2 mixture at fuel-rich condition. In the study of Zhang et al. [18], they investigated the
detonation propagation limit in C2H2-2.5O2 with high concentration Ar dilution (70% and 85%) in the same tube and observed that λ/D is equal to 1.12 and 2.04, respectively (also shown in Fig. 8 for comparison); those values are also much smaller than π, which indicates the relation λ = πD breaks down for those stable mixtures. The irregular characteristic of detonation cellular structure depends on the degree of detonation instability, and for the mixture (e.g., CH4O2) presents more irregular structure usually has a larger value of χ, and its corresponding instability is higher [45].
χ = εI
ΔI σ̇ = εI ΔI max ′ ΔR uCJ
(2)
′ represent the thermicity at where, εI is activation energy, σ̇max and uCJ maximum value and CJ velocity for particle [41]. The parameter χ as a function of equivalence ratio at different p0 is given in Fig. 9. The value of χ is found to decrease with the increase of φ. Note that Zhang et al. [27] reported the value of χ is approximately 4 for 70%Ar diluted mixture at 50 kPa, and χ should be smaller for 85% Ar diluted mixture. As suggested by our previous work [40], mixtures with smaller value of χ tend to be more insensitive to form a detonation. Fig. 6. Typical detonation cells in CH4-4O2 mixture (p0 = 40 kPa, the foil is 70 cm in length and 10 cm in width).
620
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110 100 90
Recently, Gao et al. [32] have examined the influence of tube diameter on the propagation behavior in explosive mixtures, their investigations indicated the boundary layer reaching to the same order of magnitude of the diameter scale when the inner diameter of tube is small, on this condition, the boundary effect becoming more significant than that in the macro-scale tube. In this study, the displacement thickness of boundary layer δ* was also calculated to analysis the boundary effect on detonation propagation. δ* was defined by Gooderum [47] based on the data in shock tube experiments:
10 kPa 30 kPa 50 kPa
80 70 60
δ ∗ = 0.22x 0.8 (
50 40
μe 0.2 ) ρ0 V
(2)
In which, V is the velocity of detonation, μe is the viscosity, x refers to the reaction zone thickness, and ρ0 represents the initial density. x is computed from the empirical relationship of λ, the latter is calculated from prediction model (Fig. 7). Lee [1] pointed out that, the cell length of detonation (LCell) approximately equals to the x, this is because LCell characterizes a pulsation period of detonation propagation. Gao et al. [32] also employed x = LCell to theoretically calculate the velocity deficit, they found the calculation based on the above assumption agreed well with experiment. Furthermore, the LCell is approximate 1.5λ, i.e., LCell ∼ 1.5λ, which was verified by Edwards et al. [48]. Therefore, x ∼ 1.5 λ in Eq. (2). Fig. 11 illustrates the δ*/DH as a function of p0 for CH4-2O2 mixture for an example. Fig. 11 indicates that with the reducing of p0 and the height of the channel, the value of δ* accordingly goes up. It is the enlarged displacement thickness of boundary layer in the smaller scale tube that causes more severe mass divergence into the boundary layers and furthermore the enlargement of velocity deficits, which eventually results in a longer reaction zone length and a larger cell size.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Fig. 9. The parameter χ versus equivalence ratio.
Therefore, the scaling results of Fig. 8 and the stability parameter calculation shown in Fig. 9 indicate that in the macro-channel, the detonation propagation limit criterion λ = πD is valid in the mixture with larger value of χ, but as the value of stability parameter gradually decreases, λ = πD subsequently becomes unsuitable. The results suggest the failure and propagation mechanism of the unstable and stable detonation are distinct because of the instability property of the mixture. For most explosive mixtures, detonations in those mixtures are unstable with higher instability, the boundary layer effects have minor effect on the detonation propagation since the instability is the primary control mechanism of its propagation; thus, the successful propagation of detonations in unstable mixtures originates from local areas in the failure wave, at those places, the front of detonation was sustained by the amplified detonation instabilities, however, whenever the detonation instabilities are restrained and local explosions can not further develop in the failure wave, then the detonation would eventually quench [1,46]. However, as the instability is reduced by changing the properties, e.g., increase the equivalence ratio (shown in Fig. 9) or increase in the amount of argon dilution [27], the mixture tends to more stable, and the instability lost its dominate role in detonation propagation, with the failure of the detonation related to the excessive curvature of the detonation front. Fig. 10 shows the scaling of λ/DH for different channels. It can be seen that in the micro-annular channels, the detonation propagation limits criterion λ = πD breaks down in most cases, and the proportionality coefficient between λ and DH is larger in the smaller channels, indicating the detonation cell size is significantly enlarged in the smaller channel near the limit condition. As detonation propagates in the thin annular channel, the propagation mechanism becomes strongly dependent on boundary conditions (e.g., channel scale or geometry). In the smaller channels, the velocity deficit is also increased, primarily because of heat and momentum losses and mass divergence into the boundary layers. Because heat and momentum losses associated with viscous and thermal boundary layers, Han et al. [20] have confirmed that viscous wall friction plays an important role in self-sustained propagation in micro-scale ducts, and viscous drag and heat transfer are with a linear relationship to the annulus’s circumference multiplied by the length of the reaction zone; therefore, the self-sustained detonation with higher velocity deficit has a longer reaction zone length and bigger λ near the detonation limits, which explains the results in Fig. 10. It is well established that the classical CJ theory independent of the initial and boundary conditions, but depends merely on the mixture’s thermodynamic properties. However, in the experiment, it was suggested that detonation propagation is greatly affected by the combination effect from boundary conditions and initial conditions [1,31].
4. Conclusions In this work, the critical pressure near the limits of detonation propagation of CH4-O2 was experimentally measured in macro- and micro-channels (i.e., D = 36 mm round tube; w = 2, 4.5 and 7 mm annular channels), and the scaling between detonation cell size λ and tube dimension (DH) was also analyzed to explore the failure mechanism of detonation in channels. The present experimental results indicated that in a macro-channel (e.g., D = 36 mm RT), the detonation propagation limit criterion λ = πD holds for the mixture with larger value of instability, the instability is reduced by adding more fuel, where this typical relation breaks down; therefore, the propagation limit criterion depends on the detonation instability in a macrochannel. In contrast, in micro-channels, λ = πD is unsuitable as a limit 25
0.25 0.333 0.4 0.5 0.8 1 1.33 1.6 2
20
DH
15 10 5 0
= 0 1 2 3 4 5 6 7
34
35
36
37
d 38
channel gap or tube diameter Fig. 10. The result of λ/DH for the mixtures with varying φ in different channels. 621
Fuel 239 (2019) 617–622
B. Zhang et al.
[17]
/DH , %
10
[18] [19]
[20]
1
AC (2 mm) AC (4.5 mm) AC (7 mm) RT (36 mm)
[21] [22]
0.1 0
5
10
15
20
25
30
35
40
45
[23]
50
Initial Pressure / kPa
[24]
Fig. 11. δ /DH versus p0 for detonation in RT and ACs (CH4-2O2). *
[25]
criterion for detonation because the propagation mechanism is strongly dependent on the boundary conditions. The above conclusions are indispensable supplements of typical detonation limit criterion λ = πD, which explains the detonation failure mechanism both in macro-/ micro-channels, and unstable/stable mixtures.
[26]
[27]
[28]
Acknowledgments [29]
This work is supported by the National Natural Science Foundation of China – China (Grant Nos.: 11772199, 91741114 and 11325209), and the project of State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology (Grant No.: KFJJ17-15M).
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