Fast Deflagration-to-Detonation Transition in Helical Tubes

Abstract

When designing a new type of power plants operating on pulsed detonations of gaseous or liquid fuels, the concept of fast deflagration-to-detonation transition (FDDT) is used. According to the concept, a flame arising from a weak ignition source must accelerate so fast as to form an intense shock wave at a minimum distance from the ignition source so that the intensity of the shock wave is sufficient for fast shock-to-detonation transition by some additional arrangements. Hence, the FDDT concept implies the use of special means for flame acceleration and shock wave amplification. In this work, we study the FDDT using a pulsed detonation tube comprising a Shchelkin spiral and a helical tube section with ten coils as the means for flame acceleration and shock amplification (focusing), respectively. To attain the FDDT at the shortest distances for fuels of significantly different detonability, the diameter of the pulsed detonation tube is taken close to the limiting diameter of detonation propagation for air mixtures of regular hydrocarbon fuels (50 mm). Experiments are conducted with air mixtures of individual gaseous fuels (hydrogen, methane, propane, and ethylene) and binary fuel compositions (methane–hydrogen, propane–hydrogen, and ethylene–hydrogen) at normal pressure and temperature conditions. The use of a helical tube with ten coils is shown to considerably extend the fuel-lean concentration limits of detonation as compared to the straight tube and the tube with a helical section with two coils.

1. Introduction

There are two known scenarios of deflagration-to-detonation transition (DDT) in tubes with gaseous explosive mixtures or gas suspensions of droplets and solid particles. In the first, classical DDT [1,2,3], the ignition of the mixture by a weak ignition source leads to the appearance of a flame, which, while propagating along the tube, changes its shape, increasing the combustion surface, and accelerates, forming compression and shock waves, eventually leading to the self-ignition of a shock-compressed explosive mixture and the onset of detonation. At the stage of flame acceleration, the most important role is played by gas-dynamic effects associated with the generation of turbulence ahead of the flame, for example, those produced with the help of the Shchelkin spiral [4,5,6] or regular orifice plates [7,8,9]. The second scenario was developed for the design of a new type of power plants operating on pulsed detonations and is referred to as fast DDT (FDDT) [10,11]. In this scenario, the flame arising from a weak ignition source must accelerate so fast as to form an intense shock wave at a minimum distance from the ignition source so that the intensity of the shock wave is sufficient for the fast shock-to-detonation transition caused by some additional arrangements in the detonation tube. Hence, the FDDT concept implies the use of special means for flame acceleration and shock wave amplification. Effective passive means of flame acceleration, in addition to the Shchelkin spiral and regular orifice plates, include regular or irregular obstacles of a special shape [12,13,14,15,16] and snail-shaped channels with smooth walls [17]. The obstacles of a special shape provide the fastest flame acceleration with the minimal detrimental impact on the flame-born compression and shock waves traveling ahead of the flame. In snail-shaped channels, the flame propagation velocity increases exponentially rather than linearly, as in a straight channel, due to a large transverse velocity gradient. Fluidic obstacles generating intense turbulence for flame acceleration [18,19,20] and nanosecond repetitive pulsed plasma discharges [21,22,23] can be attributed to active means of flame acceleration.

Effective passive means of shock wave amplification include a single orifice plate [24], focusing nozzles [25,26] or central bodies [27] of special shapes, one or several U-turns of the detonation tube [28,29,30,31,32], annular channels [33,34], tube coils [35,36,37,38], etc. Active means of shock wave amplification include coherent energy deposition to a traveling shock wave triggered by an additional single electrical discharge or distributed electrical discharges [39,40,41,42,43] or by creating a cloud of chemically activated gas in the path of the shock wave [44,45]. Generally speaking, the use of the FDDT concept allows one to maximize the energy potential of fuel for the fastest possible detonation initiation without applying strong ignition sources. This is possible by a proper redistribution of the available energy deposition in space and time.

The use of the FDDT concept with various means of flame acceleration and shock wave amplification allows us to develop and test a standard pulsed detonation tube (SDT-1) for ranking gaseous and liquid fuels by their detonability in mixtures with air using the measured FDDT run-up distance and time [46,47,48,49,50] as the detonability criteria, as well as to study the detonability of binary methane–hydrogen [51,52], propane–hydrogen [53,54], and ethylene–hydrogen [55,56] fuels in mixtures with air. According to Shchelkin [4], the run-up distance of DDT (if it exists) is proportional to the tube diameter; the smaller the diameter, the shorter the DDT run-up distance. The minimum (limiting) value of the tube diameter for the detonation of stoichiometric mixtures of alkane and alkene hydrocarbons larger than propane and propylene is about 50 mm, whereas lighter hydrocarbons (except for methane) and hydrogen exhibit lower limiting tube diameter values. Therefore, to minimize the size of the detonation tube in the experimental setup intended for the various hydrocarbons, we choose this value (50 mm) to cover the whole range of the hydrocarbons, at least for the mixtures of stoichiometric composition. It appears that the stoichiometric methane-air mixture is marginally detonable in the tube of such a diameter. Based on the existing knowledge, one could expect the detonability limits in the tube of larger diameter to be somewhat wider. However, keeping in mind the strong (U-shaped) dependence of the limiting tube diameter on fuel concentration, one could expect the concentration limits, starting from a certain value of tube diameter, to be independent of it. Due to the use of (i) a detonation tube with a diameter close to the limiting diameter of detonation propagation for air mixtures of regular hydrocarbon fuels (50 mm), (ii) a Shchelkin spiral, which provides fast flame acceleration, and (iii) a helical tube section with two coils, which provides the gas-dynamic focusing of the flame-born shock wave, the SDT-1 facility allows the reliable registration of FDDT in air mixtures of gaseous and liquid fuels with significantly different detonability (from methane and aviation kerosene to hydrogen) at the shortest distances (less than 2 m) and for the shortest times (less than 15 ms).

The most Important element of the SDT-1 facility is a helical tube section with two coils. This element allows the fast transition of a flame-born shock wave to a detonation at the shortest distances even in fuel-air mixtures of low detonability. The removal of the helical tube section from the SDT-1 facility, other things being equal, results in the failure of DDT for fuel-air mixtures, which do not exhibit DDT in a straight tube with a Shchelkin spiral. Calculations in [38] showed that successive reflections of the flame-born shock wave from the compressive wall in the coils produce hot spots with elevated temperature and pressure. These hot spots promote the onset of detonation. It is worth noting that similar hot spots arise during DDT and develop detonation propagation in straight tubes with Shchelkin spirals [57] or orifice plates [58]. In these cases, the hot spots originate due to reflections of the flame-born shock wave or a detonation wave from the upstream surfaces of the obstacles. However, contrary to a helical tube with smooth walls, straight tubes with obstacles exhibit a considerable hydrodynamic drag, which deteriorates the conditions for shock amplification [10].

The objectives of this work are to”test’the new version of the standard pulsed detonation tube (SDT-2), in which the helical tube section with two coils is replaced by a helical tube section with ten coils, and to study the effect of the number of coils on the FDDT in air mixtures of individual gaseous fuels (hydrogen, methane, propane, and ethylene) and binary fuel compositions (methane-hydrogen, propane-hydrogen, and ethylene-hydrogen) at normal pressure and temperature (NPT) conditions. These objectives and the obtained results are the novel and distinctive features of this study.

2. Materials and Methods

2.1. Pulsed Detonation Tubes

Figure 1 shows the schematics of SDT-1 (Figure 1a) and SDT-2 (Figure 1b) with the numbers of the measuring ports (shown in bold), the distances between the measuring ports along the axial line, and the overall dimensions of individual sections. Additionally, the total lengths of the tubes along the centerline and the overall size of the tubes are indicated.

Table 1. Schemes of installation of ionization probes (IP) and pressure sensors (PS) in SDT-1 and SDT-2.

Port#	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
SDT-1
IP	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	x
PS	x	−	−	−	+	+	+	+	+	+	+	+	+	+	+	+	−	−	−	−	−	x
SDT-2
IP	x	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+
PS	x	−	−	−	+	+	+	+	+	+	+	−	+	−	+	−	+	−	−	+	−	+

Remarks: +, yes; −, no; x, no port.

presents information on the sensors: ionization probes (Ips) and pressure sensors (PSs) installed in the measuring ports.

The SDT-2 consists of the same structural elements as the SDT-1, namely, a prechamber with a spark plug (the position of the spark gap is shown by an asterisk in Figure 1), a flame acceleration section with a Shchelkin spiral, a helical tube section for the amplification of a flame-born shock wave, and a measuring section (a straight tube with smooth walls) for measuring the reaction wave (detonation) parameters after it exits from the helical tube section. Both tubes, SDT-2 and SDT-1, have an inner diameter of 50 mm. The geometric parameters of all structural elements of SDT-2 and SDT-1, except for the helical tube section and the length of the measuring section, differ insignificantly. Thus, following the results of [51,52,53,54,55,56], the distance between the last turn of the Shchelkin spiral and the first coil of the helical tube in SDT-2 is reduced to 100 mm, whereas that of the SDT-1 is 150 mm. The Shchelkin spirals themselves are the same: they are wound from steel wire with a diameter of 6.7 mm; the spiral length is 940 mm; the outer diameter of the spiral is 49 mm; and the pitch of the turns is 24 mm. In contrast to that of SDT-1, the helical tube section in SDT-2 contains ten rather than two coils, whereas the average pitch of the coils is the same (216 mm). The measuring section in SDT-2 is approximately a factor of 2.5 shorter than that in SDT-1 (1000 mm vs. 2500 mm). Despite the fact that the length of the SDT-2 along the centerline is increased to 7075 mm (instead of 5005 mm for the SDT-1), the overall dimensions of the SDT-2 (4935 mm) and SDT-1 (4710 mm) are approximately the same. The internal volume of SDT-2 is increased by 4 dm³: 14 dm³ instead of 10 dm³ for SDT-1.

2.2. Experimental Setup and Procedure

Figure 2 shows a schematic of the experimental setup with the SDT-2. The setup consists of a fuel-air mixture preparation and supply system, ignition system, control system, and data acquisition system. All systems are identical to those used in the experimental setup with SDT-1. The mixture is prepared in a 40 dm³ mixer equipped with a fan. Before mixture preparation, the mixer is evacuated to a pressure of less than 0.1 kPa using a vacuum pump. The mixture is prepared by partial pressures. The pressure is determined with an accuracy of 0.1 kPa. Molecular mixing of the components is achieved by turning on the fan located inside the mixer for 20 min.

The SDT-2 is filled with a fuel-air mixture in the following way. First, the tube is purged with compressed air. Thereafter, the tube is purged with the fuel-air mixture from the mixer until the volume of the purged mixture is four times the volume of SDT-2. The volume of the fuel-air mixture purged through the tube is controlled by the pressure drop in the mixer, whereas the flow rate of the mixture (10–20 L/min) is controlled by the vent valve. After the vent valve is closed, the ignition is triggered with a delay of 4 s. The number of experiments (hereinafter “shots”) for fuel-air mixtures of a certain composition is normally 3. The instantaneous location and arrival time of the leading edge of the reaction front is determined using IPs [59]. The accuracy of determining the location and the arrival time of the reaction front using IPs is ± 2 mm and ± 1 μs. To record the pressure amplitude and profile in the shock and detonation waves, high-frequency piezoelectric PSs (PCB113B24) with a natural frequency of 500 kHz are used. The accuracy of determining the location of the pressure wave using PSs is ± 6 mm. From the IP and PS records, the velocities of the reaction front and leading shock wave are calculated, respectively. The apparent velocity of the leading edge of the reaction front $D_{\mathrm{f}}$ on a measuring segment is determined by the known distance between the measuring ports with IPs along the axial line and by the time interval between the instants of arrival of the reaction front at these IPs. The propagation velocity of the flame-born pressure wave $D_{\mathrm{SW}}$ is determined similarly, but, instead of the IP records, the PS records are used. The error in determining the propagation velocity of the reaction front and pressure wave at $D_{\mathrm{f}}$ > 1000 m/s and $D_{\mathrm{SW}}$ > 1000 m/s is estimated at 3%. This allows one to build “time-distance” ($t-x$) and “wave velocity-distance” ($D-x$) diagrams of the development of the process. Joint consideration of the $t–x$ and $D–x$ diagrams allows one to determine the FDDT run-up distance $L_{\mathrm{DDT}}$ and time $T_{\mathrm{DDT}}$. The values of $L_{\mathrm{DDT}}$ and $T_{\mathrm{DDT}}$ are the distance from the ignition source and the time from the ignition instant at which the reaction front velocity $D_{\mathrm{f}}$ reaches a steady-state value of the self-sustaining detonation velocity close to the thermodynamic Chapman-Jouguet (CJ) detonation velocity $D_{\mathrm{CJ}}$ for the fuel-air mixture under study. Self-sustaining detonation is understood as the steady-state reaction wave propagating at an average velocity, which is constant within ± 3%, whereas the fronts of the reaction and the leading shock wave merge within ± 6 μs. Particular attention is paid to the reproducibility of the results; a detonation wave is considered to be established if it propagates with approximately the same supersonic speed in all shots with the fuel-air mixture of a given composition. The errors in determining $L_{\mathrm{DDT}}$ and $T_{\mathrm{DDT}}$ are the maximum deviations of the values of $L_{\mathrm{DDT}}$ and $T_{\mathrm{DDT}}$ from the arithmetic mean in 3 successive shots. The minimum absolute error in determining $L_{\mathrm{DDT}}$ corresponds to half the length of the measuring segment on which the FDDT is registered. To refine the values of $L_{\mathrm{DDT}}$, the trajectory of the detonation wave recorded with the help of PSs is used.

2.3. Gases

The pressurized gases used in the experiments are air, hydrogen, methane (natural gas), propane, and ethylene at $T_0$ = 300 K. The gases provided by the manufacturer are commercial-grade (99.9% purity) hydrogen, propane, and ethylene. The room air is compressed by an oil-free air compressor (Fubag OL195/6 CM1.5, Fubag GmbH, Germany) and used as an oxidizer. The composition of the natural gas is presented in

Table 2. Natural gas composition (%vol.).

CH4	C2H6	C3H8	i-C4H10	n-C4H10	C5H12(1)	C6H14(2)	CxHy(3)	N2
94.8	2.8	0.84	0.12	0.14	0.05	0.02	0.03	1.2

⁽¹⁾ the sum of pentanes (isopentane, n-pentane, neopentane); ⁽²⁾ the sum of hexanes (n-hexane, 2,2-dimethylbutane, 2-methylpentane, and 3-methylpentane); ⁽³⁾ the sum of other hydrocarbons.

. As can be seen, the volume fraction of methane in the natural gas is 94.8%.

3. Results and Discussion

This section discusses the results of experiments on the FDDT in the SDT-2 facility in air mixtures of hydrogen, methane, propane, and ethylene, as well as in air mixtures of binary methane-hydrogen, propane-hydrogen, and ethylene-hydrogen fuels, under NPT conditions. Where possible, the experimental results are compared with those obtained in SDT-1.

3.1. Hydrogen-Air Mixtures

Experiments with hydrogen-air mixtures showed that the FDDT occurred in a very similar way in both SDT-1 and SDT-2 over a wide range of compositions determined by the value of the fuel-to-air equivalence ratio $\Phi$. Figure 3 compares the dependences of the reaction front velocity $D_{\mathrm{f}}$ and shock wave velocity $D_{\mathrm{SW}}$ measured in SDT-1 (Figure 3a) and in SDT-2 (Figure 3b) on the axial distance from the ignition source $x$ for a hydrogen-air mixture with $\Phi \approx$0.5 (hydrogen volume fraction in the mixture $x_{\mathrm{H}2}=$17.2–17.3%(vol.)). From now on, the vertical dash-dotted lines in the figures show the beginning and end of the helical tube section in SDT-1 and SDT-2, and the horizontal dashed line shows the CJ detonation velocity $D_{\mathrm{CJ}}$. The yellow vertical bar corresponds to the experimental spread of the measured FDDT run-up distance $L_{\mathrm{DDT}}$. In SDT-1, the FDDT in a mixture with $\Phi \approx$ 0.5 occurred at the exit from the helical tube section at a distance $L_{\mathrm{DDT}}$ = 2.2–2.3 m with a steady-state value of the detonation velocity in the measuring section $D\approx$ 1585±15 m/s. The CJ detonation velocity for this mixture was $D_{\mathrm{CJ}}$ = 1607 m/s. This means that the detonation propagated with the velocity deficit $\Delta D/D_{\mathrm{CJ}}=\left(D_{\mathrm{CJ}}-D\right)/D_{\mathrm{CJ}}\approx$ 1%. In SDT-2, the FDDT occurred in a similar way between the second and third coils at a distance $L_{\mathrm{DDT}}$ = 2.2–2.5 m; however, the steady-state value of the detonation velocity in the helical tube section was noticeably lower ($D$ = 1490 ± 20 m/c) than the CJ detonation velocity, so that the detonation velocity deficit was $\Delta D/D_{\mathrm{CJ}}\approx$ 7%. When the detonation wave entered the measuring section of SDT-2, the detonation velocity increased to $D=$ 1540 ± 20 m/s, and the detonation velocity deficit decreased to $\Delta D/D_{\mathrm{CJ}}\approx$ 1%.

When the hydrogen content in the hydrogen-air mixture decreased below $\Phi \approx$ 0.5, the maximum velocity of the reaction front in the Shchelkin spiral decreased, and, at a certain limiting value, $\Phi ={\Phi }^{*}$, the FDDT failed to occur. The value of ${\Phi }^{*}$ in SDT-2 turned out to be lower than in SDT-1, which means that the fuel-lean concentration limit of FDDT in SDT-2 was extended. In confirmation of this, Figure 4 shows the dependences of the propagation velocities of the reaction front and flame-born shock wave on the distance measured in SDT-1 (Figure 4a) and SDT-2 (Figure 4b) for a hydrogen-air mixture with $x_{\mathrm{H}2}$ = 12%(vol.) (${\Phi }^{*}\approx$ 0.325). In SDT-1, there was no FDDT, while in SDT-2, the FDDT occurred in the seventh coil of the helical tube section at a distance $L_{\mathrm{DDT}}\approx$ 4 m, and the steady-state value of the detonation velocity in the helical tube section was $D$ = 1240 ± 40 m/s at $D_{\mathrm{CJ}}$= 1385 m/s (velocity deficit $\Delta D/D_{\mathrm{CJ}}\approx$ 10%). The extension of the fuel-lean concentration limit of FDDT in SDT-2 was obviously caused by the longer helical tube section and a larger number of successive reflections of the flame-born shock wave from the compressive wall in the coils, leading to a secondary explosion and detonation onset. The established detonation wave was capable of propagating in a smooth-walled helical tube with a large velocity deficit above 10%, which was apparently caused by the additional stabilizing effect of the transverse shock waves formed upon reflection of the detonation wave from curved surfaces. Previously [60], a similar velocity deficit (above 10%) was detected during the propagation of transient detonation waves in short tubes under the intense action of rarefaction waves. Interestingly, when entering the straight measuring section of SDT-2, the detonation wave decayed (see Figure 4b).

Several important observations must be noted. First, the process evolution in the same segments of SDT-1 and SDT-2 facilities was about the same. This can be readily seen from Figure 3 and Figure 4 for hydrogen-air mixtures with $\Phi \approx$ 0.5 and 0.325. The evolution of the reaction front and shock wave velocities in the flame acceleration section and in the first two coils of the helical tube looked very similar. However, their further evolution looked different, apparently due to an additional effect of shock wave reflections from the curved surfaces in the longer helical tube of the SDT-2 facility. To demonstrate the effect of such reflections on DDT, let us consider Figure 5, which shows three successive snapshots of pressure isosurfaces in the course of shock wave propagation in a tube coil filled with a gaseous reactive mixture. These snapshots were obtained by calculations similar to those discussed in [38]. The incident shock wave propagates counterclockwise. As seen, shock wave propagation along the coil results in multiple reflections from the outer coil wall. One of the reflections at the 11 o’clock position results in the onset of detonation through the formation of a detonation “bubble” (see Figure 5a), its growth (see Figure 5b), and transformation to the overdriven detonation wave propagating in both directions, clockwise and counterclockwise (see Figure 5c).

Second, at the fuel-lean concentration limit of FDDT (at $\Phi ={\Phi }^{*}$), the minimum apparent propagation velocity of the reaction front at the inlet to the helical tube section was virtually constant and close to 400 m/s. As is shown later in this paper, for mixtures of hydrocarbon fuels with air, as well as for mixtures of binary hydrocarbon-hydrogen fuels, the minimum value of the apparent reaction front velocity at the fuel-lean concentration limit of FDDT (at $\Phi ={\Phi }^{*}$) was also almost constant and close to 500 m/s. Third, in SDT-1 and SDT-2, both the reaction front and the flame-born shock wave propagated at an almost constant velocity of about 800 m/s when entering the helical tube section. Fourth, in our previous experiments with spark ignition of the hydrogen-air mixture with $\Phi \approx$ 0.325 at the closed end of a straight, smooth-walled tube with the same length-to-diameter ratio, the maximum measured propagation velocity of the reaction front reached only 40–80 m/s, and no tendency of the flame to accelerate was noted. In the experiments on SDT-1 and SDT-2, the apparent flame propagation velocity in the flame acceleration section (inside the Shchelkin spiral) increased to about 600 m/s and further increased to about 800 m/s in the helical tube section. Furthermore, in SDT-2, the FDDT led to the formation of the established detonation wave propagating steadily along the helical tube section. Thus, a sufficiently extended helical tube allows one to obtain the FDDT in fuel-air mixtures of extremely fuel-lean composition, such that classical DDT is impossible neither in straight tubes with smooth walls nor in straight tubes with rough walls.

Figure 6 shows the dependences of the propagation velocities of the reaction front and the flame-born shock wave measured in different sections of SDT-2 on the fuel-to-air equivalence ratio $\Phi$ in the hydrogen-air mixture. A decrease in hydrogen volume fraction from $x_{\mathrm{H}2}$ = 29.6%(vol.) ($\Phi$=1) to $x_{\mathrm{H}2}$ = 15%(vol.) ($\Phi$ = 0.42) led to a decrease in the detonation velocity $D$ in the measuring section of SDT-2 and to an increase in the detonation velocity deficit $\Delta D/D_{\mathrm{CJ}}$ from approximately 1% to 6%. At $x_{\mathrm{H}2}$ < 15%(vol.), the detonation decayed in the measuring section of SDT-2, and the velocity of the flame-born shock wave decreased sharply from 1450 to 960 m/s when $x_{\mathrm{H}2}$ decreased from 15%(vol.) to 14%(vol.). Nevertheless, in the helical tube section, detonation propagated steadily at $x_{\mathrm{H}2}$ > 12%(vol.) ($\Phi$ = 0.325) with a velocity deficit $\Delta D/D_{\mathrm{CJ}}\approx$ 10%, which was two times higher than the detonation velocity deficit for the detonation wave propagating in the fuel-air mixture of near-stoichiometric composition ($\Delta D/D_{\mathrm{CJ}}\approx$ 5–6%). Interestingly, when the hydrogen volume fraction in the mixture approached $x_{\mathrm{H}2}$ = 15%(vol.) ($\Phi$ = 0.42), the probability of detonation failure increased in the shots with the same mixture. Thus, at $x_{\mathrm{H}2}$ = 15.6%(vol.) ($\Phi$ = 0.44), detonation failed in one of three shots. The closer the mixture composition was to $\Phi$ = 0s.42, the more shots with detonation failure were registered (see Appendix). Transition from the steadily propagating detonation to a galloping mode (the mode with significant fluctuations in the detonation velocity) always occurred through a single-headed spin [61]. The domain of the galloping mode at the fuel-lean limit for hydrogen-air mixtures found in [61] corresponded to $x_{\mathrm{H}2}$ = 12.5–14.3%(vol.) or $\Phi$ = 0.34–0.40. In our study, this domain approximately corresponded to the domain $x_{\mathrm{H}2}$ = 12–15%(vol.) ($\Phi$ = 0.325–0.42), in which we registered detonation decay in the course of detonation transition from the helical tube section to the straight measuring section with smooth walls.

Figure 7 compares the measured detonation velocities in the measuring sections of SDT-1 and SDT-2 as a function of the fuel-to-air equivalence ratio $\Phi$ in fuel-lean and near-stoichiometric hydrogen-air mixtures. In SDT-1 and SDT-2, the FDDT was registered at $\Phi$ > 0.48 and $\Phi$ > 0.32, respectively. According to the literature data, when the detonation is obtained by direct initiation, the lower concentration limit of detonation propagation in tubes of different diameter $d$ filled with the hydrogen-air mixture is $x_{\mathrm{H}2}$ = 18.3%(vol.) ($\Phi$ = 0.53 [62]) and $x_{\mathrm{H}2}$ = 12.5%(vol.) ($\Phi$ = 0.34, $d$ up to 30 cm [61]). The use of a combination of Shchelkin spiral and a helical tube section with ten coils allowed approaching the lower concentration limit of detonation reported in [61] via FDDT rather than via direct detonation initiation.

The maximum measured DDT run-up distance for hydrogen-air mixtures was attained at $\Phi =$ 0.326 (see Appendix) with a value of 3.97 m along the facility centerline, i.e., DDT always occurred far from the end of the helical tube in the SDT-2 facility.

3.2. Methane-Air Mixtures

Experiments with methane-air mixtures were conducted at $\Phi$ = 0.7, 0.8, 0.9, and 1.0 (see Appendix). Figure 8 compares the dependences of the reaction front velocity $D_{\mathrm{f}}$ and the flame-born shock wave velocity $D_{\mathrm{SW}}$ measured in SDT-1 (Figure 8a) and SDT-2 (Figure 8b) on the distance for the methane-air mixture with $\Phi$ = 1.0 (methane volume fraction $x_{\mathrm{CH}4}$ = 9.5%(vol.)). The reproducibility of experiments in SDT-2 was seen to be much better than in SDT-1. However, when switching from a mixture with $\Phi$ = 1.0 to a mixture with $\Phi$ = 0.9, detonation decayed in the measuring section of SDT-2 in two of three shots, and, when switching to a mixture with $\Phi$ = 0.8, it decayed in each shot. At $\Phi$ = 0.7, at the exit from the Shchelkin spiral, the maximum apparent propagation velocity of the reaction front reached only 150 m/s, and the flame no longer accelerated in the helical tube section.

The maximum measured DDT run-up distance for methane-air mixtures was attained at $\Phi =$ 0.8 (see Appendix) with a value of 4.04 m along the facility centerline, i.e., DDT always occurred inside the helical tube in the SDT-2 facility.

3.3. Methane-Hydrogen-Air Mixtures

The addition of methane to hydrogen led to the narrowing of the fuel-lean concentration limits of FDDT in SDT-2. Figure 9 shows an example illustrating this fact. The dilution of hydrogen with methane by 10%(vol.) (the fuel consisted of 10%(vol.) CH₄ and 90%(vol.) H₂) led to a shift in the lower concentration limit of FDDT from $\Phi$ = 0.32 for a hydrogen-air mixture to $\Phi$ = 0.40 for a 1% CH₄ + 10% H₂ + 89% air mixture. It can be seen from Figure 9 that at $\Phi$= 0.38 there was still no FDDT, and at $\Phi$ = 0.40, FDDT occurred in the helical tube section with subsequent detonation decay when it entered the straight measuring section with smooth walls.

Figure 10 shows the dependences of the propagation velocities of the reaction front and the flame-born shock wave measured in different sections of SDT-2 on the fuel-to-air equivalence ratio $\Phi$ in the blended methane-hydrogen (10% CH₄ + 90% H₂) fuel-air mixture. Similar to in hydrogen-air mixtures, there existed a range of compositions of methane-hydrogen-air mixtures in which FDDT occurred in the helical tube section, and the arising detonation wave propagated steadily in this section but decayed when entering the straight measuring section with smooth walls.

Figure 11 shows the results of measurements of the detonation velocity in the helical tube section of SDT-2 for methane-hydrogen-air mixtures with different dilutions of hydrogen with methane. The modes of steady-state detonation propagation in the helical tube section followed by detonation decay in the measuring section (D/F modes) are marked with diamonds. With an increase in the methane content and a decrease in the hydrogen content in the fuel, the domain of existence of such D/F modes in terms of $\Phi$ shrunk, while the transition region itself was quite narrow. At the fuel-lean concentration limit of FDDT, the minimum apparent propagation velocity of the reaction front at the inlet to the helical tube section was almost constant and close to 500 m/s; when hydrogen was diluted with methane by 10, 20, 30, 40, and 50%, this velocity was 510 ± 10 (${\Phi }^{*}\approx$ 0.40), 500 ± 30 (${\Phi }^{*}\approx$ 0.46), 420 ± 80 (${\Phi }^{*}\approx$ 0.50), 520 ± 20 (${\Phi }^{*}\approx$ 0.56), and 500 ± 10 (${\Phi }^{*}\approx$ 0.60) m/s, respectively (see Appendix).

According to the records of pressure sensors in the measuring section of SDT-2, detonation in methane-hydrogen-air mixtures near the fuel-lean concentration limit propagated in the spinning mode. Figure 12 shows an example of pressure records taken by three PSs in the measuring section of SDT-2 in three successive shots with the (20% CH₄ + 80% H₂)-air mixture at $\Phi$ = 0.54. As seen, the detonation decayed in the first shot and propagated steadily in the spinning mode in the second and third shots. The spinning mode of detonation can be recognized from the pressure records. When the spinning detonation wave traveled along the tube and the spin head passed over the surface of a PS, the pressure record showed a sharp pressure rise or pressure peak on the background of a weaker shock wave. Similar pressure records were obtained for the (40% CH₄ + 60% H₂)-air mixture at $\Phi$ = 0.56 (see D/F modes in Appendix).

The maximum measured DDT run-up distance was attained for (30% CH₄ + 70% H₂)-air mixture at $\Phi$ = 0.5 (see Appendix) with a value of 5.61 m along the facility centerline, i.e., DDT always occurred inside the helical tube in SDT-2 or at the exit of the tenth coil.

3.4. Propane-Hydrogen-Air Mixtures

When hydrogen was diluted with propane, the results obtained in SDT-2 were generally the same as those previously obtained in SDT-1 [33,34]. Figure 13 shows the dependences of the measured propagation velocities of the reaction front and flame-born shock wave in the measuring section of SDT-2 on the fuel-to-air equivalence ratio $\Phi$ of propane-hydrogen-air mixtures with different dilutions of hydrogen with propane (0, 10, 20, and 40%). At the fuel-lean concentration limit of FDDT, the minimum apparent propagation velocity of the reaction front at the entrance to the helical tube section was virtually constant and close to 500 m/s. When hydrogen was diluted with propane by 10% (${\Phi }^{*}\approx$ 0.46), 20% (${\Phi }^{*}\approx$ 0.53), and 40% (${\Phi }^{*}\approx$ 0.62), this velocity was equal to 510 ± 30, 400 ± 50, and 480 ± 60 m/s, respectively. Note that FDDT in a propane-air mixture (without hydrogen) occurs only at $\Phi$ ≥ 0.72 ($x_{\mathrm{C}3\mathrm{H}8}\ge$ 2.9%(vol.)), and there is no FDDT in the leaner mixtures. In the propane-air mixture, the lower concentration limit of FDDT in SDT-2 (2.9%(vol.) C₃H₈) turned out to be slightly narrower than the lower detonability limits reported in [61] and [60] (2.3%(vol.) C₃H₈ and 2.6%(vol.) C₃H₈) for tubes 160 and 70 mm in diameter, respectively, but slightly wider than the limit of detonation reinitiation after it transitioned from a rough tube (a tube with a Shchelkin spiral) to a smooth-walled tube (3.5%(vol.) C₃H₈ [61]).

The maximum measured DDT run-up distance for propane-air mixtures was attained at $\Phi =$ 0.71 (see Appendix) with a value of 5.8 m along the facility centerline, i.e., DDT always occurred inside or shortly outside the helical tube of the SDT-2.

3.5. Ethylene-Hydrogen-Air Mixtures

Figure 14 shows the dependences of the propagation velocities of the reaction front and the flame-born shock wave in the measuring section of SDT-2 on the fuel-to-air equivalence ratio $\Phi$ of ethylene-hydrogen-air mixtures with dilutions of hydrogen with ethylene by 0, 10, 20, 30, and 40%. With a decrease in the hydrogen content in the fuel, the fuel-lean concentration limit of FDDT shifted to higher values of $\Phi$ from 0.32 (100% H₂) to 0.50 (60% H₂). In SDT-1, the FDDT in ethylene-air mixtures (without hydrogen) was registered at $\Phi$ > 0.6, while, in the leaner mixtures, the FDDT was not obtained. Note that for ethylene-air mixtures, the lower detonability limit was 3.5%(vol.) C₂H₄ ($\Phi$ = 0.52), as reported in [61].

Figure 15 shows an example demonstrating the extension of the concentration limits of the FDDT in SDT-2 compared to SDT-1 for a fuel containing 30%(vol.) C₂H₄ and 70%(vol.) H₂ at $\Phi$ = 0.50. In the initial stages, flame acceleration in SDT-1 and SDT-2 looked almost the same up to distances of 1.7–1.8 m from the ignition source. Thereafter, the FDDT occurred in SDT-2 between the second and third coils of the helical tube section, whereas the FDDT failed in SDT-1; the flame-born shock wave exited the coils at a velocity below 1200 m/s and decayed. When $\Phi$ decreased to 0.48, the FDDT was still registered in SDT-2. At the fuel-lean concentration limit of FDDT, the minimum apparent propagation velocity of the reaction front at the inlet to the helical tube section was almost constant and close to 500 m/s. When diluting hydrogen with ethylene by 10, 20, 30, and 40%, this velocity was 450 ± 60 m/s (${\Phi }^{*}\approx$ 0.40), 470 ± 50 m/s (${\Phi }^{*}\approx$ 0.44), 450 ± 70 (${\Phi }^{*}\approx$ 0.48), and 410 ± 90 m/s (${\Phi }^{*}\approx$ 0.50), respectively.

The maximum measured DDT run-up distance was attained for (20% C₂H₄ + 80% H₂)-air mixture at $\Phi$ = 0.44 (see Appendix) with a value of 4.69 m along the facility centerline, i.e., DDT always occurred inside the helical tube of the SDT-2.

4. Conclusions

In this manuscript, we continued the study of the FDDT in tubes filled with gaseous explosive mixtures. The concept of FDDT implies the adoption of special means for flame acceleration and the amplification of the flame-born shock wave. For flame acceleration we used a Shchelkin spiral, and for shock wave amplification we used a helical tube section providing gas-dynamic focusing of the flame-born shock wave. Based on our previous studies and the results obtained in the pulsed detonation tube SDT-1, we developed a new pulsed detonation tube, SDT-2, with a longer helical tube section containing ten rather than two coils. Using SDT-2, we studied the influence of the number of coils in the helical tube section on the FDDT in air mixtures of hydrogen, methane, propane, and ethylene, as well as binary methane-hydrogen, propane-hydrogen, and ethylene-hydrogen fuels, under normal conditions. Despite FDDT mainly occurring inside the helical tube of SDT-2, it was designed for studies of FDDT in both gaseous and heterogeneous fuel-air mixtures, which could potentially have a larger DDT run-up distance. Therefore, SDT-2 was designed with some reserve in terms of the helical tube length. Similar to SDT-1, the SDT-2 facility was used to rank the various gaseous and heterogeneous fuel-air mixtures by their detonability in terms of DDT run-up distance and time. This ranking is different from the common approach based on ranking fuel-air mixtures by their detonability in terms of the detonation cell size; the former accounts for both the low-temperature and high-temperature reactivity of fuels during flame propagation and preflame self-ignition, while the latter is relevant only to high-temperature reactivity of fuels during propagation of developed detonations.

In this study, the following new results were obtained:

(1)

The possibility of a significant reduction in the detonation run-up distance in the studied fuel-air mixtures in comparison with that of the SDT-1 was demonstrated;

(2)

The possibility of a significant extension of the fuel-lean concentration limit of the fast deflagration-to-detonation transition in the studied fuel-air mixtures in comparison with that of the SDT-1 was demonstrated; in sufficiently long helical tubes, fast deflagration-to-detonation transition is possible in explosive mixtures of such a composition, for which classical DDT is impossible in straight tubes with smooth or rough walls;

(3)

The minimum apparent propagation velocity of the reaction front at the entrance to the helical tube section at the fuel-lean concentration limit of fast deflagration-to-detonation transition turned out to be virtually constant and close to 400–500 m/s for all studied fuel-air mixtures;

(4)

For all studied fuel-air mixtures, steady-state propagation of detonation in the helical tube section with a velocity deficit above 10% was recorded, which is impossible when detonation propagates in a straight tube with smooth walls.