Flame acceleration and explosion safety applications

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Proceedings of the

Proceedings of the Combustion Institute 33 (2011) 2161–2175

Combustion Institute www.elsevier.com/locate/proci

Flame acceleration and explosion safety applications Sergey B. Dorofeev ⇑ FM Global, Research Division, 1151 Boston-Providence Turnpike, P.O. Box 910, Norwood, MA 02062, USA Available online 12 October 2010

Abstract Accidental explosions of flammable gases and reactive gas mixtures remain a significant concern in process industries. The present paper reviews the basic mechanism of Flame Acceleration (FA), the results of recent studies on FA, and their application to explosion safety. A short overview of the physical phenomena involved in FA is followed by the description of FA and flame propagation regimes in smooth tubes, obstructed channels, unconfined and semi-confined mixtures with or without congestion. A framework is summarized that may be used to evaluate the potential for FA and the onset of detonation. Advances made over recent years in the understanding of FA are discussed. These advances include: new theoretical models that address various aspects of the phenomena involved in FA, high resolution numerical simulations and new detailed experimental studies. A combination of the results of these studies is shown to allow for a significant improvement of our ability to address practical problems. Areas where additional research is required to improve reliability of predictions of FA for industrial safety applications are also discussed. Ó 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Deflagration; Flame acceleration; DDT; Explosion safety

1. Introduction Accidental explosions of flammable gases and reactive gas mixtures remain a significant concern in process industries, including: chemical, petrochemical, mining, transport, power generation, nuclear, and other industries. Explosion hazards associated with combustible gases require the development and maintenance of various safety measures aimed at the prevention and mitigation of accidental explosions. Almost universally, uncontrolled explosions start by the ignition of a flame from either an electrical spark, or the autoignition of a mixture in contact with a hot surface. This event results in a relatively slow ⇑ Fax: +1 781 255 4024.

E-mail address: [email protected] (S.B. Dorofeev).

(often laminar) flame, which, under certain conditions, can accelerate and undergo transition to detonation. For stoichiometric fuel–air mixtures, the pressure ratio across a detonation wave is in the range of 15–20 resulting in about a 20 bar overpressure for an atmospheric initial pressure. Generally, prior to the onset of detonation, the flame accelerates to supersonic speeds relative to a fixed observer, generating a lead shock ahead of the flame with a corresponding pressure ratio in the order of 10. This level of overpressure is typically well above acceptable limits in the process industries. In addition, the supersonic speeds of explosion propagation in either the fast flame or detonations regimes do not allow for the successful application of explosion mitigation methods, such as explosion venting, or explosion suppression. Thus, the fast supersonic flames that can develop due to flame acceleration are not only the prerequisite

1540-7489/$ - see front matter Ó 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2010.09.008

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for the onset of detonation, but also represent significant hazard in their own right. It is mainly due to the reasons outlined above that the topic of Flame Acceleration (FA) and Deflagration to Detonation Transition (DDT) attracts continuing research interest. Over the years, there have been significant advances made in the understanding of FA and DDT that are outlined in several reviews [1–5]. Thanks to experimental, theoretical and advanced numerical studies, the basic mechanisms involved in flame acceleration have been identified and the basic DDT process is fairly well understood. Among these mechanisms, processes involved in FA are particular to the specific initial and boundary conditions of the problem. While the onset of detonation appears to be a relatively universal phenomenon, different physical mechanisms dominate FA depending on the case studied, such as in obstructed channels, smooth tubes, large volumes filled with combustible mixtures, or unconfined vapor clouds. Due to this variety, the research interest in FA has broadened recently and valuable new insights have been obtained. The objective of the present paper is to review the basic mechanism of FA, the results of recent studies on FA and their application to explosion safety. The following description is divided into four main parts: the first part gives a short overview of the physical phenomena involved in FA, the second part deals with FA and flame propagation regimes in smooth tubes, the third focuses on the same phenomena in channels equipped with obstacles and the fourth addresses FA in unconfined and semi-confined mixtures, with or without congestion. This is followed by a summary of the framework, which may be used to evaluate the potential for FA and the onset of detonation, and concluding remarks. This description of the process of FA is applicable to gaseous mixtures of various fuels, and is accompanied by practical examples. 2. Physical phenomena in flame acceleration Basic flame properties, such as the laminar burning velocity, SL, the ratio of densities between reactants and products, r, the thermal flame thickness, d, and the flame sensitivity to strain and curvature and the corresponding Markstein numbers are important parameters for the characterization of flame behavior. Flame instabilities, and their interactions with confinement, obstructions and turbulence, may result in FA through a number of mechanisms that are specific to the particular initial and boundary conditions of the problem. More details on basic flame properties and their relation to FA may be found in [5] and the references therein. The processes immediately following a weak ignition in a combustible mixture are characterized as deflagrations, which propagate at a subsonic

speed into the unburnt mixture. A freely expanding flame, however, is intrinsically unstable. The initially smooth surface of a laminar flame can become wrinkled due to the Landau–Darrieus (LD) instability [6,7], which can be stabilized or destabilized by thermal-diffusion effects. This can result in a cellular structure on the laminar flame. The LD instability has been observed in freely propagating spherical flames [8] and, in absence of other factors, it results in an increase of the flame speed with the flame radius, RF, as R1=3 F . Generally, if confinement and/or obstructions are present, several powerful instabilities may strongly influence flame propagation. These are the well known Kelvin–Helmholtz (KH) and Rayleigh–Taylor (RT) instabilities. The first of these two is a shear instability while the second is developed when a flame is accelerated towards the unburned gas [9–12]. Both the KH and RT instabilities are triggered when the flame is suddenly accelerated over an obstacle or through a vent. In compressible flows, when shocks and pressure waves are formed, the Richtmyer–Meshkov (RM) instability also affects flame surface evolution. Flame propagation in closed vessels and ducts generates acoustic waves that can interact with the flame front and develop flame perturbations through a variety of instability mechanisms. In these cases, flame pulsations and the associated cellular structure caused by acoustics, may be strongly coupled with the RT instability [13–15] and can be further amplified by satisfying the Rayleigh criteria [16] for thermo-acoustics. While the LD and diffusive instabilities are relatively weak, the KH, RT and RM instabilities represent powerful mechanisms that are mainly responsible for the increase of the flame surface and generation of turbulence in channels with obstacles. When turbulence is present in the flow or is generated by the flame, interactions between the flame and turbulence become an intrinsic part of FA. Numerous experimental, analytical, and numerical studies have focused on the description of the behavior of premixed turbulent flames, the details of which can be found in comprehensive reviews on turbulent flames [17–21 and others]. Turbulent burning velocities have been systematically measured and correlations have been suggested by many investigators, as described in [5]. These correlations allow for the determination of turbulent burning velocities as a function of dimensionless parameters characterizing turbulence intensity, scale, and mixture properties. 3. Flame acceleration in smooth tubes 3.1. Process of FA in smooth tubes As already indicated, the initially smooth surface of a laminar flame can be wrinkled due to the

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Landau–Darrieus instability, which can be stabilized or destabilized by the thermal-diffusion effects. This can result in the formation of a cellular flame. With the development of the cellular instability the flame surface grows and the flow generated due to the expansion of the combustion products accelerates. However, the growth of perturbations arising from the LD instability can only provide a relatively slow growth of flame surface area. Thermal expansion of the combustion products produces movement in the unburned gas. The flow interaction with the confinement causes an increase of the flame surface. This results in a moderate increase of the flow velocity and flame speed relative to a fixed observer. This initially laminar phase of FA was analyzed by Bychkov et al. [22,23] and Akkerman et al. [24]. The effects related to the flame surface evolution may lead to a slight deceleration of the flame when the flame reaches the tube walls. The same effects may also be responsible for the temporary development of a “tulip” shape of the laminar flame, which is different from the original meaning of a “tulip” shape of turbulent flames described in the following discussion. Shchelkin [25], Soloukhin [26] and Salamandra et al. [27] demonstrated that flame acceleration in relatively smooth tubes is strongly affected by wall roughness. Initially, the flow may have a smooth axial velocity distribution in the compression wave ahead of the flame. In a sufficiently long tube, the velocity profile is expected to get steeper with time and at some stage a shock forms ahead of the flame. The flow interaction with the tube walls results in the formation of a turbulent boundary layer. The boundary layer starts to appear in the compression wave and continues to grow after the lead shock is formed. Interactions of the flame with the boundary layer results in a significant increase of the burning rate near the tube walls, and a characteristic “tulip” flame shape is formed [27]. A sequence of photographs showing the propagation of a “tulipshaped” flame is provided in Fig. 1 [28]. While the flame propagates along the tube, turbulence is also generated in the core flow. All these processes may result in a variety of flame shapes in the tube depending on the mixture properties and the tube size. The tulip shape may be formed and destroyed at a later stage of flame propagation. The roughness of the tube wall is an important parameter that controls the rate of flame acceleration. In obstructed tubes with a relatively small blockage ratio (BR), approximately 10% or less, the characteristic features of the flame acceleration process and the flame structures generated are similar to those in smooth tubes. 3.2. Run-up distances to supersonic flames In sufficiently long tubes and in typical fuel–air mixtures, flames accelerate up to speeds of about

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Fig. 1. Sequence of shadow photographs (0.1 ms between frames) showing boundary layers ahead of accelerated flame. Flame propagates from left to right; speed of the leading edge of flame is about 320 m/s. Wall roughness is 1 mm; mixture is stoichiometric H2–O2 at 0.6 bar initial pressure.

600–1000 m/s [5]. If the tube diameter is sufficiently large, transition to detonation can be observed as soon as the flame reaches a speed of the order of the sound speed of the combustion products. Although there are situations when additional propagation distance is necessary to establish the detonation regime, most of the data available on the run-up distances refers to the run-up distance to detonation. There are some early experimental data on the effects of tube diameter, initial pressure, and temperature on the run-up-distance to detonation, XD, for tubes without obstacles. These data were discussed in [5]. More recent studies [28–32] give a detailed description of the flame acceleration process so that flame speed as a function of distance is available. These data made it possible to isolate the FA process and to analyze the effect of mixture properties, tube size and roughness on the run-up distance to supersonic flames [33]. The distance necessary to reach a flame speed equal to the sound speed in combustion products, ap, was evaluated in the framework of a simple analytical model. The model considers a flame in a tube with a diameter D and a wall roughness h at a distance X from the ignition point, where a boundary layer of thickness D is formed ahead of the flame. The flame propagates in the boundary layer with a turbulent velocity ST relative to the unburned mixture and with a velocity ST + V in the laboratory frame, where V is the flow speed ahead of the flame. The burning velocity in the core of the flow is lower than the value of ST in the boundary

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layer. The thickness of the boundary layer grows with time while the flow interacts with the wall, resulting in an increase of the boundary layer thickness as measured at various flame positions along the tube, as shown in [28,31]. The model utilizes a general expression for the flow balance in the tube (without accounting for compressibility effects), the turbulent velocity correlation of Bradley et al. [34], and a description for the thickness of the boundary layer, D, at the flame’s position along the tube [28]. The run-up distance, XS, is defined as the distance the flame has propagated when its speed reaches the sound speed in the combustion products. This approach results in the following expression for the run-up distance:     XS c 1 D ¼ ln c þK ; ð1Þ D C j h where j, K and C are constants independent of mixture composition: j = 0.4; K = 5.5; and C = 0.2; D/h = 2/(1  (1  BR)1/2); and c = D/D is given by: 1 "  1=3 #2mþ7=3 ap d c¼ ; ð2Þ gðr  1Þ2 S L D where g and m are two unknown parameters, which were determined from experimental data: g = 2.1 and m = 0.18. Note that most of the data used were for Le close to 1. Although several simplifications were made and compressibility was not taken into account, the model was shown to describe the experimental data by predicting the run-up distances with an accuracy of about ±25%.

Recently, Bychkov et al. have extended their theoretical analysis of flame acceleration in smooth tubes to include compressibility effects [35]. This analysis is based on non-slip adiabatic tube walls with no boundary layer or turbulence and an inverted flame when compared to that in Fig. 1. The conclusion of the theory is that compressibility moderates FA and may result in stationary high speed flame propagation. The results of this analysis have not been compared with experimental data. Comparison with numerical simulations have shown qualitative agreement. 4. Flame acceleration in obstructed channels 4.1. Process of FA in obstructed channels A sequence of the processes typical for FA in obstructed channels is shown in Fig. 2 using shadow photographs of the flame taken at different times [36,37]. Initially, a smooth, or wrinkled, laminar flame develops depending on the mixture properties. While the cellular instability (if it develops, as in Fig. 2) can provide for a slow growth of the flame surface with flame propagation, obstacles located along the path of the expanding flame can cause a rapid increase of the flame surface. Thermal expansion of the hot combustion products produces movement in the unburned gas. The flow around the obstacles leads to a rapid increase of the flame surface (see Fig. 2). This effect results in a further increase of the flow velocity, the flame speed relative to a fixed observer, and the flame surface. This feedback loop can result in continuous flame accelera-

Fig. 2. Sequence of shadow photographs of flame propagation in 10% H2–air mixture with BR = 0.6. Channel width is 80 mm. Times after ignition are 105, 112, and 113 ms.

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tion. This acceleration mechanism is not related to any turbulence effect. In addition to the ‘geometrical’ increase of the flame surface, flame instabilities may lead to an additional flame surface increase. As noted in high resolution numerical simulations, [38–41] the flame may wrinkle as it passes through the orifice of the obstacle due to the Rayleigh–Taylor instability caused by the flow acceleration. This, however, was not noticeable in shadow photos [36,37], nor in the most recent experimental study [42]. The Kelvin–Helmholtz instabilities are observed both in tests and simulations at the flame surface when a shear layer forms downstream of the obstacle. An impressive example of the flame evolution obtained in high resolution 2D numerical simulations is shown in Fig. 3 [41]. Johansen and Ciccarelli [43] have visualized the development of the unburned gas flow field ahead

Fig. 3. Schematic for numerical simulation of flame evolution in a channel filled with a stoichiometric mixture of hydrogen and air at atmospheric pressure and 298 K. Geometric parameters: d/2 = 2 cm, S = 4 cm. Bottom: two temperature scales, one for hot products and other for cold reactants (courtesy of Elaine Oran).

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of the flame propagating in a square duct equipped with two-dimensional obstacles. They have shown that the sudden onset of the flow produces a set of laminar eddies on the downstream side of each obstacle. These eddies grow into large recirculation zones that eventually occupy the volume between adjacent obstacles. In time, a turbulent shear layer develops separating the recirculation zone and the core flow which leads to the randomization of the flow ahead of the flame, and the generation of turbulence. Turbulence increases the local burning rate by increasing both the surface area of the flame and the transport of local mass and energy. A higher overall burning rate, in turn, produces a higher flow velocity in the unburned gas. This ‘turbulent’ feedback loop can result in further flame acceleration. At the same time, the ‘geometrical’ increase of the flame surface continues to play a role. The increase of the burning velocity, ST, with turbulence level is generally limited and a saturation of the ST is usually reached at a level of 10–20 times the laminar burning velocity, SL. Further increase of the turbulence intensity can cause local quenching of the combustion process, which results in a decrease in the effective energy release rate. Depending on the mixture properties and boundary conditions, the interaction of the flame with turbulence in the unburned gas can lead to either weak FA, resulting in relatively slow unstable turbulent flame regimes, or to strong FA, resulting in fast flames propagating at supersonic speed relative to a fixed observer. Detailed shadow photographs of the flame structure in these regimes are given in [5] and in a more recent study [42]. Here we present examples for slow flames (Fig. 4, [36,37]) and fast flames (Fig. 5, [42]). The slow unstable turbulent flames, as shown in Fig. 4, propagate through a highly turbulent media at conditions close to quenching. Incomplete combustion is often observed, or the reaction quenches globally at some stage of flame propagation. Also, these flames move back and forth locally, consuming unreacted material that is left behind the flame. The fast turbulent flames (Fig. 5), which result from strong FA, are supersonic combustion waves consisting of a lead shock, or a system of shocks, followed by a turbulent flame brush. The overall energy release rate in the turbulent flame brush is high enough to result in supersonic flame propagation relative to a fixed observer. The flame speed in this regime, which is referred to as the “choked flame” or “choking regime”, is often close to the speed of sound in the combustion products. The shock waves ahead of the flame undergo periodic reflections off the obstacles and the tube walls and thus are dispersed in all directions (see Fig. 5 and photographs in [42,44,45]). The propagation velocity of the fast flame depends on the obstacle geometry. The larger

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gration in an obstacle laden tube. Teodorczyk et al. [44,45] suggested that this fast flame regime can be distinguished from quasi-detonation by a relatively low strength lead shock which is not sufficient to cause mixture autoignition. According to Refs. [44–46], the mechanism responsible for ignition in high speed deflagrations is turbulent mixing of hot combustion products and reactants promoted by the obstacles. Chue et al. [47] proposed that this complex shock–flame structure can be modeled as a CJ deflagration. They argue that in the choked regime flame propagation is governed more by the mixture energy content than the turbulent transport rates. Veser et al. [48] suggested that the relatively low burning velocity of a regular turbulent flame (i.e., ST  10SL) can provide the required burning rate because of the development of a sufficiently large flame surface area (see the flame brush in Fig. 2). Three-dimensional numerical simulations of flame propagation in a tube with orifice plate obstacles using a range of prescribed constant ST were made in [48]. It was shown that the surface of the flame adjusts itself in such a way that it is higher for lower ST values and vice versa, while the flame speed in the laboratory frame remains almost independent of the value of ST and close to ap. More details on the development of supersonic flames in obstacle laden channels have been obtained recently by Ciccarelli et al. [42]. Using high resolution shadow photography, they showed that shock–flame interactions are the key to maintaining the high propagation speed in the final quasi-steady explosion propagation regime. Each shock–flame interaction produces flame distortions due to baroclinic effects and the Richtmeyer–Meshkov instability. Repeated flame–shock interactions result in the appearance of the turbulent flame front. This observation is in accord with the high resolution numerical simulation results [38–41] and with an earlier experimental study on shock– flame interaction [49]. Fig. 4. Shadow photographs showing structure of slow deflagration in 10% H2–air mixture with BR = 0.6. Flame tong propagates trough the central part of the channel (with a speed of about 100 m/s relative to fixed observer) leaving unreacted material behind. Then flame propagates back through partially unreacted material. Channel width is 80 mm. Left boundary is 1.6 m from ignition, times after ignition are 130.7, 131.6, 132.8, 134, and 135.8 ms.

the blockage ratio (up to about BR = 0.75), the more distinctive the choking regime [30]. This regime is also more pronounced for the orifice plate obstacles, than with less regular obstacle configurations, as suggested by Chao and Lee [46]. Historically, there have been a number of points of view on the description of the propagation mechanism of a high speed turbulent defla-

4.2. Weak and strong FA In the studies of Kuznetsov et al. on the behavior of turbulent flames in mixtures based on hydrogen [30], and hydrocarbon fuels [50], it was found that there is a well-defined difference in flame behavior between slow, subsonic flames and fast combustion phenomena, such as choked flames and detonations. This was true in tubes with diameters from 80 to 520 mm containing different configurations of obstacles. Figure 6 shows, for example, the flame propagation velocities versus distance along tubes with a BR = 0.3 for hydrogen–air mixtures. More examples are given in [5,30,50]. Generally, flame acceleration results in one of three different propagation regimes: slow subsonic flames, fast supersonic flames (choked

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Fig. 5. Shadow photographs showing structure of fast deflagration stoichiometric methane–air at an initial pressure of 47 kPa absolute and room temperature with BR = 0.33. Turbulent flame is preceded with system of shocks. Channel width is 76 mm. Camera operated at a speed 30,000 frames/s (courtesy of Gaby Ciccarelli).

2000

1600

BR=0.3 (air)

80 mm

174 mm 10%H 2

10%H 2

11%H 2

13%H2

520 mm 10%H 2

12%H 2

13%H 2

13%H 2 15%H 2 17.5%H 2

v, m/s

1200

800

Fast flames 400

Slow flames 0

0

10

20

30

40

50

60

70

x/D

Fig. 6. Flame velocities for lean hydrogen–air mixtures versus dimensionless distance along tubes of 174, and 520 mm id, and square channel of 80  80 mm, BR = 0.3.

flames), and quasi-detonations. In some cases of slow flames, global quenching is observed. The most pronounced difference is observed between the cases of slow and fast flames. This is true for all mixtures and blockage ratios tested in [30,50]. The difference between the fast flames and quasi-detonations is not well defined for

BR = 0.3 as seen in Fig. 6. For the larger BRs of 0.6 and 0.9, the slow flames were more unstable compared to BRs < 0.6. Global flame quenching was not observed for BRs 6 0.1. The sharp difference between the slow and fast flames is a consequence of the difference in the FA process, which can be characterized as weak or strong. In the case of weak FA, a significant increase of the flame speed from initial values characterized by the laminar flame is observed, but the propagation velocity remains well below the sound velocity in the reactants. The energy release rate, defined by the flame surface area, chemistry and turbulence, remains limited in the case of weak FA, and the flame is not able to generate significant compression waves during its propagation. In the cases of strong FA, the energy release rate appears to be sufficiently high, such that the flames are able to generate strong compression and shock waves ahead, resulting in fast supersonic flames and, possibly, transition to detonation. With the aim to evaluate the boundary between cases of weak and strong FA, a set of parameters that can influence the flame-flow feedback mechanism in obstructed ducts was considered in [51]. The parameters that were correlated with the flame behavior included flame properties, such as SL, d, r, the characteristic sound speeds, the ratio of specific heats and the Lewis (Le),

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Markstein (Mab) and Zeldovich numbers (b ¼ Ea ðT b  T u Þ=T 2b ), where Ea is the effective activation energy, Tu and Tb are initial and adiabatic flame temperatures). These parameters can be determined a priori and define the potential for flame acceleration. An important conclusion reached in [51] was that the mixture expansion ratio is the key parameter that defines the border between weak and strong FA. A sufficiently large expansion ratio was found necessary for the development of fast flames. The corresponding critical conditions were suggested in the form: r > r*(b, LT/d), where LT is the integral scale of turbulence. It was found that for a sufficiently large scale ratio, LT/d, exceeding two orders of magnitude, the critical conditions become independent of scale and can be reduced to a function of the mixture composition only r > r*(b). Figure 7 shows, as an example, the combustion regimes as a function of expansion ratio, r, and b, where the critical r increases with b. The effect of the dimensionless activation energy b on the critical conditions for strong FA was related in [51] with quenching of products/ reactants eddies mixed by turbulence. Qualitatively, a high b provides a stronger ability for the turbulent mixing to locally quench the combustion process. This problem was revisited in [52] with a more detailed analysis of quenching for a range of sizes of products/reactants eddies mixed by turbulence. It was found that the critical conditions for quenching/re-ignition of the largest mixed eddies appear to be independent of any parameters characterizing turbulence:

9 CH - fuels

8 7

σ

6 5 4 Ma > 0 slow flames choked flames and detonations

3 H2 mixtures

+- 8% deviation

2 1

3

4

5

6

7

8

9

10

11

12

β Fig. 7. Combustion regime as a function of expansion ratio r and b for mixtures with Mab > 0. Black points – fast flames and detonations (strong FA) and gray points – slow combustion regimes (weak FA).

r2 b2 ðb=2  1Þn e1b=2 ¼ 1; 6Len Cnþ1 l

ð3Þ

where l  20 is a constant, n is reaction order and C is the Gamma function. Only mixture properties, such as r, b, n, and Le, affect the thermal regime of the largest mixed eddies. This is a remarkable result, which shows that the mixture properties may prescribe certain types of flame behavior in turbulent flows. It was also shown that all mixed eddies are expected to be quenched in sub-critical mixtures (left hand side of Eq. (3) less than unity) in any turbulent flows with a Karlovitz number Ka P 1. If one considers the process of FA, its effectiveness depends on the feedback between the flame generated flow of the unburned gas and the flame itself. Turbulence generated in the flow ahead of the flame interacts with the flame. Initially, this interaction results in a positive feedback through the increase of the flame surface, flow speed, and turbulence level. At a certain stage, the Ka-number may become greater than unity, and mixed reactants/products pockets may be formed in the flow. Then, if quenching of the largest mixed pockets dominates (left-hand side of Eq. (3) less than unity), all mixed eddies are quenched, the overall rate of energy release is suppressed, and the positive feedback is destroyed. If re-ignition of the largest mixed eddies dominates, mixing of products and reactants by turbulence does not significantly affect the rate of energy release, and the positive feedback is sustained. This argument supports the view that the critical conditions defined by Eq. (3) may be interpreted as the border line between cases of weak and strong FA. The critical conditions given by Eq. (3) were compared with experimental data in [52]. It was shown that the order of magnitude, the actual values and the general trend of the function r(b) are close to those obtained experimentally. Both the model and the data show that the critical mixture expansion ratio increases with b and Le. This means that unstable mixtures (Le < 1) and mixtures with low dimensionless effective activation energy (such as lean hydrogen mixtures at high initial temperatures) will show strong FA in compositions with low energy content (r of about 2 and more). At the same time, stable mixtures and mixtures with high dimensionless effective activation energy (such as rich methane mixtures at low initial temperatures) can only sustain strong FA in compositions with high energy content (r of about 7 and more). 4.3. Run-up distances to supersonic flames Historically, the run up distance to detonation was the primary interest of most of the DDT studies. Veser et al. [48] were the first who specifically addressed the run-up distance to supersonic

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flames. The problem of the minimum run-up distance for the flame to accelerate to supersonic combustion regimes in tubes with obstacles was studied both experimentally and numerically. A simple model was proposed, which describes the evolution of the flame shape in a channel containing obstacles with a relatively high BR. The model assumes that the flame has the shape of a deformed cone (as in Fig. 2), which stretches until the moment the speed of the flame head reaches the sound speed, ap, with respect to the combustion products. The position of the flame head at this time gives an estimate for the run-up distance. Then the flame cone cannot stretch any further and moves down the tube at a quasi-steady velocity. The dimensionless flame acceleration distance was determined in the model, taking into account mixture properties, such as SL, r, and ap. It was assumed that for a relatively high BR, the turbulent burning velocity reaches a maximum saturation value of the order of ST  10SL at the initial stage of FA, and from that point on FA is mainly due to the increase of the surface of the turbulent flame brush as it is shown in Fig. 2. The flame acceleration distance XS was expressed as a function of BR: X S 10S L ðr  1Þ 1  BR a ; R ap 1 þ b  BR

ð4Þ

where R is the tube radius and a and b are unknown parameters of the model. The scaling of the run-up distance with mixture properties, as in the left-hand side of Eq. (4), was evaluated using a wide range of experimental data and the results of 3D numerical simulations. It was shown that the grouping on the left-hand side of Eq. (4) collapses the data to within ±25%. The effect of BR given by the right-hand side of Eq. (4) was found to correlate well with the data for a = 2 and b = 1.5, for a BR range of 0.3–0.75. Recently, Card et al. [53], Ciccarelli et al. [54] and Sorin et al. [55] investigated flame acceleration and DDT in fuel–air mixtures and compared their results with Eq. (4). The comparison is reviewed in detail in [5]. Most recently Valiev et al. [56] addressed FA in tubes with obstacles

theoretically as well as computationally. The influence of gas compression on FA was considered and it was shown numerically that the flame acceleration rate decreases with increasing Mach number, and that the velocity eventually saturates at a value that correlates with the CJ deflagration speed. The analytical model [56] considers incompressible flow with constant flame speed and yields the dependence of the flame tip speed as a function of distance for several configurations of obstacles, as was also done in model [48]. In the notation adopted here, the flame speed increase in the model [56] is proportional to distance: X S T ðr  1Þ ; ð5Þ R 1  BR which is similar to the model [48]. The proportionality coefficient in Eq. (5) depends on the obstacle configuration. Practical applications of the model were not considered in [56] and no comparison with experimental data was provided. Equation (4) currently remains perhaps the only practical scaling relation for the run-up distances to supersonic flame, which takes into account the duct geometry, i.e., tube diameter and BR, as well as mixture properties, i.e., laminar burning velocity, expansion ratio and isobaric sound speed in the combustion products. The accuracy of the predictions of Eq. (4) is estimated to be within a factor of 2. However, in cases where the value of SL is unknown, i.e., at elevated initial temperatures or with sensitized mixtures, one should be cautious with estimates based on Eq. (4). A comparison of the run-up distances predicted using Eqs. (1) and (2) (BR 6 0.1) with those predicted by Eq. (4) (BR P 0.3) for stoichiometric mixtures of methane, propane, ethylene, and hydrogen with air is presented, as an example, in Fig. 8. It is seen that the run-up distances in smooth tubes are significantly higher than the run-up distances in obstructed tubes. In the range of BR between 0.1 and 0.3, neither of the two models is applicable. For practical applications, one may “bridge” the range of BR from 0.1 to 0.3 as shown by dashed lines in Fig. 8.

V f  rS T /

D=1m

H2 BR<0.1 C2H4 BR<0.1 C3H8 BR<0.1 CH4 BR<0.1 H2 BR>0.3 C2H4 BR>0.3 C3H8 BR>0.3 CH4 BR>0.3

XS/D model

100 90 80 70 60 50 40 30 20 10 0 0.01

2169

0.1

1

BR

Fig. 8. Run-up distances over tube diameter as a function of BR for D = 1 m.

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Figure 9 shows, as an illustration, the dimensionless run-up distance versus BR for a stoichiometric hydrogen/air mixture at 298 K and 1 bar initial temperature and pressure in tubes with diameters from 0.01 to 10 m. For relatively smooth tubes with a BR between 0.01 and 0.1, XS/D decreases with the tube diameter. For obstructed tubes with a BR > 0.3, XS/D does not depend on the tube diameter, and the run-up distance, XS, is proportional to D. Many practical applications involve cases where combustible mixtures are at elevated initial temperatures and/or pressures. The elevated initial temperature and/or pressure changes the fundamental properties of a combustible mixture, such as the laminar burning velocity, flame thickness, and isobaric sound speed. These changes can result in significant variations of the critical runup distances as predicted by Eqs. (1), (2) and (4). It is very important that the initial temperatures and pressures be properly taken into account for the determination of the critical run-up distances. 5. Unconfined and semi-confined mixtures 5.1. Unconfined spherical deflagrations without obstacles The initial dynamics of unconfined expanding spherical flames with no obstacles is similar to that described above for flames in channels. As the flame grows in time and the flame stretch reduces, a cellular structure develops due to the thermal diffusion and Landau–Darrieus instabilities. Eventually, in situations without confinement and obstacles, the LD instability dominates the flame surface morphology. The hydrodynamic cells continue to develop on the flame surface, increasing the flame surface wrinkling of a globally spherical flame. This results in the continuous increase of the energy release rate and the flame speed in relation to a fixed observer.

D=0.01m BR<0.1 D=0.1m BR<0.1

70

XS /D model

60

H2/air

50

D=1m BR<0.1 D=10m BR<0.1 D=0.01m BR>0.3 D=0.1m BR>0.3

40

20 10 0 0.01

0.1 BR

RF ¼ R0 þ Atn ;

1

Fig. 9. Run-up distances in stoichiometric hydrogen–air mixtures at 298 K and 1 bar initial pressure as a function of BR for various tube diameters.

ð6Þ

where R0, A and n are constants. At large radii Eq. (6) reduces to a self-similar acceleration law, RF  Atn , with exponent n = 1.5. Gostintsev et al. [8] suggested that this self-similar regime of flame propagation is due to the fractal nature of the LD flame instability. This led to the following expression for the flame speed as a function of radius for mixtures with a typical expansion ratio r  8: 1=3 1=3 V f ¼ 0:4  S 4=3 RF ; L v

ð7Þ

where v is the thermal diffusivity of the mixture and the coefficient 0.4 was determined from experimental data. There have been several studies on the fractal nature of the LD instability and self-similar flame acceleration [57–62]. The exponent n was found to be in the range 1.25–1.5 in these studies. Another approach to estimate flame acceleration of an unconfined spherical flame was suggested in [63,64]. It was assumed that the flame accelerates through self-turbulization due to the generation of the flow ahead of the flame. It was further assumed that the turbulent intensity, u0 , is a fraction of the flow speed ahead of the flame. This self-induced turbulence increases the burning rate of the flame brush, ST. Application of the Bradley correlation [34] for the turbulent burning velocity yields the following expression:  1=3 LT : ð8Þ V f / ðr  1ÞS L d The only value with a dimension of length, which affects LT, is the flame radius, RF. Thus, LT may be considered to be proportional to RF, and the flame speed increases as R1=3 F . Using the approximate expression, d  v=S L , and the empirical coefficient determined in [63,64], Eq. (8) may be rewritten as (r  8): 1=3 1=3 V f ¼ 0:47  S 4=3 RF : L v

D=1m BR>0.3 D=10m BR>0.3

30

An analysis of a representative set of experimental data on unconfined spherical deflagrations, presented by Gostintsev et al. [8], included mixtures of C2H2, H2, C2H4, C2H4O, C3H8 and CH4 with air and oxygen. Experiments were made with initial mixture radii ranging from 0.3 to 10 m. It was found that the flame radius increases with time according to the relation:

ð9Þ

It is seen that the functional dependences of the flame speed in Eqs. (7) and (9) are the same, and the coefficients are similar. The difference in their value is less than the uncertainty in the two empirical coefficients in Eqs. (7) and (9). It is remarkable that the approach in [63,64], based on the assumption of flame self-turbulization, and the analysis in [8], based on the LD

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instability and its fractal nature, both resulted in similar functional dependences of the flame speed on the flame radius and mixture properties. It is interesting to note that the similar functional behavior pointed out above can be only observed with the exponent n in Eq. (6) equal to 1.5. The increase of the flame speed for unconfined flames without obstacles is very weak. For example, for a flame to accelerate to a speed of only 80 m/s, flame propagation distances of approximately 6200, 2600, and 13 m are needed for stoichiometric mixtures of methane, propane, and hydrogen in air, respectively. This is in good agreement with the largest known unconfined test performed at Fraunhofer ICT [65,66], where experiments were made with stoichiometric hydrogen– air mixtures in a hemi-spherical plastic envelope, cut just before ignition, with a radius of 10 m. A maximum flame speed of 80 m/s and maximum air blast wave overpressure of 6 kPa were reported. It may be concluded that, in industrial situations, it is practically impossible to expect FA to supersonic flame speeds and DDT in fuel–air clouds without any obstacles or confinement. Most recently, the analysis of self-accelerating spherical flames was extended by Akkerman et al. [67] to include compressibility effects. The solutions for the flame speeds and the conditions for explosions triggering upstream of the flame were derived. It was concluded that the mechanism of self-similar flame acceleration under unconfined conditions is weak and it is unlikely to result in DDT in terrestrial situations, in accordance with the estimates presented above. 5.2. Unconfined deflagrations with obstacles The effect of obstacles on the propagation speed of an unconfined spherical flame was studied theoretically in [63,64]. In that model, an initially spherical flame is considered, which propagates from an ignition source through an obstructed area. The flame speed increases due to the increase of the flame area (flame folding) in the obstacle field and due to the increase of the turbulent burning rate during flame propagation. The latter effect should also describe the increase of the flame speed with distance in a system without obstacles as described in the previous section. The flame folding effect can be approximately described by applying simple geometrical considerations. A uniform obstacle field is assumed, with distance between obstacles, x, and characteristic size, y. This model gives the following estimate for the flame speed as a function of distance:  2  1=3 4 ry RaF RF V f ¼ a2 brðr  1ÞS L 1 þ ; a d 3 x ðrxÞ ð10Þ

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where a2b and a are unknown parameters. Equation (10) gives an approximate description of the flame speed as a function of distance in an area with or without obstacles. It is seen that a significant increase of the flame speed with distance can be only expected in a system with obstacles, y – 0. Mixture properties are accounted for in Eq. (10) through the values of SL and r. The two unknown parameters a2b and a were evaluated using experimental data on flame speed as a function of distance inside obstacle arrays [68,69] and data on flames with no obstacles [70,71]. It was found that a2b = 8.5e3, a = 0.63. Data for relatively high-reactivity fuels, including hydrogen, ethylene and propylene, were selected for a wide range of distances and flame speeds. Another set of experimental data with [72,73] and without obstacles [8,66] was used here for model validation. It was shown that this correlation and the predictions of blast effect based on it both agree well with this set of data. Figure 10 shows, as an example, the maximum flame speed for the four fuels with a medium level of congestion represented by y/x = 0.33 and x = 1 m. It is seen that the fuel properties strongly affect the flame speeds. For the flame radius of 3 m and medium congestion, flames reach the speeds of about 50, 75, 130, and 400 m/s for methane, propane, ethylene and hydrogen mixtures with air. The effect of obstructions is so strong, that flame self-acceleration discussed in the previous section can be totally neglected in practical applications. This leads to the notion of the Potential Explosion Source (PES) widely used to evaluate vapor cloud explosion hazards for industrial safety applications, where only the congested part of the cloud, the PES, is used for blast estimates. It should be noted, however, that although the uncongested part of the cloud cannot contribute to the maximum level of the blast overpressure defined by the maximum flame speed, it can significantly increase the impulse of the blast wave and overall damage potential. 5.3. FA in vented explosions Explosion venting can be used to prevent or minimize damage to an enclosure by relieving the pressure generated within the volume and is an important engineering loss prevention solution. Analytical models and empirical correlations have been developed, a number of which have been included in engineering standards and guidelines [74,75]. These correlations, however, often have conflicting recommendations. This is due to the complex nature of the process itself and the influence of many factors that can affect the peak overpressure, such as size and shape of the enclosure, the mixture being burned, the type of vent and vent deployment pressure, congestion or obstacles inside the chamber and the ignition location.

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Flame speed, m/s

350 300 250

H2 C2H4 C3H8 CH4

200 150 100 50 0 0

2

4

6

8

10

12

14

16

Distance, m

Fig. 10. Flame speeds for stoichiometric fuel–air mixtures as a function of flame propagation distance for medium congestion.

Overpressure (bar)

0.06 Unfiltered 80 Hz Low Pass

0.04 0.02 0.00 -0.02 0.0

0.2

0.4

0.6

0.8

1.0

Time (s) 0.3

Overpressure (bar)

The initial dynamics of FA in vented explosion is similar to that described in the previous sections. However, as the flame approaches the vent, specific features, characteristic to vented explosion, are revealed. It has been found that vented deflagrations exhibit a wide range of physical phenomena which directly influence the dynamics of the process [76–78] and are of fundamental interest for combustion science. These phenomena include Helmholtz oscillations [76,79], the external explosion [76,80], flame instabilities [76,78], flame–acoustic interactions [76–78], and the generation of turbulence [78,81]. In addition to these factors, the Rayleigh–Taylor instability has been observed in vented explosions. In particular, it has been observed to develop on the flame surface inside the chamber during both Helmholtz oscillations [76,79], and near the vent after the flame exits the chamber [82]. Most recently [83,84] it was suggested that the RT instability may be a key parameter affecting the rate of pressure generation throughout the vented deflagration process. This analysis was based on a series of vented explosion experiments performed for stoichiometric propane–air mixtures in a 63.7-m3 explosion test chamber. It was found that the effect of the RT instability developing outside of the vessel, during the external explosion, is an important factor in the internal pressure build-up. Previous studies [85,86] have found that conventional factors, such as turbulence generation, have failed to provide an adequate description of the dynamics of the external explosion and the associated pressure transient as the one shown in Fig. 11 (top). Helmholtz oscillations and acoustics of various frequencies in the chamber were also found to be strongly coupled with the RT instability [84]. It was shown that the length scale of the flame perturbations closely match the minimum RT unstable wavelengths for each of the oscillation frequencies in the tests. It was also found that the highest frequency oscillations generated the strongest pressure transient as the one shown in Fig. 11 (bottom).

Unfiltered 80 Hz Low Pass

0.2 0.1 0.0 -0.1 0.0

0.4

0.8

1.2

1.6

Time (s)

Fig. 11. Experimental pressure traces for a 4.0 vol.% propane–air explosion in 64 m3 chamber with 5.4 m2 vent, ignited on the wall opposite the vent (top) and 2.73 m2 vent, ignited near the vent (bottom).

Thus the RT instability was found to be present throughout the combustion process: outside the vessel during the external explosion, as well as in the development of acoustics within the chamber. In both cases, the coupling between the flame instabilities and the combustion process produced significant peak over-pressures and was responsible for the overall peak pressures seen in all of the tests studied. The implication of this result is that the RT instability must be considered when developing correlations and models predicting explosion severity and the consequences of vented explosions. One of the simplified models for predicting the pressure transient in vented

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explosions developed in [83,87] shows good correlation with experimental data for a zero vent deployment pressure. Situations when the vent panel deploys at a finite pressure are more difficult to model. At vent deployment, a rarefaction wave enters the chamber and accelerates the flame in the RT unstable direction. This acceleration, may be very strong, results in the rapid growth of instabilities, which may, under certain conditions, lead to the onset of detonations [88,89]. Generally, the available correlations and engineering models for the prediction of the flame dynamics and pressure transients in vented explosions are less reliable relative to the models for FA in ducts and unconfined clouds. This is due to the wider range of physical phenomena involved and the complex interactions between them. This area, however, is very important for safety and requires additional research effort. 6. Evaluation of potential for flame acceleration and DDT As the processes involved in FA are quite complex, it is difficult to present a detailed description of all scenarios of FA. There are several analytical models available that address various aspects of the phenomena. Excellent progress has been made during the last decade in high resolution numerical simulations. Their results are very useful to gain insights into the physics involved and provide a qualitative picture of the flame dynamics and DDT. While in some cases a reasonably good agreement between simulations and experiments has been demonstrated, the quantitative predictions of the best available numerical models still do not seem sufficiently reliable for practical safety applications. As a result, the tools available to address practical safety problems are essentially based on experimental data. The understanding of the physics involved in FA allows for the formulation of simple models that, after their calibration against a representative range of experimental data, can be used to address practical safety problems. These models form a framework of criteria to evaluate the range of accessible flame speeds and corresponding explosion hazards. Most of these criteria are formulated as necessary conditions for FA and DDT, although there are several models that give an estimate of the actual flame speeds as a function of the initial and boundary conditions. The main ideas of this framework are outlined in the following discussion. FA follows all cases of a weak ignition of a combustible gas. However, there are cases when FA is weak and supersonic combustion regimes cannot be developed and flame speeds remain well below the sound speed in reactants. In order for FA to be strong, a sufficiently large expansion

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ratio r must exist between the unburned and burned gas. The critical expansion ratio depends on the mixture composition and initial temperature and pressure. In addition, the transverse size of the mixture should be at least two orders of magnitude larger than the laminar flame thickness for strong FA to be possible. If strong FA is possible, a range of high flame speeds can be observed. A sufficiently large runup distance is necessary for actual development of supersonic combustion regimes, or so called fast flames. The minimum run-up distances depend on mixture properties (such as the laminar burning velocity, laminar flame thickness, and isobaric sound speed in the combustion products), initial conditions, obstacle configuration and/or duct size. The supersonic combustion regimes must develop before the conditions for the onset of detonation can be reached. If the supersonic combustion regime is developed, then detonation initiation may only occur if the physical size of a duct or mixture volume, L, is sufficiently large compared with a length scale that characterizes the reactivity of the mixture. The usual choice of the reactivity length scale is the detonation cell size k, which is a function of mixture composition and thermodynamic conditions. Results of numerous DDT studies reviewed in detail in [5] have demonstrated that the physical dimension of a duct or mixture volume, L, has to be greater than some multiple of the cell width, k, in order for the onset of detonation to be possible. Uncertainties in the determination of the critical conditions, including critical values of mixture expansion ratio, uncertainties in the detonation cell size data, the laminar burning velocity, the laminar flame thickness and the effect of the Lewis number must be taken into account in practical applications. There are issues with respect to changes of the thermodynamic state of unburned mixture during flame acceleration, which can change the critical conditions for FA and DDT, and should also be taken into account in practical applications. 7. Concluding remarks An overview of the physical phenomena involved in flame acceleration following weak ignition of a combustible gas has been presented. Over recent years, significant advances have been made in the understanding of FA and DDT. These advances include new theoretical models that address various aspects of the phenomena involved in FA, high resolution numerical simulations, and new detailed experimental studies. Combining the results of these studies allowed for a significant improvement of our ability to address practical problems. Yet, there are areas where our ability to provide reliable predictions of FA for safety applications is limited.

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In the area of theoretical models, there is still a need of experimental data to calibrate these models. In the development of numerical models, there is a significant lack of resolution available to address problems at industrial scales. Sub-grid models are still required to resolve not only the range of turbulent and chemical scales, but also the flame instabilities that result in the growth of the flame surface. Currently, only 2D models can provide a resolution approaching the required minimum at industrial scales. There are also gaps in the availability of experimental data on flame properties for a number of gases used in industry. Many of the models are not applicable to practice because the required values of the laminar burning velocities are unknown. Even such a well studied fuel as hydrogen continues to present a challenge. Particularly, more detailed data on the basic properties of lean hydrogen flames are required and the effect of the Le number on the turbulent burning velocity must be better understood for the development and validation of models describing FA in hydrogen mixtures. From the point of view of safety applications, the area of vented explosions needs significant improvement. Available engineering models and correlations adopted in various industrial guidelines and standards are not sufficiently reliable for the prediction of maximum pressure loads and vent sizing. The deviation of the predicted maximum overpressures from test data may exceed an order of magnitude when the available guidelines are used.

Acknowledgement The author greatly acknowledges FM Global for the support provided during preparation of this review.

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