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A simple and effective approach for evaluating unconfined hydrogen/air cloud explosions Liang Pu a,b, Xiangyu Shao a,*, Qiang Li a, Yanzhong Li a,b a b
School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China State Key Laboratory of Technologies in Space Cryogenic Propellants, Beijing 100028, China
article info
abstract
Article history:
The topic of hydrogen safety assessment has been focused by many researchers. The
Received 10 February 2018
overpressure evaluation of vapor cloud explosion (VCE), is an important issue for both
Received in revised form
designing and evaluating on chemical plants, as well as buildings. Unknown flame radius
7 April 2018
history limits the original acoustic approximation model's application. The objective of this
Accepted 7 April 2018
work is to develop an achievable model for hydrogen/air deflagration assessment in en-
Available online xxx
gineering applications, and the model should have high computational efficiency. A tentative scheme that starts from flame/piston speed history solving was adopted, and the
Keywords:
flame/piston radius and acceleration history will be obtained subsequently. Thus, the
Hydrogen
overpressure history for far field could be gotten based on the acoustic approximation
Safety assessment
model. A simplified scheme was employed for the region inside the flame cloud. The model
Vapor cloud explosion (VCE)
proposed in this paper could be solved in several seconds, because there are no differential
Acoustic approximation model
equations but only algebraic equations. The model was verified by hydrogen/air deflagra-
Overpressure
tion tests from small scale to large scale. Compared with the experimental data, the model appeared well agreements in the medium and large scale cases. In the small scale cases, the model obtained acceptable solutions. © 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Hydrogen is a promising energy carrier. Because of the flammability of hydrogen, its safety utilization has always been a key concern [1e4]. Not only hydrogen, the leakage of combustible gas or liquefied combustible gas (with a low boiling point at ambient temperature), like methane, ethylene, propane etc., will form flammable cloud in the atmosphere [5]. Once the flammable cloud meets ignition source, the VCE (Vapor Cloud Explosions) accident may be occurs, which is a main hazardous issue in energy and chemical industries [6]. It is reported that approximately 174 VCE accidents occurred
during 1940e2010 in the world [7,8], which resulted in huge personal and property losses. The large overpressure load created by the flammable cloud explosion, could cause serious damage. Thus, it is necessary to investigate the pressure wave of potential accidents, for the safety assessment of existing and designed chemical plants and buildings [9]. And this topic has attracted much attention by researchers for several decades. There are three categories models to predict the overpressure of VCE accidents [10e12]: empirical models (TNT equivalence model [13], TNO Multi-Energy model [14,15], Baker-Strehlow model [16], CAM [17], etc.), phenomenological models (CLICHE model [18] and SCOPE model [19]) and CFD
* Corresponding author. E-mail addresses: [email protected] (L. Pu), [email protected] (X. Shao). https://doi.org/10.1016/j.ijhydene.2018.04.041 0360-3199/© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Pu L, et al., A simple and effective approach for evaluating unconfined hydrogen/air cloud explosions, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.04.041
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Nomenclatures af ai c0 C D D2 M p R Ra R0 S' SL t T v V x y
flame/piston acceleration (m/s2) empirical parameter in Eq. (5) sonic speed (m/s) molecular fraction (%) flame/piston speed in Eqs. (4) and (5) (m/s) maximum flame speed in Eq. (4) (m/s) molecular weight pressure (Pa) distance from ignition center (m) affected distance from ignition center (m) initial radius of hemispherical unburnt mixture cloud (m) average flame speed in Eqs. (5), (20) and (23) (m/ s) laminar flame speed in Eq.(18)e(21), (23) (m/s) time (s) temperature (ºC) flame speed in Eq.18 and 19 (m/s) volume of unburnt mixture (m3) distance between obstacles in Eq. (18) (m) characteristic size in Eq. (18) (m)
Greek symbols a reciprocal of b b expansion coefficient in Eq. (7) d flame thickness in Eq. (21) (m) m dynamic viscosity (Pa$s) r density (kg/m3) t shifted time (s) 4 equivalence ratio c thermal diffusivity in Eq. (19) (m2/s) Subscripts 0 initial ex external f flame or final i inflexion in internal k index of component max maximum s start
models. Compared with the first two models, an appropriate CFD model can obtain much accurate overpressure time history and more details of the flow field. However, the great demand for time and computer resource of CFD models makes it impractical for engineering applications, especially for complex geometries [10,20e23]. At this aspect, the empirical models and phenomenological models have their advantages, but these models also have their weaknesses. For empirical models, the weaknesses includes: the drawbacks of weak accuracy (TNT, Baker-Strehlow), only positive phase peak pressure can be solved (TNT, Multi-Energy), and not good realizability (Multi-Energy, CAM), etc. The sophistication of phenomenological models is between empirical models and complex CFD models, but few information of the flow field
could be provided in phenomenological models [10,11]. The detailed comments of the previous models can be seen in Ref. [10] and Ref. [12]. In addition, Deshaies [24] proposed an acoustic approximation model, in which the flow field outside the flame was divided into two zones, incompressible source flow (near the flame front) and acoustic approximation of source flow (far from the flame front). The overpressure time history at a distance of R from the ignition center was given as follows:near the flame front: pðR; tÞ p0 ¼
( 2 2 r0 dRf ðtÞ d Rf ðtÞ ð1 aÞ 2Rf ðtÞ þ R2f ðtÞ dt dt2 R ) 2 1 a R4f ðtÞ dRf ðtÞ R3 2 dt
(1)
and far from the flame front: ( 2 2 ) r0 dRf ðtÞ d Rf ðtÞ 2 þ Rf ðtÞ pðR; tÞ p0 ¼ ð1 aÞ 2Rf ðtÞ dt dt2 R t¼t
r c0
(2)
(3)
The acoustic approximation model can provide the pressure-time history with a varied distance R, and it has been checked by several deflagration experiments [25,26]. As can be seen in Eqs. (1) and (2), the flame radius history Rf(t) is necessary for overpressure calculation, but it is unknown before the experiment. Thus, this drawback restricts the practical use of the acoustic approximation model. Moreover, the acoustic approximation model can't obtain the pressure wave inside the flame sphere. However, the length of the flame may be hundreds of meters to several kilometers, and the affected zone is very broad in large scale VCE accidents. Based on the discussions mentioned previously, the weaknesses of the models motivate the authors to develop a simple and effective model for engineering application of safety assessment. In the present study, a model on the basis of the acoustic approximation model is proposed, in which an independent flame history function improves the availability of the original model. In Section Theory hypothesis, the pressure wave shapes are classified with findings from hydrogen/air deflagration tests. The calculation model of overpressure is presented in Section Method, and in Section Result and discussion it will be validated.
Theory hypothesis Findings in gas deflagration experiments In 1983, the Fraunhofer Institute for Propellants and Explosives (Germany) conducted a series of large scale unconfined vapor cloud explosion (UVCE) tests in open area, with H2/Air mixtures [27]. This program was part of the Prototype Plant Nuclear Process Heat project (PNP). Table 1 lists the experimental details of some tests in the series. The unconfined flammable clouds were considered as hemispherical shape,
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Table 1 e Partial experimental conditions and results of some tests [27]. Test No. GHT GHT GHT GHT
25 40 11 34
CH2(% vol.)
T0 (ºC)
p0 (kPa)
R0(m)
V (m3)
SL (m/s)
S' (m/s)
Smax (m/s)
29.5 29.5 31.0 29.7
8 6 8 10
98.1 98.5 100.7 98.9
1.53 2.88 5.00 10.00
7.5 50.0 261.7 2094.0
2.35 2.32 2.50 2.39
46.2 57.0 62.0 73.9
45.0 54.0 59.0 83.6
and were simulated by the balloons that made of thin polyethylene foil. The balloons were filled with nearly stoichiometric H2/Air mixtures, and their radii were 1.53 m, 2.88 m, 5 m and 10 m. The clouds were ignited at the center of the balloon on the ground level. Among the tests, GHT 34 is the largest scale experiment, and is also the largest H2/Air deflagration test in the literature to date. In this test, the maximum flame propagating speed was Smax ¼ 83.6 m/s, and the maximum value of the flame radius Rfmax was approximately 20 m [28]. To make it easy to compare, the overpressure histories at different distances R are set together in Fig. 1. And the peak values of positive overpressure in different scale tests are set in Fig. 2. On the basis of the experimental data, some findings could be concluded as follows: 1). When the distance R less than Rfmax, with R varying, the peak values of positive overpressure at different distances are very close to each other, and the negative peak values are close too. 2). When the distance R larger than Rfmax, the overpressure value decays with distance, and the arrival time rises with distance. 3). If the distance R not larger than Rfmax, when the positive value increases to the peak value, it changes slightly. That is to say, there is a sustained period of the positive peak value. In addition, the more close to the ignition center, the more of the sustained period is. Note that, these findings only suitable for the case that a homogenous cloud deflagrates in open atmosphere.
Fig. 1 e Overpressure wave histories of H2/Air deflagration Test at various distances [28].
Fig. 2 e Pressure-distance relation of H2/Air deflagration tests [27].
Process of the flame propagation There are two types of gas explosion mechanism: deflagration and detonation. A difference between the two mechanisms is the combustion wave propagating speed, the former is subsonic velocity, and the latter is supersonic velocity (up to 2000 m/s) [20]. However, for the unconfined hydrocarbon-air mixtures explosion, except for directly initiated by a powerful explosive, or deflagration to detonation transition (DDT) caused by obstacles and confinement, detonation is unlikely [20,24]. In the case of deflagration, most of the previous investigations adopted the spherical piston theory, in which the expanding flame front is assumed as a propagating piston. In the present study, the explosion of UVCE deflagration is considered, and the piston theory is adopted too. As Fig. 3a) shows, Rin and Rex are employed to denote the distances which less than or larger than Rfmax, respectively. Fig. 3b) shows the one dimensional propagating process of the piston, and the flow field is divided into three zones: burnt zone (Zone 01), incompressible source field (Zone 02) and acoustic source filed (Zone 03). If R ¼ Rin, a point should be firstly in Zone 02, and subsequently in Zone 01 with the flame front propagating. In the case of R ¼ Rex, a point should be always in Zone 03. Outside the Zone 01, there is a thin reaction zone, and this is the flame front too. The expansion of the combustion products in the reaction zone drives the piston propagating in the atmosphere. Thus, the burnt gas in Zone 01 is at rest, the overpressure sustains the peak value, and the gas in Zone 02 and Zone 03 is affected by the piston, obtained an increasing overpressure. When the flame front arrives in the region that
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Fig. 3 e Process of the flame propagation.
the maximum flame radius is (Rf ¼ Rfmax), the overpressure of a point justly comes to the peak value, and decreases with the flame extinguishing subsequently. At this aspect, for the case of Rf ¼ Rfmax, there is no sustained period of peak overpressure in the positive phase, and the more the distance R close to the Rfmax, the shorter the sustained period is. After the flame quenching, the piston does not disappear, there is a piston induced by the inertia of gas, until its momentum dissipates to zero. Moreover, the pressure wave can be assumed as sonic wave spreads in the atmospheric, so there is a delay time of pressure wave associated with distance R. Now, it is concluded that, the findings in Section Findings in gas deflagration experiments could be explained based on the previous discussions.
Fig. 4 e Classification of the pressure wave.
Classification of the pressure wave shape The typical pressure wave is “N” shape, as shown in Fig. 4a). As can be found in Fig. 4a), there is a positive phase and negative phase, and þDpmax and -Dpmax denote the peak values in the two phases, respectively. Furthermore, tþ and ti are employed to represent the respective time of þDpmax and -Dpmax occurred, and the start and finish time of the pressure wave is denoted by ts and tf, respectively. Based on the discussions mentioned in Section Findings in gas deflagration experiments and Section Process of the flame propagation, the shapes of pressure waves in different zones of the flow field can be summarized in Fig. 4b). The curve 2 denotes the case of R ¼ Rfmax, and the curves of 1, 3 denote the case of R < Rfmax and R > Rfmax, respectively. The superscripts are used to distinguish the times of different curves. In addition, for
curve 1, t'þþ denotes the time of the overpressure firstly come to the positive peak value.
Method As Deshaies stated in Ref. [24], for the original acoustic approximation model, if the complete flame radius history Rf(t) is unknown, it is impossible to obtain the prediction of the pressure signal. In this section, we seek to develop a tentative solution that begins with the flame/piston speed history D(t), which is independent of experimental data. If D(t) is known, the flame/piston history Rf(t) can be solved by integrating. It should be noted that, the authors consider that the flow field is induced by a real flame piston, and a virtual piston caused by
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gas inertia, as mentioned in Section Process of the flame propagation. Thus the time histories of the two ones constitute a complete flame/piston history Rf(t).
Flame/piston speed In the past, the flame speed has once been assumed as constant value, however, it is not likely occurs in real situations. Guibert-Duplantier et al. [29] found an empirical relationship of the flame/piston speed history D(t), as shown in Eq. (4): DðtÞ ¼
D2 ½1 tanh ai ðt ti Þ 2
(4)
where, D2 denotes the maximum flame speed, ai and ti are empirical parameters. As can be seen in Fig. 5 [29], when the flame speed calculated by the empirical relationship (Eq. (4)) compared with that by the acoustic approximation model (Eq. (2), Dp (R,t) is known), there is a good accuracy. However, in Eq. (4), there are four issues appear puzzled: 1) The first is the maximum flame speed D2. As can be seen in Fig. 5, the flame speed isn't a monotonic increasing function, and the maximum value does not at the time t ¼ tþ (þDpmax generated), but there is a confused state in Ref. [29]. In addition, in Fig. 5, it can be found that the flame speed fluctuates on the average value, which can also be found in the tests [30]. So, in the present study, it is thought more appropriate to apply the average flame speed S0 in Eq. (4). 2) The second is the instant of the flame vanishes. The researchers thought the flame speed decay at the end of the explosion when the combustion vanishes [29]. As mentioned in Section Process of the flame propagation, the combustion quenches at t ¼ tþ, and the piston vanishes at t ¼ tf. So, it should be reminded that D(t) denotes the flame/ piston speed history in this study. 3) The third is the experimental data Dp (R, t) adopted to compute D(t). The pressure signal data Dp(t) used to calculate D(t) by Eq. (2) was at a distance of R ¼ 0.35 m (R0 ¼ 0.05 m) in Ref. [29], and several different D(t) were obtained at various distances [25]. That is to say, if different Dp(t) were adopted, different flame front histories will be obtained from Eq. (4). Nevertheless, in the whole period of explosion, it is clear that only one flame front exists, as well as the piston front. Only with a fixed distance Rfmax, the pressure signal Dp(t) represents the pressure wave induced
5
by deflagration (undergoes the flame vanishing and the gas piston generating subsequently), and in the case of R > Rfmax, Dp(t) is thought as sonic wave propagating in environment. 4) The fourth is the meaning of ti. In Ref. [29], the authors stated ti is closed to the time of -Dpmax generated, and did not explicitly explain the distance of the Dp(t) occurred. Obviously, Dp(t) varies with R, and ti is variable value in that case. On the basis of 3), it is thought at the distance R ¼ Rfmax in the present study. To make it clear, several statements could be summarized based on the discussions mentioned above: 1) In Eq. (4), it is more appropriate to apply the average flame speed S0 instead of the maximum flame speed D2. 2) D(t) denotes the flame speed history (t tþ) and the piston speed history (tþ
S0 ½1 tanh ai ðt ti Þ 2
(5)
here, ti denotes the shifted time of -Dpmax generated (R ¼ Rfmax, and takes into account the acoustic wave propagation delay time Rfmax/c0). The maximum flame radius Rfmax could be solved by Refs. [24,29]: Rfmax ¼ R0 b1=3
(6)
here, b represents for the expansion coefficient of combustion products [29]: b ¼ r0 =rb
(7)
The variables of r0, rb denote the density of unburnt mixture and burnt gas, respectively.
Empirical parameters ai and ti In Ref. [29], the authors found that ai is a linear function of D2, and ti/R0 is a decreasing function of D2. Note that, these functions were obtained based on small scale experiments of hydrocarbon/air mixtures (C2H4 and C3H8). In the present study, on the basis of small scale and large scale H2/Air explosion tests, the function of ai and ti/R0 with S0 are expanded, as shown in Fig. 6 and Fig. 7, respectively.
Flame/piston radius and acceleration histories When the speed history of the flame/piston D(t) is known, Rf(t) could be obtained by integrating the speed history D(t), the relationship is given by: Fig. 5 e Comparison between the flame speed calculated by Eq. (2) (solid line) and Eq. (4) (dotted line) [29].
Rf ðtÞ ¼
S0 1 1 t ln½cosh½ai ðt ti Þ þ ln½coshðai ti Þ ai ai 2
(8)
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Fig. 6 e Parameter ai versus the average flame speed S′ for different fuel/air mixtures.
Fig. 8 e History of the flame/piston acceleration.
Eq. (2) can be changed into the form of Eq. (10). The parameter a in Eq. (10) represents for the reciprocal of expansion coefficient b. DpðR; tÞ ¼
Fig. 7 e Parameter ti, ti versus the average flame speed S′ for different fuel/air mixtures.
h i r0 ð1 aÞ 2Rf ðtÞD2 ðtÞ þ R2f ðtÞaf ðtÞ R
(10)
To take into account the delay time of acoustic wave, Eq. (3) should be jointly solved for different R. The sound speed c0 is assumed as a constant of 340 m/s. For the regions of Zone 01 and Zone 02 (R < Rfmax), we propose a simplified scheme to calculate the overpressure history (curve 1 in Fig. 4b)), as can be seen in Fig. 9. If the instants of t's, t'þþ, t'þ, t'i and t'f are solved, as well as the overpressure values at these moments, the simplified overpressure history can be obtained. It is obvious that, the overpressure values at t ¼ t's, and t ¼ t'f are equal to zero. Furthermore, the overpressure values at the moments of t'þþ and t'þ are equal to þDpmax, and equal to Dpmax at the instant
and the derivative of D(t) can also be obtained as the acceleration history of the flame/piston af(t), the detailed equation is given as follows: af ðtÞ ¼
dDðtÞ d2 Rf ðtÞ S0 ai ¼ ¼ dt dt2 2 cosh2 ½ai ðt ti Þ
(9)
Fig. 8 displays the general form of the af(t) curve. As can be seen in Fig. 8, af(t) descends slightly before the Dp(t) comes to þDpmax, with the quenching of the combustion, af(t) descends dramatically, and Dp(t) comes to -Dpmax when the maximum value of the af(t) obtained. In addition, if t>ti, af(t) increases rapidly until it approaches to zero, as well as the overpressure Dp. The effect of flame/piston acceleration on the overpressure can also be seen in Eq. (2).
Overpressure of different flow field Based on the discussions in previous sections, the overpressure history with a distance of R in Zone 03 of the flow field (R Rfmax), could be solved by Eq. (2). To make it concise,
Fig. 9 e Schematic diagram of rough overpressure history (R < Rfmax).
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Fig. 10 e Validation with large scale H2/Air deflagration test.
of t'i. The times and the corresponding overpressure values can be solved by Eq. (11)e(17). R c0
(11)
t0þ ðRÞ ¼ tþ
(12)
t0s ðRÞ ¼
t0þþ ðRÞ ¼ t0þ ðRÞ
Rfmax R S0
(13)
t0i ðRÞ ¼ ti
Rfmax R c0
(14)
t0f ðRÞ ¼ tf
Rfmax R c0
(15)
þDpmax ðRÞ ¼ r0 ðS0 Þ ð1 aÞ 2
3a 2
Dpmax ðRÞ ¼ Dpmax Rfmax
(16) (17)
Note that, it is assumed that the solutions of t'þ, t'i, t'f, and -Dpmax are based on the values of the case of R ¼ Rfmax in the present study.
Result and discussion Validation of the method To validate the method developed in Section Method, a large scale [28,31,32] and a series of small scale [30,33] H2/Air deflagration experiments are employed. The calculated data of the former one are compared with both experimental and CFD predicted data, and the latter ones are contrasted with experimental data only. Note that, in these validations, the experimental data of average flame speed S0 are adopted (Eq. (5)). Fig. 10 displays the comparison of experimental data, numerical simulation data [28] and calculated data of GHT 34 Test [28], with various distances. In this experiment, the test point of R ¼ 18 m is close to the maximum flame radius Rfmax (~20 m), the test point of R ¼ 5 m is in the region of Zone 01 and Zone 02 (R < Rfmax), and the distances of 35 m and 80 m are beyond Rfmax (R > Rfmax). Therefore, the case of R ¼ 5 m is solved by the simplified scheme (Eq. 11e17), and solutions of the 18 m, 35 m and 80 m cases are computed by Eq. (10). In Fig. 10, it can be seen that, for large scale deflagration, the model presented in this study yields a satisfactory
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consistency when compared with the experimental data and CFD predicted data. The University of Tokyo conducted a series of small scale H2/Air deflagration tests with various equivalence ratios 4 in 2013 [30,33]. Fig. 11 displays the comparison of calculated data and the experimental data, at a fixed position (R ¼ 0.3 m). As can be seen in Fig. 11, the values of Dp(t) in the negative phase have acceptable consistence, and in the positive phase present unsatisfactory values. A possible explanation for this is that, the peak values in the various equivalence ratio cases may be not the real overpressure induced by explosion, which probably caused by the rupture of the bubble. It is clear that, with an increased equivalence ratio, the peak values of Dp(t) changes slightly, at a little overpressure level of 100 Pa. Thus, it is reasonable to deduce that, the peak values at a level of 100 Pa in the tests may be mainly caused by the disturbance of the bubble's rupture. At this aspect, the values of Dp(t) in the negative phase probably appear the real values induced by deflagration in the test. Because the rupture of the bubble occurs in a transient time, and the flow field may be led by the piston quickly. As stated in section Empirical parameters ai and ti, the functions of ai and ti/R0 with S' (Figs. 6e7) are obtained based on the small scale and large scale H2/Air explosion tests. To
further validate the effectiveness of this model, an independent validation with medium scale tests are necessary. However, the fact is that the deflagration tests of H2/Air mixture in unconfined space are very little in the literature. And we have not found another medium scale tests which have enough information, including: weather conditions, flame front histories, and overpressure histories, etc., except for GHT test series (from 7.5 m3 to 260 m3). Unfortunately, the whole overpressure histories of the medium scale GHT tests have not been found in literature. Therefore, an independent validation with the positive peak overpressure values of the medium scale GHT tests is conducted, as shown in Fig. 12. In the validation, R0, T0, P0, CH2 are necessary initial parameters, and experimental values of S0 are employed to calculate ai and ti from Figs. 6e7, respectively. Then the solution could be obtained using the model stated in Section Method, and the positive peak overpressure values could be gotten. In Fig. 12, it is obvious that the calculated data of GHT 25 and GHT 40 cases display good agreements with the experimental data. For GHT 11 case, there is a larger discrepancy compared with the others, however, the maximum deviation value is 15%. Based on the discussions mentioned previously, we can conclude that the model presented in Section Method is effective for H2/Air deflagration evaluating.
Fig. 11 e Validation with small scale H2/Air deflagration tests.
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According to Eq. (18), the average flame speed S0 for unconfined mixtures (y ¼ 0) could be given by: S0 ¼ 0:006bðb 1ÞSL
1=3 Rfmax d
(20)
The flame thickness equation is [34e36]: d¼
m0 r0 SL
(21)
here, r0, b and SL can be obtained from CHEMKIN-Pro 18.0 code, and the dynamic viscosity of mixture can be solved by: P m0 ¼
Fig. 12 e Independent validation with medium scale H2/Air deflagration tests (hollow: Experimental [27]; solid: Calculated).
Comparison of average flame speed correlations The validation in Section Validation of the method demonstrates that, when an experimental data of S0 is employed (as Table 2 shows), the model in this study appears good consistence with medium and large scale deflagration tests, and has acceptable solution in the small scale cases. To obtain a complete independent model, an independent correlation of S0 is necessary. Dorofeev [34e36] proposed two flame speed correlations respectively for semi-confined and unconfined gas mixtures. For the semi-confined mixtures, the flame speed is given by: vf ¼ 0:0085bðb 1ÞSL
4 by R0:63 f 1þ 3 x ðbxÞ0:63
!2 1=3 Rf d
(18)
where, vf and Rf respectively denote the flame speed and radius, x and y represent for characteristic sizes of obstacle, SL is laminar flame speed, and d is the thickness of flame. And for unconfined mixtures, the flame speed is given as follows: 1=3 1=3 Rf vf ¼ 0:47S4=3 L c
(19)
where, c is thermal diffusivity.
1=2
Ck m0;k Mk P Ck M1=2 k
(22)
The dynamic viscosity of each component m0,k could be obtained from NIST Standard Reference Database 23 [37], Ck is molecular fraction of component k, and Mk is molecular weight of component k. In addition, when the flame speed vf is considered as the average flame speed S0 , and the flame radius Rf as maximum flame radius Rfmax, Eq. (19) becomes the following form: 1=3 1=3 Rfmax S0 ¼ 0:47S4=3 L c
(23)
where, c can also be obtained from NIST Standard Reference Database 23. Table 2 lists a comparison of the computed values and experimental data of average flame speed S'. It can be seen that, the solutions of Eq. (20) agree well with the experimental data of large scale deflagration test, and appear conservative in the small scale cases. For Eq. (23), the solutions display radical for both small and large scale deflagration tests. Thus, for accurate prediction of Dp (R, t), it is reasonable to adopt a more precise S0 correlation.
Flow chart In Fig. 13, a flow chart for computing the overpressure history Dp (R, t) is given. The initial conditions, such as R0, T0, P0, and Cfuel are necessary. These parameters can be imported into CHEMKIN and the NIST database, to obtain r0, rb and SL from the former, and m0,k from the latter. Then a, b, Rfmax, m0, d, and S0 could be solved by the relative equations. When S0 is known, ai and ti can be obtained by Figs. 6e7, and D(t) can also be solved subsequently. Based on the known D(t), Rf(t) and af(t)
Table 2 e Comparison of average flame speed. 4
Test
UTokyo [30]
GHT [27]
No.25 No.40 No.13 No.34
1.0 1.8 3.0 4.0 0.996 0.996 0.832 1.005
R0(m)
0.05
1.53 2.88 5.00 10.00
S' (m/s)
Deviation (%)
Test
Eq. (20)
Eq. (23)
Eq. (20)
Eq. (23)
15.5 18.6 11.1 7.6 46.2 57.0 49.5 73.9
9.9 11.6 5.3 2.6 44.6 55.0 42.2 83.5
19.3 26.2 16.2 9.8 64.8 79.4 73.2 124.2
36.1 37.6 52.3 65.8 3.5 3.5 14.7 13.0
24.5 40.9 45.9 28.9 40.3 39.3 47.9 68.0
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Fig. 13 e Flow chart of overpressure history calculation.
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 8 ) 1 e1 2
could be gotten. Then if R Rfmax, Dp (R, t) could be solved by Eq. (10), otherwise, a rough Dp (R, t) can be got by the simplified scheme. In the end, the delay time of acoustic wave for various distances R should be considered, and Dp (R, t) transforms to Dp (R, t) for R Rfmax case.
Conclusion In this study, an improved acoustic approximation model for UVCE was proposed. For the large scale VCE accidents, a simplified scheme was also presented for overpressure evaluating of the area inside the flame cloud. The model could be used for engineering application of safety assessment on hydrogen energy, and has a high computational efficiency compared with CFD models, because there are no differential equations but only algebraic equations. The model appears good accuracy in the medium and large scale H2/Air deflagration case. In the small scale cases, the model obtained acceptable solutions. However, to obtain a high accuracy prediction, a more precise correlation of average flame speed should be further developed in future. In addition, the effect of obstacles should be considered, and the model can also be used to more fuels based on validation.
Acknowledgement This research work was jointly supported by the National Natural Science Foundation of China (No. 51641608) and the State Key Laboratory of Technologies in Space Cryogenic Propellants of China (SKLTSCP1511, SKLTSCP1809).
references
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A simple and effective approach for evaluating unconfined hydrogen/air cloud explosions Liang Pu a,b, Xiangyu Shao a,*, Qiang Li a, Yanzhong Li a,b a b
School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China State Key Laboratory of Technologies in Space Cryogenic Propellants, Beijing 100028, China
article info
abstract
Article history:
The topic of hydrogen safety assessment has been focused by many researchers. The
Received 10 February 2018
overpressure evaluation of vapor cloud explosion (VCE), is an important issue for both
Received in revised form
designing and evaluating on chemical plants, as well as buildings. Unknown flame radius
7 April 2018
history limits the original acoustic approximation model's application. The objective of this
Accepted 7 April 2018
work is to develop an achievable model for hydrogen/air deflagration assessment in en-
Available online xxx
gineering applications, and the model should have high computational efficiency. A tentative scheme that starts from flame/piston speed history solving was adopted, and the
Keywords:
flame/piston radius and acceleration history will be obtained subsequently. Thus, the
Hydrogen
overpressure history for far field could be gotten based on the acoustic approximation
Safety assessment
model. A simplified scheme was employed for the region inside the flame cloud. The model
Vapor cloud explosion (VCE)
proposed in this paper could be solved in several seconds, because there are no differential
Acoustic approximation model
equations but only algebraic equations. The model was verified by hydrogen/air deflagra-
Overpressure
tion tests from small scale to large scale. Compared with the experimental data, the model appeared well agreements in the medium and large scale cases. In the small scale cases, the model obtained acceptable solutions. © 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Hydrogen is a promising energy carrier. Because of the flammability of hydrogen, its safety utilization has always been a key concern [1e4]. Not only hydrogen, the leakage of combustible gas or liquefied combustible gas (with a low boiling point at ambient temperature), like methane, ethylene, propane etc., will form flammable cloud in the atmosphere [5]. Once the flammable cloud meets ignition source, the VCE (Vapor Cloud Explosions) accident may be occurs, which is a main hazardous issue in energy and chemical industries [6]. It is reported that approximately 174 VCE accidents occurred
during 1940e2010 in the world [7,8], which resulted in huge personal and property losses. The large overpressure load created by the flammable cloud explosion, could cause serious damage. Thus, it is necessary to investigate the pressure wave of potential accidents, for the safety assessment of existing and designed chemical plants and buildings [9]. And this topic has attracted much attention by researchers for several decades. There are three categories models to predict the overpressure of VCE accidents [10e12]: empirical models (TNT equivalence model [13], TNO Multi-Energy model [14,15], Baker-Strehlow model [16], CAM [17], etc.), phenomenological models (CLICHE model [18] and SCOPE model [19]) and CFD
* Corresponding author. E-mail addresses: [email protected] (L. Pu), [email protected] (X. Shao). https://doi.org/10.1016/j.ijhydene.2018.04.041 0360-3199/© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Pu L, et al., A simple and effective approach for evaluating unconfined hydrogen/air cloud explosions, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/j.ijhydene.2018.04.041
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Nomenclatures af ai c0 C D D2 M p R Ra R0 S' SL t T v V x y
flame/piston acceleration (m/s2) empirical parameter in Eq. (5) sonic speed (m/s) molecular fraction (%) flame/piston speed in Eqs. (4) and (5) (m/s) maximum flame speed in Eq. (4) (m/s) molecular weight pressure (Pa) distance from ignition center (m) affected distance from ignition center (m) initial radius of hemispherical unburnt mixture cloud (m) average flame speed in Eqs. (5), (20) and (23) (m/ s) laminar flame speed in Eq.(18)e(21), (23) (m/s) time (s) temperature (ºC) flame speed in Eq.18 and 19 (m/s) volume of unburnt mixture (m3) distance between obstacles in Eq. (18) (m) characteristic size in Eq. (18) (m)
Greek symbols a reciprocal of b b expansion coefficient in Eq. (7) d flame thickness in Eq. (21) (m) m dynamic viscosity (Pa$s) r density (kg/m3) t shifted time (s) 4 equivalence ratio c thermal diffusivity in Eq. (19) (m2/s) Subscripts 0 initial ex external f flame or final i inflexion in internal k index of component max maximum s start
models. Compared with the first two models, an appropriate CFD model can obtain much accurate overpressure time history and more details of the flow field. However, the great demand for time and computer resource of CFD models makes it impractical for engineering applications, especially for complex geometries [10,20e23]. At this aspect, the empirical models and phenomenological models have their advantages, but these models also have their weaknesses. For empirical models, the weaknesses includes: the drawbacks of weak accuracy (TNT, Baker-Strehlow), only positive phase peak pressure can be solved (TNT, Multi-Energy), and not good realizability (Multi-Energy, CAM), etc. The sophistication of phenomenological models is between empirical models and complex CFD models, but few information of the flow field
could be provided in phenomenological models [10,11]. The detailed comments of the previous models can be seen in Ref. [10] and Ref. [12]. In addition, Deshaies [24] proposed an acoustic approximation model, in which the flow field outside the flame was divided into two zones, incompressible source flow (near the flame front) and acoustic approximation of source flow (far from the flame front). The overpressure time history at a distance of R from the ignition center was given as follows:near the flame front: pðR; tÞ p0 ¼
( 2 2 r0 dRf ðtÞ d Rf ðtÞ ð1 aÞ 2Rf ðtÞ þ R2f ðtÞ dt dt2 R ) 2 1 a R4f ðtÞ dRf ðtÞ R3 2 dt
(1)
and far from the flame front: ( 2 2 ) r0 dRf ðtÞ d Rf ðtÞ 2 þ Rf ðtÞ pðR; tÞ p0 ¼ ð1 aÞ 2Rf ðtÞ dt dt2 R t¼t
r c0
(2)
(3)
The acoustic approximation model can provide the pressure-time history with a varied distance R, and it has been checked by several deflagration experiments [25,26]. As can be seen in Eqs. (1) and (2), the flame radius history Rf(t) is necessary for overpressure calculation, but it is unknown before the experiment. Thus, this drawback restricts the practical use of the acoustic approximation model. Moreover, the acoustic approximation model can't obtain the pressure wave inside the flame sphere. However, the length of the flame may be hundreds of meters to several kilometers, and the affected zone is very broad in large scale VCE accidents. Based on the discussions mentioned previously, the weaknesses of the models motivate the authors to develop a simple and effective model for engineering application of safety assessment. In the present study, a model on the basis of the acoustic approximation model is proposed, in which an independent flame history function improves the availability of the original model. In Section Theory hypothesis, the pressure wave shapes are classified with findings from hydrogen/air deflagration tests. The calculation model of overpressure is presented in Section Method, and in Section Result and discussion it will be validated.
Theory hypothesis Findings in gas deflagration experiments In 1983, the Fraunhofer Institute for Propellants and Explosives (Germany) conducted a series of large scale unconfined vapor cloud explosion (UVCE) tests in open area, with H2/Air mixtures [27]. This program was part of the Prototype Plant Nuclear Process Heat project (PNP). Table 1 lists the experimental details of some tests in the series. The unconfined flammable clouds were considered as hemispherical shape,
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Table 1 e Partial experimental conditions and results of some tests [27]. Test No. GHT GHT GHT GHT
25 40 11 34
CH2(% vol.)
T0 (ºC)
p0 (kPa)
R0(m)
V (m3)
SL (m/s)
S' (m/s)
Smax (m/s)
29.5 29.5 31.0 29.7
8 6 8 10
98.1 98.5 100.7 98.9
1.53 2.88 5.00 10.00
7.5 50.0 261.7 2094.0
2.35 2.32 2.50 2.39
46.2 57.0 62.0 73.9
45.0 54.0 59.0 83.6
and were simulated by the balloons that made of thin polyethylene foil. The balloons were filled with nearly stoichiometric H2/Air mixtures, and their radii were 1.53 m, 2.88 m, 5 m and 10 m. The clouds were ignited at the center of the balloon on the ground level. Among the tests, GHT 34 is the largest scale experiment, and is also the largest H2/Air deflagration test in the literature to date. In this test, the maximum flame propagating speed was Smax ¼ 83.6 m/s, and the maximum value of the flame radius Rfmax was approximately 20 m [28]. To make it easy to compare, the overpressure histories at different distances R are set together in Fig. 1. And the peak values of positive overpressure in different scale tests are set in Fig. 2. On the basis of the experimental data, some findings could be concluded as follows: 1). When the distance R less than Rfmax, with R varying, the peak values of positive overpressure at different distances are very close to each other, and the negative peak values are close too. 2). When the distance R larger than Rfmax, the overpressure value decays with distance, and the arrival time rises with distance. 3). If the distance R not larger than Rfmax, when the positive value increases to the peak value, it changes slightly. That is to say, there is a sustained period of the positive peak value. In addition, the more close to the ignition center, the more of the sustained period is. Note that, these findings only suitable for the case that a homogenous cloud deflagrates in open atmosphere.
Fig. 1 e Overpressure wave histories of H2/Air deflagration Test at various distances [28].
Fig. 2 e Pressure-distance relation of H2/Air deflagration tests [27].
Process of the flame propagation There are two types of gas explosion mechanism: deflagration and detonation. A difference between the two mechanisms is the combustion wave propagating speed, the former is subsonic velocity, and the latter is supersonic velocity (up to 2000 m/s) [20]. However, for the unconfined hydrocarbon-air mixtures explosion, except for directly initiated by a powerful explosive, or deflagration to detonation transition (DDT) caused by obstacles and confinement, detonation is unlikely [20,24]. In the case of deflagration, most of the previous investigations adopted the spherical piston theory, in which the expanding flame front is assumed as a propagating piston. In the present study, the explosion of UVCE deflagration is considered, and the piston theory is adopted too. As Fig. 3a) shows, Rin and Rex are employed to denote the distances which less than or larger than Rfmax, respectively. Fig. 3b) shows the one dimensional propagating process of the piston, and the flow field is divided into three zones: burnt zone (Zone 01), incompressible source field (Zone 02) and acoustic source filed (Zone 03). If R ¼ Rin, a point should be firstly in Zone 02, and subsequently in Zone 01 with the flame front propagating. In the case of R ¼ Rex, a point should be always in Zone 03. Outside the Zone 01, there is a thin reaction zone, and this is the flame front too. The expansion of the combustion products in the reaction zone drives the piston propagating in the atmosphere. Thus, the burnt gas in Zone 01 is at rest, the overpressure sustains the peak value, and the gas in Zone 02 and Zone 03 is affected by the piston, obtained an increasing overpressure. When the flame front arrives in the region that
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Fig. 3 e Process of the flame propagation.
the maximum flame radius is (Rf ¼ Rfmax), the overpressure of a point justly comes to the peak value, and decreases with the flame extinguishing subsequently. At this aspect, for the case of Rf ¼ Rfmax, there is no sustained period of peak overpressure in the positive phase, and the more the distance R close to the Rfmax, the shorter the sustained period is. After the flame quenching, the piston does not disappear, there is a piston induced by the inertia of gas, until its momentum dissipates to zero. Moreover, the pressure wave can be assumed as sonic wave spreads in the atmospheric, so there is a delay time of pressure wave associated with distance R. Now, it is concluded that, the findings in Section Findings in gas deflagration experiments could be explained based on the previous discussions.
Fig. 4 e Classification of the pressure wave.
Classification of the pressure wave shape The typical pressure wave is “N” shape, as shown in Fig. 4a). As can be found in Fig. 4a), there is a positive phase and negative phase, and þDpmax and -Dpmax denote the peak values in the two phases, respectively. Furthermore, tþ and ti are employed to represent the respective time of þDpmax and -Dpmax occurred, and the start and finish time of the pressure wave is denoted by ts and tf, respectively. Based on the discussions mentioned in Section Findings in gas deflagration experiments and Section Process of the flame propagation, the shapes of pressure waves in different zones of the flow field can be summarized in Fig. 4b). The curve 2 denotes the case of R ¼ Rfmax, and the curves of 1, 3 denote the case of R < Rfmax and R > Rfmax, respectively. The superscripts are used to distinguish the times of different curves. In addition, for
curve 1, t'þþ denotes the time of the overpressure firstly come to the positive peak value.
Method As Deshaies stated in Ref. [24], for the original acoustic approximation model, if the complete flame radius history Rf(t) is unknown, it is impossible to obtain the prediction of the pressure signal. In this section, we seek to develop a tentative solution that begins with the flame/piston speed history D(t), which is independent of experimental data. If D(t) is known, the flame/piston history Rf(t) can be solved by integrating. It should be noted that, the authors consider that the flow field is induced by a real flame piston, and a virtual piston caused by
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gas inertia, as mentioned in Section Process of the flame propagation. Thus the time histories of the two ones constitute a complete flame/piston history Rf(t).
Flame/piston speed In the past, the flame speed has once been assumed as constant value, however, it is not likely occurs in real situations. Guibert-Duplantier et al. [29] found an empirical relationship of the flame/piston speed history D(t), as shown in Eq. (4): DðtÞ ¼
D2 ½1 tanh ai ðt ti Þ 2
(4)
where, D2 denotes the maximum flame speed, ai and ti are empirical parameters. As can be seen in Fig. 5 [29], when the flame speed calculated by the empirical relationship (Eq. (4)) compared with that by the acoustic approximation model (Eq. (2), Dp (R,t) is known), there is a good accuracy. However, in Eq. (4), there are four issues appear puzzled: 1) The first is the maximum flame speed D2. As can be seen in Fig. 5, the flame speed isn't a monotonic increasing function, and the maximum value does not at the time t ¼ tþ (þDpmax generated), but there is a confused state in Ref. [29]. In addition, in Fig. 5, it can be found that the flame speed fluctuates on the average value, which can also be found in the tests [30]. So, in the present study, it is thought more appropriate to apply the average flame speed S0 in Eq. (4). 2) The second is the instant of the flame vanishes. The researchers thought the flame speed decay at the end of the explosion when the combustion vanishes [29]. As mentioned in Section Process of the flame propagation, the combustion quenches at t ¼ tþ, and the piston vanishes at t ¼ tf. So, it should be reminded that D(t) denotes the flame/ piston speed history in this study. 3) The third is the experimental data Dp (R, t) adopted to compute D(t). The pressure signal data Dp(t) used to calculate D(t) by Eq. (2) was at a distance of R ¼ 0.35 m (R0 ¼ 0.05 m) in Ref. [29], and several different D(t) were obtained at various distances [25]. That is to say, if different Dp(t) were adopted, different flame front histories will be obtained from Eq. (4). Nevertheless, in the whole period of explosion, it is clear that only one flame front exists, as well as the piston front. Only with a fixed distance Rfmax, the pressure signal Dp(t) represents the pressure wave induced
5
by deflagration (undergoes the flame vanishing and the gas piston generating subsequently), and in the case of R > Rfmax, Dp(t) is thought as sonic wave propagating in environment. 4) The fourth is the meaning of ti. In Ref. [29], the authors stated ti is closed to the time of -Dpmax generated, and did not explicitly explain the distance of the Dp(t) occurred. Obviously, Dp(t) varies with R, and ti is variable value in that case. On the basis of 3), it is thought at the distance R ¼ Rfmax in the present study. To make it clear, several statements could be summarized based on the discussions mentioned above: 1) In Eq. (4), it is more appropriate to apply the average flame speed S0 instead of the maximum flame speed D2. 2) D(t) denotes the flame speed history (t tþ) and the piston speed history (tþ
S0 ½1 tanh ai ðt ti Þ 2
(5)
here, ti denotes the shifted time of -Dpmax generated (R ¼ Rfmax, and takes into account the acoustic wave propagation delay time Rfmax/c0). The maximum flame radius Rfmax could be solved by Refs. [24,29]: Rfmax ¼ R0 b1=3
(6)
here, b represents for the expansion coefficient of combustion products [29]: b ¼ r0 =rb
(7)
The variables of r0, rb denote the density of unburnt mixture and burnt gas, respectively.
Empirical parameters ai and ti In Ref. [29], the authors found that ai is a linear function of D2, and ti/R0 is a decreasing function of D2. Note that, these functions were obtained based on small scale experiments of hydrocarbon/air mixtures (C2H4 and C3H8). In the present study, on the basis of small scale and large scale H2/Air explosion tests, the function of ai and ti/R0 with S0 are expanded, as shown in Fig. 6 and Fig. 7, respectively.
Flame/piston radius and acceleration histories When the speed history of the flame/piston D(t) is known, Rf(t) could be obtained by integrating the speed history D(t), the relationship is given by: Fig. 5 e Comparison between the flame speed calculated by Eq. (2) (solid line) and Eq. (4) (dotted line) [29].
Rf ðtÞ ¼
S0 1 1 t ln½cosh½ai ðt ti Þ þ ln½coshðai ti Þ ai ai 2
(8)
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Fig. 6 e Parameter ai versus the average flame speed S′ for different fuel/air mixtures.
Fig. 8 e History of the flame/piston acceleration.
Eq. (2) can be changed into the form of Eq. (10). The parameter a in Eq. (10) represents for the reciprocal of expansion coefficient b. DpðR; tÞ ¼
Fig. 7 e Parameter ti, ti versus the average flame speed S′ for different fuel/air mixtures.
h i r0 ð1 aÞ 2Rf ðtÞD2 ðtÞ þ R2f ðtÞaf ðtÞ R
(10)
To take into account the delay time of acoustic wave, Eq. (3) should be jointly solved for different R. The sound speed c0 is assumed as a constant of 340 m/s. For the regions of Zone 01 and Zone 02 (R < Rfmax), we propose a simplified scheme to calculate the overpressure history (curve 1 in Fig. 4b)), as can be seen in Fig. 9. If the instants of t's, t'þþ, t'þ, t'i and t'f are solved, as well as the overpressure values at these moments, the simplified overpressure history can be obtained. It is obvious that, the overpressure values at t ¼ t's, and t ¼ t'f are equal to zero. Furthermore, the overpressure values at the moments of t'þþ and t'þ are equal to þDpmax, and equal to Dpmax at the instant
and the derivative of D(t) can also be obtained as the acceleration history of the flame/piston af(t), the detailed equation is given as follows: af ðtÞ ¼
dDðtÞ d2 Rf ðtÞ S0 ai ¼ ¼ dt dt2 2 cosh2 ½ai ðt ti Þ
(9)
Fig. 8 displays the general form of the af(t) curve. As can be seen in Fig. 8, af(t) descends slightly before the Dp(t) comes to þDpmax, with the quenching of the combustion, af(t) descends dramatically, and Dp(t) comes to -Dpmax when the maximum value of the af(t) obtained. In addition, if t>ti, af(t) increases rapidly until it approaches to zero, as well as the overpressure Dp. The effect of flame/piston acceleration on the overpressure can also be seen in Eq. (2).
Overpressure of different flow field Based on the discussions in previous sections, the overpressure history with a distance of R in Zone 03 of the flow field (R Rfmax), could be solved by Eq. (2). To make it concise,
Fig. 9 e Schematic diagram of rough overpressure history (R < Rfmax).
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 8 ) 1 e1 2
7
Fig. 10 e Validation with large scale H2/Air deflagration test.
of t'i. The times and the corresponding overpressure values can be solved by Eq. (11)e(17). R c0
(11)
t0þ ðRÞ ¼ tþ
(12)
t0s ðRÞ ¼
t0þþ ðRÞ ¼ t0þ ðRÞ
Rfmax R S0
(13)
t0i ðRÞ ¼ ti
Rfmax R c0
(14)
t0f ðRÞ ¼ tf
Rfmax R c0
(15)
þDpmax ðRÞ ¼ r0 ðS0 Þ ð1 aÞ 2
3a 2
Dpmax ðRÞ ¼ Dpmax Rfmax
(16) (17)
Note that, it is assumed that the solutions of t'þ, t'i, t'f, and -Dpmax are based on the values of the case of R ¼ Rfmax in the present study.
Result and discussion Validation of the method To validate the method developed in Section Method, a large scale [28,31,32] and a series of small scale [30,33] H2/Air deflagration experiments are employed. The calculated data of the former one are compared with both experimental and CFD predicted data, and the latter ones are contrasted with experimental data only. Note that, in these validations, the experimental data of average flame speed S0 are adopted (Eq. (5)). Fig. 10 displays the comparison of experimental data, numerical simulation data [28] and calculated data of GHT 34 Test [28], with various distances. In this experiment, the test point of R ¼ 18 m is close to the maximum flame radius Rfmax (~20 m), the test point of R ¼ 5 m is in the region of Zone 01 and Zone 02 (R < Rfmax), and the distances of 35 m and 80 m are beyond Rfmax (R > Rfmax). Therefore, the case of R ¼ 5 m is solved by the simplified scheme (Eq. 11e17), and solutions of the 18 m, 35 m and 80 m cases are computed by Eq. (10). In Fig. 10, it can be seen that, for large scale deflagration, the model presented in this study yields a satisfactory
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consistency when compared with the experimental data and CFD predicted data. The University of Tokyo conducted a series of small scale H2/Air deflagration tests with various equivalence ratios 4 in 2013 [30,33]. Fig. 11 displays the comparison of calculated data and the experimental data, at a fixed position (R ¼ 0.3 m). As can be seen in Fig. 11, the values of Dp(t) in the negative phase have acceptable consistence, and in the positive phase present unsatisfactory values. A possible explanation for this is that, the peak values in the various equivalence ratio cases may be not the real overpressure induced by explosion, which probably caused by the rupture of the bubble. It is clear that, with an increased equivalence ratio, the peak values of Dp(t) changes slightly, at a little overpressure level of 100 Pa. Thus, it is reasonable to deduce that, the peak values at a level of 100 Pa in the tests may be mainly caused by the disturbance of the bubble's rupture. At this aspect, the values of Dp(t) in the negative phase probably appear the real values induced by deflagration in the test. Because the rupture of the bubble occurs in a transient time, and the flow field may be led by the piston quickly. As stated in section Empirical parameters ai and ti, the functions of ai and ti/R0 with S' (Figs. 6e7) are obtained based on the small scale and large scale H2/Air explosion tests. To
further validate the effectiveness of this model, an independent validation with medium scale tests are necessary. However, the fact is that the deflagration tests of H2/Air mixture in unconfined space are very little in the literature. And we have not found another medium scale tests which have enough information, including: weather conditions, flame front histories, and overpressure histories, etc., except for GHT test series (from 7.5 m3 to 260 m3). Unfortunately, the whole overpressure histories of the medium scale GHT tests have not been found in literature. Therefore, an independent validation with the positive peak overpressure values of the medium scale GHT tests is conducted, as shown in Fig. 12. In the validation, R0, T0, P0, CH2 are necessary initial parameters, and experimental values of S0 are employed to calculate ai and ti from Figs. 6e7, respectively. Then the solution could be obtained using the model stated in Section Method, and the positive peak overpressure values could be gotten. In Fig. 12, it is obvious that the calculated data of GHT 25 and GHT 40 cases display good agreements with the experimental data. For GHT 11 case, there is a larger discrepancy compared with the others, however, the maximum deviation value is 15%. Based on the discussions mentioned previously, we can conclude that the model presented in Section Method is effective for H2/Air deflagration evaluating.
Fig. 11 e Validation with small scale H2/Air deflagration tests.
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According to Eq. (18), the average flame speed S0 for unconfined mixtures (y ¼ 0) could be given by: S0 ¼ 0:006bðb 1ÞSL
1=3 Rfmax d
(20)
The flame thickness equation is [34e36]: d¼
m0 r0 SL
(21)
here, r0, b and SL can be obtained from CHEMKIN-Pro 18.0 code, and the dynamic viscosity of mixture can be solved by: P m0 ¼
Fig. 12 e Independent validation with medium scale H2/Air deflagration tests (hollow: Experimental [27]; solid: Calculated).
Comparison of average flame speed correlations The validation in Section Validation of the method demonstrates that, when an experimental data of S0 is employed (as Table 2 shows), the model in this study appears good consistence with medium and large scale deflagration tests, and has acceptable solution in the small scale cases. To obtain a complete independent model, an independent correlation of S0 is necessary. Dorofeev [34e36] proposed two flame speed correlations respectively for semi-confined and unconfined gas mixtures. For the semi-confined mixtures, the flame speed is given by: vf ¼ 0:0085bðb 1ÞSL
4 by R0:63 f 1þ 3 x ðbxÞ0:63
!2 1=3 Rf d
(18)
where, vf and Rf respectively denote the flame speed and radius, x and y represent for characteristic sizes of obstacle, SL is laminar flame speed, and d is the thickness of flame. And for unconfined mixtures, the flame speed is given as follows: 1=3 1=3 Rf vf ¼ 0:47S4=3 L c
(19)
where, c is thermal diffusivity.
1=2
Ck m0;k Mk P Ck M1=2 k
(22)
The dynamic viscosity of each component m0,k could be obtained from NIST Standard Reference Database 23 [37], Ck is molecular fraction of component k, and Mk is molecular weight of component k. In addition, when the flame speed vf is considered as the average flame speed S0 , and the flame radius Rf as maximum flame radius Rfmax, Eq. (19) becomes the following form: 1=3 1=3 Rfmax S0 ¼ 0:47S4=3 L c
(23)
where, c can also be obtained from NIST Standard Reference Database 23. Table 2 lists a comparison of the computed values and experimental data of average flame speed S'. It can be seen that, the solutions of Eq. (20) agree well with the experimental data of large scale deflagration test, and appear conservative in the small scale cases. For Eq. (23), the solutions display radical for both small and large scale deflagration tests. Thus, for accurate prediction of Dp (R, t), it is reasonable to adopt a more precise S0 correlation.
Flow chart In Fig. 13, a flow chart for computing the overpressure history Dp (R, t) is given. The initial conditions, such as R0, T0, P0, and Cfuel are necessary. These parameters can be imported into CHEMKIN and the NIST database, to obtain r0, rb and SL from the former, and m0,k from the latter. Then a, b, Rfmax, m0, d, and S0 could be solved by the relative equations. When S0 is known, ai and ti can be obtained by Figs. 6e7, and D(t) can also be solved subsequently. Based on the known D(t), Rf(t) and af(t)
Table 2 e Comparison of average flame speed. 4
Test
UTokyo [30]
GHT [27]
No.25 No.40 No.13 No.34
1.0 1.8 3.0 4.0 0.996 0.996 0.832 1.005
R0(m)
0.05
1.53 2.88 5.00 10.00
S' (m/s)
Deviation (%)
Test
Eq. (20)
Eq. (23)
Eq. (20)
Eq. (23)
15.5 18.6 11.1 7.6 46.2 57.0 49.5 73.9
9.9 11.6 5.3 2.6 44.6 55.0 42.2 83.5
19.3 26.2 16.2 9.8 64.8 79.4 73.2 124.2
36.1 37.6 52.3 65.8 3.5 3.5 14.7 13.0
24.5 40.9 45.9 28.9 40.3 39.3 47.9 68.0
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Fig. 13 e Flow chart of overpressure history calculation.
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 8 ) 1 e1 2
could be gotten. Then if R Rfmax, Dp (R, t) could be solved by Eq. (10), otherwise, a rough Dp (R, t) can be got by the simplified scheme. In the end, the delay time of acoustic wave for various distances R should be considered, and Dp (R, t) transforms to Dp (R, t) for R Rfmax case.
Conclusion In this study, an improved acoustic approximation model for UVCE was proposed. For the large scale VCE accidents, a simplified scheme was also presented for overpressure evaluating of the area inside the flame cloud. The model could be used for engineering application of safety assessment on hydrogen energy, and has a high computational efficiency compared with CFD models, because there are no differential equations but only algebraic equations. The model appears good accuracy in the medium and large scale H2/Air deflagration case. In the small scale cases, the model obtained acceptable solutions. However, to obtain a high accuracy prediction, a more precise correlation of average flame speed should be further developed in future. In addition, the effect of obstacles should be considered, and the model can also be used to more fuels based on validation.
Acknowledgement This research work was jointly supported by the National Natural Science Foundation of China (No. 51641608) and the State Key Laboratory of Technologies in Space Cryogenic Propellants of China (SKLTSCP1511, SKLTSCP1809).
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