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Journal of Loss Prevention in the Process Industries 25 (2012) 114e126

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Journal of Loss Prevention in the Process Industries

journal homepage: www.elsevier .com/locate/ j lp

Parametric approach of the domino effect for structural fragments

Dongliang Sun a,b, Juncheng Jiang a,*, Mingguang Zhang a, Zhirong Wang a,Guangtuan Huang b, Jianjiang Qiao b

a Jiangsu Key Laboratory of Urban and Industrial Safety, Institute of Safety Engineering, School of Urban Construction and Safety Engineering,Nanjing University of Technology, Nanjing 210009, Jiangsu, Chinab State Environmental Protection Key Laboratory of Environmental Risk Assessment and Control on Chemical Process, School of Resources and Environmental Engineering,East China University of Science and Technology, Shanghai 200237, China

a r t i c l e i n f o

Article history:Received 31 August 2010Received in revised form23 June 2011Accepted 26 June 2011

Keywords:Domino effectIndustrial explosionMonte-Carlo simulationsNumber of fragmentsParametric approach

* Corresponding author. Mail Box 13, No. 200 ZhoUniversity of Technology, Nanjing 210009, China. Tel.25 83239973.

E-mail addresses: jcjiang@njut.edu.cn, j_c_jiang@1

0950-4230/$ e see front matter � 2011 Published bydoi:10.1016/j.jlp.2011.06.029

a b s t r a c t

More specific and accurate probabilistic models of the numbers of fragments generated respectively byBoiling Liquid Expanding Vapor Explosions (BLEVEs), Mechanical Explosions (MEs), Confined Explosions(CEs), and Runaway Reactions (RRs) of a horizontal cylindrical vessel were developed using themaximum entropy principle based on historical accident data. The theoretical results from the fourprobability density functions were compared to the observed data, and the numbers of fragments fol-lowed discrete exponential distributions in the interval [1, 9]. Beside the summary of the probabilisticdistributions of the other random variables in the process of fragment projection, the effects on thefragment trajectory and target terms were investigated using a parametric approach. The results showedthat using the complete model, wind shear, turbulence, and absence of fragment rotation caused thefragments to impact within shorter distances; fragment rotation and lack of wind decreased the prob-ability of impact within a given distance, but the rupture probability of the target was not affected byfragment rotation or wind. The probabilistic confidence intervals of fragment range, impact, and targetpenetration became narrower with the number of simulation runs, but the accuracy of the resultsincreased. The probability of fragment impact increased with the volume of the target vessel and thedegree of filling of the explosion vessel, but did not depend on the kind of explosion. The probability oftarget rupture increased slowly with the degree of filling of the explosion vessel, but was little influencedby the volume of the target vessel or the kind of explosion.

� 2011 Published by Elsevier Ltd.

1. Introduction

In chemical process industries, the domino effect is a well-known cause of major accidents (Antonioni, Spadoni, & Cozzani,2009; Cozzani, Antonioni, & Spadoni, 2006; Nguyen, Mébarki,Ami Saada, Mercier, & Reimeringer, 2009). An accidental eventwhich starts at one unit may damage another through heat radia-tion, blast waves, or projectiles. In reality, a sudden explosion cangenerate many fragments which can be projected over longdistances, threaten other sites located in the vicinity, and lead tomore severe consequences due to the nature of the domino effect.Fragment projection in an explosive accident is one importantcause of the domino effect on chemical process equipment(Pietersen, 1988). The overall domino effect caused by fragments iscomposed of a set of elementary cycles, and each cycle includes

ngshan North Road, Nanjing: þ86 25 83587421; fax: þ86

63.com (J. Jiang).

Elsevier Ltd.

three detailed steps: the source term (explosion and generation ofthe fragments), the fragment trajectory term (angles, velocities, anddisplacements from the source), and the target term (impact of andinteraction between the fragments and the target).

2. Analysis of previous work

Some research on the three components described above hasbeen performed in previous work (Abbasi & Abbasi, 2007; Bahman,Abbasi, Rashtchian, & Abbasi, 2010; Baum, 1988, 1995, 1998, 1999a,1999b, 2001; Bukharev & Zhukov, 1995; CCPS, 1994; Genova,Silvestrini, & Leon Trujillo, 2008; Gubinelli & Cozzani, 2009a,2009b; Gubinelli, Zanelli, & Cozzani, 2004; Hauptmanns, 2001a,2001b; Holden, 1988; Holden & Reeves, 1985; Lees, 1996;Lepareux et al., 1989; Mébarki, Mercier, Nguyen, & Ami Saada,2009; Mébarki, Nguyen, Mercier, 2009; Mébarki et al., 2007;Mébarki, Nguyen, Mercier, Ami Saada, & Reimeringer, 2008;Neilson, 1985; Nguyen et al., 2009; Qian, Xu, & Liu, 2009; Scilly &Crowther, 1992; Stawczyk, 2003; Tulacz & Smith, 1980; Van den

Table 1Accident data for horizontal tank from Gubinelli and Cozzani.

Source: Gubinelli and Cozzani(2009a,b)

Number of fragments

1 2 3 4 [5e9]

Explosion categoryBLEVENumber of events 5 56 35 3 0

MENumber of events 0 6 1 1 0

CENumber of events 0 9 0 0 1

RRNumber of events 2 3 1 0 1

Table 2Accident data for BLEVE of horizontal tank from Hauptmanns, Holden, and Mébarki.

Explosion category Number of fragments

BLEVE 1 2 3 4 5 6 7 8 9

Source Number of eventsHauptmanns

(Baker et al., 1977;Hauptmanns, 2001a;Holden & Reeves, 1985)

9 17 11 5 2 1 1 e e

Holden and Reeves(Abbasi & Abbasi, 2007)

8 7 9 3 e e e e e

Holden and Reeves(Holden & Reeves, 1985;Nguyen et al., 2009)

11 8 11 6 e e 1 e e

Mébarki et al.(Mébarki, Mercier, et al.,2009)

17 10 12 7 1 1 1 0 1

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126 115

Bosch & Weterings, 1997; Zhang & Chen, 2009). In recent work(Mébarki, Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009;Mébarki et al., 2007, 2008; Nguyen et al., 2009), the mechanicaland kinetic features of the source terms (random variables such asnumber of fragments, shape, and mass) were investigated, and thecorresponding probabilistic distributions were developed using themaximum entropy principle for the source terms. In the fragmenttrajectory term, trajectory equations for the fragments wereproposed, and the ground distributions of the fragments wereassessed. In the target term, probabilistic models of fragmentimpact were developed, a calculation of the impact probability wascarried out, and its effects on the probability of impact were eval-uated. As for target damage, a simplified plastic model for evalu-ating the probability of rupture with high reliability was proposed,and its influence on penetration depth was investigated. However,in the analysis described above (Mébarki, Mercier, et al., 2009;Mébarki, Nguyen, et al., 2009; Mébarki et al., 2007, 2008; Nguyenet al., 2009), for the source terms, i.e. the development of a proba-bilistic model for the number of fragments from a horizontalcylindrical vessel explosion, available accident datawere scarce, andonly BLEVEs (Boiling Liquid Expanding Vapor Explosions) resultingin fragment projection had been considered; for a spherical vesselexplosion, a uniformdistribution of the number of fragmentswithinthe interval from 1 to 19 was assumed. Furthermore, the charac-teristics of fragment flight, impact, and penetration into nearbyfacilities, i.e., the fragment trajectory and target terms, still need tobe improved. Generally speaking, the accuracy of quantitative riskanalysis for industrial sites relies intimately on the hypotheses andthe adequacy of the models developed for the whole domino-effectsequence. On the basis of these findings (Mébarki, Mercier, et al.,2009; Mébarki, Nguyen, et al., 2009; Mébarki et al., 2007, 2008;Nguyen et al., 2009), improvements was made to define morespecific and accurate probabilistic models of the number of frag-ments from a horizontal cylindrical vessel explosion by collectingand analyzing data from past accidents leading to fragmentprojection. The objectives were to recommend a more reasonableprobability density function for the number of fragments froma spherical vessel explosion, to reachmore specific conclusions afterreviewing the reference works on the source terms, and then toexplore the effects of the algorithms (movement approach, frag-ment rotation, wind, and number of simulation runs) on the frag-ment trajectory and target terms (the ground distributions offragments, the probability of impact between the fragments and thetarget, and the rupture probability of the impacted target) and theinfluence of the calculation parameters (the objective volume, thedegree of filling of the source vessel, and the kind of explosion) onthe target term (the probability of fragment impact and the ruptureprobability of the target) using Monte-Carlo simulations includingthe improved source terms, the kinematics of projectiles, andprobabilistic models of fragment impact, penetration, and damage.

3. Source terms

An industrial explosion may generate many fragments withvarious features, which can be considered as random variables:number of fragments (N), shape and size (fP), mass (m), initialvelocity at departure (vO), initial departure angles (horizontal andvertical angles, q and 4), aerodynamic coefficients (lift and dragcoefficients, CL and CD), and degree of filling of the source vessel (f).

3.1. Number of fragments, N

3.1.1. Case of horizontal cylindrical vessel explosionIn recent work (Mébarki, Mercier, et al., 2009; Mébarki, Nguyen,

et al., 2009; Mébarki et al., 2007, 2008; Nguyen et al., 2009), the

maximum entropy principle was used to establish the probabilitydensity function (p.d.f.) of the number of fragments. However, fewexperimental or accident data were used, and only the BLEVEsgenerating the fragments were investigated. In fact, for a horizontalcylindrical vessel, many other accidental scenarios (e.g., aMechanicalExplosion (ME), Confined Explosion (CE), or Runaway Reaction (RR))can also cause fragment projection. Therefore, more data on thenumber of fragments generated by various experimental and acci-dent scenarios shouldbe collected so thatmore specificmodels of thenumber of fragments can be developed. Based on the work ofGubinelli and Cozzani (2009a, 2009b), the primary scenarios fora horizontal cylindrical vessel are BLEVE, ME, CE and RR, based onresearch on data sources of past accidents leading to fragmentprojection. The relations between the number of fragments from thevessel explosion, the tank shape, the type of primary scenario, andthe fracture patterns and mechanics were discussed in detail.Moreover, the number of fragments was determined with high reli-ability for each type of primary scenario for the tank. The data fromGubinelli and Cozzani (2009a, 2009b) were collected for horizontalcylindrical vessels and are shown inTable 1. Simultaneously, the datafrom previous authors (Abbasi & Abbasi, 2007; Baker et al., 1977;Hauptmanns, 2001a; Holden & Reeves, 1985; Mébarki, Mercier,et al., 2009; Nguyen et al., 2009) for horizontal cylindrical vesselswere comprehensively collected and are shown in Table 2. The datafor each type of primary scenario inTables 1 and 2 are incorporated inTable 3. The number of BLEVEs is larger than in previous work.Generally speaking, the reliability of the results depends on theamountof data used in theprobability analysis. The number of eventswith respect to the observedprobabilities of the numberof fragmentsfor different accidental events can be obtained by statistics and isshown inTable 3. For CEs and RRs, the number in the interval [5e9] isconsidered as a random variable following a uniform distribution.Therefore, each value in [5e9] has the same observed frequency. In

Table 3Overall number of events for the four explosion categories for horizontal tanks and the corresponding observed probabilities of number of fragments.

Explosion category Number of fragments

1 2 3 4 5 6 7 8 9

BLEVENumber of events 50 98 78 24 3 2 3 0 1Observed frequency 0.1931 0.3784 0.3012 0.0927 0.0116 0.0077 0.0116 0 0.0037

MENumber of events 0 6 1 1 0 0 0 0 0Observed frequency 0 0.7500 0.1250 0.1250 0 0 0 0 0

CENumber of events 0 9 0 0 1Observed frequency 0 0.9000 0 0 0.0200 0.0200 0.0200 0.0200 0.0200

RRNumber of events 2 3 1 0 1Observed frequency 0.2857 0.4286 0.1429 0 0.02856 0.02856 0.02856 0.02856 0.02856

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126116

general, the interval representing the number of fragments from thefour categories of explosions is [1e9], with the numbers of fragments1e4 accounting for the larger proportions for each event. However,the numbers of fragment and their observed probabilities aredifferent for different accidental events.

The probabilistic distributions corresponding to the number offragments for BLEVE, ME, CE, and RR have been developed based onthe available data summarized in Table 3 according to themaximum entropy principle. In general, the model obtained usingthe maximum entropy principle, which is the most rational andstochastic way to predict unknown data under certain availableinformation, is the only unbiased way to account for all the avail-able information, because all the other models impose additionalconstraints and assumptions which cannot be derived from theavailable information. Consequently, the maximum entropy prin-ciple is the best choice for developing the models. The probabilitydensity function of the maximum entropy principle can beexpressed as shown in Eq. (1), because the only variable involved isthe number of fragments N:

PðNÞ ¼ expð � l0 � l1$g1ðNÞ � l2$g2ðNÞ �/� ln$gnðNÞÞ (1)

where N is the number of fragments, P() the discrete probabilitydensity function, l0, l1, ., ln the Lagrangian multipliers, and g1(N),g2(N), ., gn(N) the available information.

The data in Table 3 consist of the large sample (the full set ofdata on BLEVEs) and the small sample (the smaller set of data onMEs, CEs, and RRs). However, the numbers of fragments for the fourexplosion categories are all between 1 and 9, and the interval D istherefore defined as [1e9]. For the four explosion categories ina horizontal cylindrical vessel, the available information from thedata in Table 3 includes E1 (the average of the number of fragments)and E2 (the variance of the number of fragments), which areexpressed respectively in Eqs. (2) and (3), where PðiÞObserved is theobserved frequency of the number of fragments i. On the basis ofthe available information, the probability density function for thenumber of fragments can be expressed as in Eq. (4). Equation (5) isobtained by combining Eqs. (2)e(4), and the Lagrangian multipliersl0, l1, ., ln are obtained based on the data in Table 3. Finally, theprobability density functions are determined, and the theoreticalfrequencies corresponding to the number of fragments can becalculated using the probabilistic distributions.

E1 ¼X9i¼1

i$PðiÞObserved (2)

E2 ¼X9

ði� E1Þ2PðiÞObserved (3)

i¼1

PðNÞ ¼ exp � l0 � l1N � l2N2 (4)

� �PDexp

�� l0 � l1i� l2i2

�¼ 1

PDi$exp

�� l0 � l1i� l2i2

�¼ E1P

Dði� E1Þ2$exp

�� l0 � l1i� l2i2

�¼ E2

8>>>>><>>>>>:

(5)

The probability density functions of the numbers of fragmentsfrom BLEVE, ME, CE, and RR have been developed as shown in Eqs.(6)e(9). The results of the probabilistic distributions of the numberof fragments for the four explosion categories are shown in Fig. 1.All the theoretical histograms are compared to the accident(observed) frequencies presented in Table 3. This means that thenumbers of fragments from BLEVE, ME, CE, and RR follows thediscrete exponential distribution within the interval [1, 9].

PðNÞ ¼ exp�� 2:16þ 0:97N � 0:24N2

�ðp:d:f : of the number of fragments from BLEVEÞ ð6Þ

PðNÞ ¼ exp�� 6:26þ 4:80N � 1:01N2

�ðp:d:f : of the number of fragments from MEÞ ð7Þ

PðNÞ ¼ exp�� 0:93� 0:11N � 0:05N2

�ðp:d:f : of the number of fragments from CEÞ ð8Þ

PðNÞ ¼ exp�� 0:20� 0:72N þ 0:03N2

�

ðp:d:f : of the number of fragments from RRÞ ð9Þ

3.1.2. Case of spherical vessel explosionIn recentwork (Mébarki et al., 2007, 2008;Mébarki,Mercier, et al.,

2009; Mébarki, Nguyen, et al., 2009; Nguyen et al., 2009), a uniformdistribution for the number of fragments within the interval [1e19]was assumed. In fact, because a crack can start anywhere in theexplosion of a spherical vessel, the larger the source vessel volume,the greater will be the number of fragments generated by theexplosion, because higher vessel volumes and higher vessel surfaceareas correspond to a higher probability of fracture branching.Therefore, the volume of the source vessel has an effect on thenumber of fragments. On this topic, a linear correlation between thenumberof fragmentsNand the source vessel volumeVwasoriginallyproposed: N ¼ �3:77þ 0:96� 10�2V (Holden and Reeves, 1985).Later, a correlation between the number of fragments N and the

Fig. 1. Theoretical and accident probabilities versus number of fragments from BLEVE, ME, CE, and RR.

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126 117

source vessel volume V was obtained: N¼�0.425þ 6.115�10�3Vover the interval [3e19] of the number of fragments, with an averageof eight fragments as determined from data sources from past acci-dents giving rise to fragment projection (thedata sources involved13BLEVEs, and there were no other accidental scenarios causing frag-ment projection) (Gubinelli & Cozzani, 2009a, 2009b; Holden, 1986;Westin, 1971). In Holden and Reeves’s work, the scarcity of the dataavailable resulted in a relevant uncertainty of the correlation, whichmight result in larger errors. In Gubinelli and Cozzani’s work, moredata were collected, and therefore higher model accuracy could beachieved, and the established correlation gave more reasonableresults for low source vessel volumes (the number of fragmentsapproached zero for source volumes near to zero). Therefore, themodel developed by Gubinelli and Cozzani, i.e.,N¼�0.425þ 6.115�10�3V, has been used here.

3.2. Other features of source terms

In recent research by (Mébarki, Mercier, et al., 2009; Mébarki,Nguyen, et al., 2009; Mébarki et al., 2007, 2008; Nguyen et al.,2009), other features of the source terms (random variables) havebeen investigated and discussed, including fragment shape and size(fP),mass (m), initial velocityatdeparture (vO), initial departureangles(horizontal andvertical angles,qand4), andaerodynamic coefficients(lift and drag coefficients, CL and CD). Based on these previous results,the same probabilistic distributions of these features of the sourceterms are used here, as shown in Table 4 (Mébarki, Mercier, et al.,2009; Mébarki, Nguyen, et al., 2009; Nguyen et al., 2009).

The degree of filling of the source f at the time of the accident,which is important for calculation of the total energy imparted

to the fragments, is practically never known for real accidents(Hauptmanns, 2001a, 2001b). It is therefore considered to bea feature of the source terms (i.e., a random variable) followinga uniform distribution within [0, 1].

All the features of the source terms are summarized in Table 4(Gubinelli & Cozzani, 2009b; Mébarki, Mercier, et al., 2009;Mébarki, Nguyen, et al., 2009; Nguyen et al., 2009).

4. Fragment trajectory and target terms

4.1. Methods

4.1.1. Kinematics of projectilesAs discussed in the references (Mébarki, Mercier, et al., 2009;

Mébarki, Nguyen, et al., 2009; Mébarki et al., 2007, 2008; Nguyenet al., 2009), the total energy E can be calculated using Eq. (10)from Baum (1988) when the explosion of a tank occurs. After-ward, the kinetic energy Ek of the fragments can be derived usingEq. (11). The initial velocity of the generated fragments, vO, can bederived from their kinetic energy Ek and mass m (Eq. (12)).

E ¼�1�

�p0p1

�ðg�1Þ=gþðg� 1Þp0

p1

�p1

g� 1fV (10)

Ek ¼ cE (11)

vO ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Ek=m

p(12)

where p0 is the atmospheric pressure, p1 the failure pressure of thesource vessel, g the specific heat ratio, f the degree of filling of the

Table 4Features of source terms and their probabilistic distributions.

Features of source terms P.d.f. in case of horizontal cylindrical vessel P.d.f. in case of spherical vessel

N: Number of fragments Discrete exponential distribution for any accidentamong BLEVE, ME, CE and RR

Requires the vessel volume for BLEVE accidents(Gubinelli & Cozzani, 2009b)

fp: Frequency of eachfragment shape and size

Uniform distribution for any form among end-cap,oblong end-cap, and flattened fragment, the size beinguniformly distributed (Mébarki, Mercier, et al., 2009; Mébarki,Nguyen, et al., 2009; Nguyen et al., 2009)

Uniform distribution for any form between end-capand flattened fragment, the size being uniformly distributed(Mébarki, Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009)

m: Fragment mass Beta distribution (Mébarki, Mercier, et al., 2009) The same mass for all the fragments (Mébarki, Mercier, et al., 2009)vO: Departure velocity

(multiplicative factor c)Exponential distribution (Mébarki, Mercier, et al., 2009;Mébarki, Nguyen, et al., 2009)

Exponential distribution (Mébarki, Mercier, et al., 2009; Mébarki,Nguyen, et al., 2009)

q: Horizontal departureangle

Uniform distribution: 20% in [30�e150�], 30% in[150�e210�], 20% in [210�e330�], 30% in [330�e30�](Mébarki, Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009;Nguyen et al., 2009)

Uniform distribution within the interval [0�e360�] (Mébarki,Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009)

4: Vertical departure angle Uniform distribution within the interval [�90� to 90�](Mébarki, Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009;Nguyen et al., 2009)

Uniform distribution within the interval [�90� to 90�] (Mébarki,Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009)

CL: Lift coefficient Uniform distribution within the interval [0.351e0.468]for any form between end-cap and oblong end-cap;the effect can be neglected for plate (Nguyen et al., 2009)

Uniform distribution within the interval [0.351e0.468] for end-cap;and the effect can be neglected for plate (Nguyen et al., 2009)

CD: Drag coefficient Uniform distribution within the interval [0.8e1.1] forany form between end-cap and oblong end-cap, anduniform distribution within the interval [1.1e1.8] forplate (Nguyen et al., 2009)

Uniform distribution within the interval [0.8e1.1] for end-cap, anduniform distribution within the interval [1.1e1.8] for plate(Nguyen et al., 2009)

f: Degree of filling ofsource

Uniform distribution within the interval [0e1] Uniform distribution within the interval [0e1]

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126118

source vessel, V the volume of the source vessel, and c a multipli-cative factor.

The trajectory of the fragments projected from the explodedvessels results from the combined effects of inertia, gravitation, andaerodynamics (drag and lift). Let (O, X, Y, Z) be the set of systemaxes used for the trajectory description, and let O be the departurepoint for a generated fragment. The fragment trajectory can bedescribed as shown in Fig. 2 (Mébarki, Nguyen, et al., 2009). Afterthe fragment has been projected, an impact is possible with anypotential target that crosses its trajectory.

For fragment movements, the complete movement approach isused (Mébarki, Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009;Mébarki et al., 2007, 2008; Nguyen et al., 2009):

� kD$cosð4Þ$cosðqÞ � ð�1Þq$kL$sinð4Þ$cosðqÞ$�_x2 þ _y2 þ _z2

�� €x ¼ 0� kD$cosð4Þ$sinðqÞ � ð�1Þq$kL$sinð4Þ$sinðqÞ

$�_x2 þ _y2 þ _z2

�� €y ¼ 0� ð�1Þq$kD$sinð4Þ þ kL$cosð4Þ

$�_x2 þ _y2 þ _z2

�� €z� g ¼ 0

8>>><>>>:

(13)

where q¼ 1 for the descending part; q¼ 2 for the ascending part; x,y, and z are the fragment center coordinates; kD is the drag factor,kD ¼ rairCDSD=2m; kL is the lift factor, kL ¼ rairCLSL=2m; rair is thespecific density of air; SD is the frontal surface; and SL is the hori-zontally projected surface. The solution of the fragment motionequations can be derived under the following set of hypothesesproposed by the authors (Mébarki, Mercier, et al., 2009; Mébarki,Nguyen, et al., 2009; Mébarki et al., 2007, 2008; Nguyen et al.,2009):

- Initial conditions: at departure, the fragment is located at thesystem origin (point O). The pulse produced by the blast (vesselexplosion) produces the initial velocity of the fragment (vOx,vOy, vOz).

- When the projectile reaches its trajectory upper position (toppoint), its vertical velocity becomes zero.

- Final conditions: when the fragment crashes on the ground, inthe descending part of its trajectory, its vertical coordinatebecomes zero.

Hereafter, the distributions of the fragments crashing on theground will be evaluated by means of Monte-Carlo simulationsusing the complete movement approach.

4.1.2. Fragment impact, penetration, and damageWhen the trajectory of a fragment has been completely identi-

fied, its possible impacts on a given target can be easily determinedby its position on the fragment trajectory. As discussed in detail inthe references (Mébarki, Mercier, et al., 2009; Mébarki, Nguyen,

et al., 2009; Mébarki et al., 2007, 2008; Nguyen et al., 2009), theimpact probability between the projectiles and the potential targetcan be calculated. Monte-Carlo simulations were run to evaluatethe probability of impact, Pimp, in the light of Eq. (14):

Pimp ¼PNsim

s¼1PN

j¼1nðj; sÞN

Nsimand nðj; sÞ

¼ 1 if�VtargetXVfragmentsf

�0 otherwise

((14)

where Nsim is the total number of Monte-Carlo simulations; n isa parameter that indicates whether the projectile impacts thetarget; Vfragment is the fragment trajectory; and Vtarget is the targetvolume with a given location, dimensions, and shape. The numer-ical algorithm for the Monte-Carlo method for fragment impact has

Table 5Characteristics of source and target vessels in Nanjing Chemical Industry Park ofChina.

Source A Source B Target A Target B

Type Horizontalcylindricalvessel

Sphericalvessel

Sphericalvessel

Sphericalvessel

Volume/m3 200 1000 800 2400Vessel diameter/m 2.8 6.2 5.8 8.3Failure pressure/MPa 1.20 1.01 0.98 1.26Atmospheric

pressure/MPa0.1 0.1 0.1 0.1

Mass/kg 25,429 121,886 88,761 22,4301Wall thickness/m 0.024 0.026 0.020 0.029Center coordinates (0, 0, 0) (40 m, 0, 0) (0, 40 m, 6 m) (0, �40 m, 9 m)Critical residual

thickness/me e 0.004 0.006

Fig. 2. Description of fragment trajectory.

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126 119

been described in detail by (Mébarki, Mercier, et al., 2009; Mébarki,Nguyen, et al., 2009; Mébarki et al., 2007, 2008; Nguyen et al.,2009).

When an impact between the fragment and a target occurs, itmay cause partial or complete damage (penetration or perforation)to the target and then result in the explosion of the target. As dis-cussed in the references (Mébarki, Mercier, et al., 2009; Mébarki,Nguyen, et al., 2009; Mébarki et al., 2007, 2008; Nguyen et al.,2009), the penetration depth of the fragment can be expressedusing Eq. (15) or (16), and the rupture probability of the impactedtarget, Prup, can be expressed as in Eq. (17):

hP ¼�dP coshþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdP coshÞ2þ

4ptanh

�Ecfu3u

�2=3s

2tanhfor the casehs0

(15)

hP ¼ 1pdP

�Ecfu3u

�2=3

for the case h ¼ 0 (16)

Prup ¼ PðEe � 0Þ ¼

PNsimui¼1

PNimp

j¼1qðj; iÞNimp

Nsimuand

qðj; iÞ¼ 1 if ðEe � 0Þ0 otherwise

; Ee ¼ ðet � hPÞ � ecr

�(17)

where dp is the fragment diameter; h is the incidence angle of thefragment; fu is the ultimate strength of the target constitutivematerial and 3u its ultimate strain; Ec is the kinetic energy expendedwhen the penetration process occurs; Ee is the limit state function;Nimp is the number of the fragments impacting the target, obtainedthrough Eq. (14) in each simulation; Nsimu is the total number ofMonte-Carlo simulations indicating that

PNj¼1 nðj; sÞ=N in Eq. (14) is

not zero in each simulation; et is the target thickness; hp is thepenetration depth; and ecr is the critical plate thickness. Use of theMonte-Carlo method for evaluation of the rupture probability ofthe impacted target has been described in detail in the references(Mébarki, Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009;Mébarki et al., 2007, 2008; Nguyen et al., 2009).

4.1.3. Characteristics of source and target vesselsAs a case study, the industrial site installed in Nanjing Chemical

Industry Park of China was considered, as described in Table 5(Shen, 2010). A horizontal cylindrical vessel and a spherical vessel(Source A and Source B) were considered as the sources (bothaccidents are BLEVEs), and two spherical vessels (Target A and

Target B) were considered as the targets. The threshold thickness ecris shown only for Target A and Target B, and the correspondingvalues for the calculation using Eq. (17) are provided in Table 5.Using the probabilistic distributions summarized in Table 4, thesource terms in Table 5, and Eqs. (10)e(12), the fragment features(e.g. number of fragments, mass, energy, departure velocity, andangles) at departure can be obtained. Then the trajectory and theground distributions of fragments can be derived using Eq. (13). Forthe calculation of the fragment impact probability and the ruptureprobability of the impacted target, because Source B was notlocated at the system origin (point O) (see the center coordinates inTable 5), the simulations were carried out by coordinate translationfor Source B, Target A, and Target B.

4.2. Investigation of the effects of various factors on grounddistribution of fragments

The factors include movement approach, fragment rotation,wind, and number of simulation runs. The confidence intervalapproach is preferable when dealing with probabilities andconvergence analysis.

4.2.1. Movement approachTwo kinds of movement approaches were considered: the

complete model as given in Eq. (13) and the simplified model (thecomplete model in the absence of air resistance (kD¼ 0 in Eq. (13))).The resulting fragment distributions were obtained by thecomplete and simplified movement approaches without fragmentrotation or wind (see Fig. 3). Fig. 3 shows that the complete modelpredicted that the fragments would impact in shorter distancesthan the predicted by the simplified model for Sources A and B.

4.2.2. Fragment rotationFragment rotation around the center of mass was discussed in

detail in the references (Mébarki, Mercier, et al., 2009; Mébarki,Nguyen, et al., 2009; Mébarki et al., 2007, 2008; Nguyen et al.,2009). On this basis, results for the effect of fragment rotation onthe distributions were obtained without any wind effects (Fig. 4).From these results, it can be seen that absence of rotation causedthe fragments to impact in shorter distances for Sources A and B.

4.2.3. WindWind shear and turbulence are the two main effects that

a variable wind field can have on the flight of fragments (Liu, Wang,& Jia, 2006). These characteristics of wind were considered as thedynamics of a moving air mass, i.e., as wind velocity and direction.Therefore, the models used for wind shear and turbulence can besummarized as follows.

Fig. 3. Theoretical simulations using complete and simplified models versus range.

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126120

The induced velocity of a single-line vortex can be expressed asin Eq. (18) (Liu et al., 2006):

VWiðri; tÞ ¼ G0i2pri

1� exp

� r2i4yt

!!(18)

where G0i is the vortex strength with t¼ 0; ri is the circumferentialdistance from the vortex center; y is the kinematic viscosity coef-ficient of the fluid surface; and t is the lifespan of the vortex. Thenumber, position, and strength of vortices can be confirmed bycomparison with the actual situation. The horizontal and verticalcomponents of the wind induced by n line vortexes can beexpressed as follows (Liu et al., 2006):

uW ¼Xns

i¼1

xWi

riVWi (19)

wW ¼ �Xns

i¼1

zWi

riVWi (20)

where xWi ¼ x� xWi0, zWi ¼ z� zWi0, ri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2Wi þ z2Wi

q, and xWi0

and zWi0 are the coordinates of the vortex center.The time spectrum of the turbulent velocity can be calculated

using Dryden’s atmospheric turbulence model (Liu et al., 2006):

Fig. 4. Theoretical simulations with no rot

FuuðuÞ ¼ s2uLupvT

1

1þ ½ðLu=vTÞu�2

FvvðuÞ ¼ s2vLvpvT

1þ 12½ðLv=vTÞu�2n1þ 4½ðLv=vTÞu�2

o2FwwðuÞ ¼ s2w

LwpvT

1þ 12½ðLw=vTÞu�2n1þ 4½ðLw=vTÞu�2

o2

8>>>>>>>>>>><>>>>>>>>>>>:

(21)

where Lu, Lv, and Lw are respectively the turbulence scales of thethree directions; su, sv, and sw are the turbulence intensities in thethree directions; vT is the velocity of the fragment when it passesthrough the turbulence field; and u is the time frequency.

In the models described above, the wind velocity can beassumed constant because the velocity of the fragment is muchlarger than the wind velocity and its variations. Therefore, varia-tions in the wind field can be neglected according to the “frozenfield” hypothesis (Liu et al., 2006).

As discussed above, the results for the distributions of thefragments on the ground with no wind and with wind shear andturbulence were obtained without fragment rotation (Fig. 5). Theresults show that wind shear and turbulence caused the fragmentsto impact in shorter distances than under no-wind conditions forSources A and B, and that the ground distributions of the fragmentswith wind shear were almost the same as those with turbulence.

ation and with rotation versus range.

Fig. 5. Theoretical simulations with no wind, wind shear, and turbulence versus range.

Fig. 6. Theoretical simulations with different numbers of simulation runs versus range.

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126 121

4.2.4. Number of simulationsThe influence of the number of simulations on the fragment

distributions was investigated without fragment rotation or wind.The number of simulations (Nsim) was successively chosen as 103, 104,105, 106, 107, 108, 109, and 1010 simulations. The results are shown inFig. 6, which shows that therewas almost no change in the theoreticalcalculations with increasing number of simulations for Sources A andB. The confidence intervals of the probabilities in the range of0e100 mwere investigated. Using the methods mentioned in OnlineMBA Library (2010), the confidence intervals of the probabilities in the

Fig. 7. Comparison of probabilities of impa

range of 0e100m corresponding to 103, 104, 105, 106, 107, 108, 109, and1010 simulations for Source A were: [0.13104, 0.16818], [0.14466,0.15645], [0.14844, 0.15217], [0.14964, 0.15082], [0.14975, 0.15012],[0.15017, 0.15029], [0.15013, 0.15017], and [0.15038, 0.15039]; theconfidence intervals of the probabilities in the range of 0e100mcorresponding to 103, 104, 105, 106, 107, 108, 109, and 1010 simulationsfor Source B were: [0.15376, 0.17617], [0.16141, 0.16849], [0.16373,0.16598], [0.16452, 0.16523], [0.16487, 0.16509], [0.16495, 0.16502],[0.16483, 0.16485], and [0.16484, 0.16484]. These results demonstratethat the probabilistic confidence interval became narrower with

ct with no rotation and with rotation.

Fig. 8. Comparison of rupture probabilities of the target with no rotation and with rotation.

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126122

increasing number of simulations, meaning that the accuracy of theprobabilities increased due to the larger number of samples corre-sponding to a larger number of simulations.

4.3. Investigation of the effects of various factors on the impactprobability of fragments and the rupture probabilityof the impacted target

4.3.1. Fragment rotationThe effects of fragment rotation on the probability of fragment

impact and the rupture probability of the target were investigated.The results for the impact probability and the rupture probability ofthe target without rotation and with rotation were obtainedwithout wind (Figs. 7 and 8). These figures show that the proba-bility of impact with rotation was smaller than that without rota-tion for Sources A and B, and that there was almost no differencebetween the rupture probabilities of the target with and withoutrotation for Sources A and B.

4.3.2. WindThe effects of wind were considered as discussed in Section

4.2.3. The results for the impact probability and the rupture prob-ability of the target with no wind, with wind shear, and withturbulence were obtained without fragment rotation (Figs. 9 and10). These figures show that the probability of impact with windshear or turbulence was greater than that with no wind for SourcesA and B, and that there was almost no difference between the

Fig. 9. Comparison of probabilities of impact with n

probabilities of impact with wind shear and with turbulence forSources A and B. Furthermore, there was almost no differencebetween the rupture probabilities of the target with no wind, withwind shear, and with turbulence for Sources A and B.

4.3.3. Number of simulationsThe influence of the number of simulations was considered

without fragment rotation or wind, and Target B was chosen asthe target. The results are shown in Figs. 11 and 12, which showsthat 104 simulations provided enough accuracy for the proba-bility of impact for Sources A and B. There was almost nodifference between the rupture probabilities of the target for103, 104, 105, 106, 107, 108, 109, and 1010 simulations for Sources Aand B. The confidence intervals for the impact probabilities andthe rupture probabilities of the target were also investigated.Using the methods mentioned in Online MBA Library (2010), theconfidence intervals of the impact probabilities for 103, 104, 105,106, 107, 108, 109, and 1010 simulations for Source A were:[0.01331, 0.03169], [0.01744, 0.02296], [0.01952, 0.02128],[0.02002, 0.02058], [0.02011, 0.02029], [0.02027, 0.02033],[0.02029, 0.02031], and [0.02029, 0.02030]; the confidenceintervals of the impact probabilities for 103, 104, 105, 106, 107, 108,109, and 1010 simulations for Source B were: [0.00425, 0.01695],[0.00537, 0.00863], [0.00668, 0.00772], [0.00684, 0.00716],[0.00705, 0.00715], [0.00718, 0.00722], [0.00719, 0.00721], and[0.00719, 0.00720]; the confidence intervals for the ruptureprobabilities of the target for 103, 104, 105, 106, 107, 108, 109, and

o wind, with wind shear, and with turbulence.

Fig. 11. Probability of impact versus number of simulations.

Fig. 10. Comparison of rupture probabilities of the target with no wind, with wind shear, and with turbulence.

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126 123

1010 simulations for Source A were: [0.22767, 0.63233], [0.37155,0.50845], [0.41846, 0.46154], [0.42319, 0.43681], [0.41785,0.42215], [0.43932, 0.44068], [0.43978, 0.44022], and [0.42993,0.43007]; the confidence intervals of the rupture probabilities ofthe target corresponding to 103, 104, 105, 106, 107, 108, 109, and1010 simulations for Source B were: [0.08468, 0.65532],[0.24755, 0.47245], [0.32494, 0.39506], [0.35869, 0.38131],

Fig. 12. Rupture probability of target

[0.35647, 0.36353], [0.34889, 0.35110], [0.35965, 0.36035], and[0.35989, 0.36011]. These results demonstrate that the proba-bilistic confidence interval became narrower with increasingnumber of simulations and that the accuracy of the probabilityof impact and the rupture probability of the target increasedwith the larger number of samples corresponding to the largernumber of simulations.

versus number of simulations.

Fig. 13. Top view of fragment trajectory and impact.Fig. 15. Rupture probability of target versus degree of filling of source.

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126124

4.4. Investigation of the effects of the calculation parameters on theimpact probability of fragments and the rupture probability of theimpacted target

The effects of the distance from the source to the target and theobjective orientation on the impact probability of fragments andthe rupture probability of the target have been researched in thereferences (Mébarki, Mercier, et al., 2009; Mébarki, Nguyen, et al.,2009; Mébarki et al., 2007, 2008; Nguyen et al., 2009; Zhang &Chen, 2009; Qian et al., 2009). In this work, the calculationparameters, including the target volume, the degree of filling of thesource vessel, and the kind of explosion were considered, and theirinfluence was investigated using Monte-Carlo simulations. Thevessels described in Table 5 were used.

4.4.1. Target volumeBased on the results shown in Figs. 7e10, the probability of

impact increased with greater target volume (the probability ofimpact on Target B was greater than that on Target A under thesame conditions such as fragment rotation and wind (Figs. 7 and9)), and the rupture probability was little affected by the targetvolume (the rupture probability of Target B was almost the same asthat of Target A under the same conditions (Figs. 8 and 10)).

As stated above, the larger the target volume, the larger was theimpact probability. The larger volume corresponds to a larger targetsurface area, and therefore it increases the possibility of impact. Asdiscussed previously by (Mébarki, Mercier, et al., 2009; Mébarki,

Fig. 14. Probability of impact versus degree of filling of source.

Nguyen, et al., 2009; Mébarki et al., 2007, 2008; Nguyen et al.,2009), in Fig. 13 (Mébarki, Nguyen, et al., 2009), which showsa top view of the fragment trajectory and impact on the targethorizontal cylindrical vessel, the sum of the projected area of thetarget in the direction from target C to origin O (surface area 1) andthe upper area of the target (surface area 2) is the total area (totalsurface area, St) in which impact may occur.

Therefore, St¼ Surface area 1þ Surface area 2¼ [(apþ bt)sin jþ (apþ at)cos j]� (apþ ct)þ (apþ bt)� (apþ at), where ap isthe maximal distance from the fragment rotation center to itsboundary, and at, bt, and ct are the cubic dimensions of the hori-zontal cylindrical target. As stated in the references (Mébarki,Mercier, et al., 2009; Mébarki, Nguyen, et al., 2009; Mébarki et al.,2007, 2008; Nguyen et al., 2009), an impact may occur when thefragment center crosses the cube with dimensions[(apþ at)� (apþ bt)� (apþ ct)], where O is the origin of thegenerated fragment, C the target center, I the impact point, and j

the angle between the major axis of the target and the source. Theprobability of fragment impact depends on St. Moreover, fora spherical vessel target, St¼ (apþ at)2� (sin jþ cos jþ 1) underat¼ bt¼ ct.

4.4.2. Degree of filling of the source vesselThe accident scenarios for both Source A and Source B in Table 5

are considered as BLEVEs, and Target A is chosen as the target. Asstated in Section 3.2, the degree of filling of source f followsa uniform distribution within [0, 1]. The effect of the degree of

Fig. 16. Probability of impact versus kind of explosion.

Fig. 17. Rupture probability of target versus kind of explosion.

D. Sun et al. / Journal of Loss Prevention in the Process Industries 25 (2012) 114e126 125

filling of the source vessel on the impact probability of fragmentsand the rupture probability of the target was investigated forf-values of 0.2, 0.5, and 0.8. The results are shown in Figs. 14 and 15.Fig.14 shows that the probability of impact increases with degree offilling for Sources A and B, and Fig. 15 shows that the ruptureprobability of the target increases slowly according to the degree offilling for Sources A and B. This can be explained by the fact thata larger kinetic energy of fragments can be derived from a sourcewith a larger degree of filling when a vessel explosion occurs. Afterthe flight of the fragments, the kinetic energy Ec when penetrationoccurs, the penetration depth hp, and the rupture probability of thetarget Prup can be increased according to Eqs. (15)e(17).

4.4.3. Kind of explosionAs stated in Section 3.1.1, for horizontal cylindrical vessel

explosions, four types of accident scenarios (BLEVE, ME, CE, and RR)can cause fragment projection. Therefore, the effects of the fourkinds of explosions of Source A on the impact probability of frag-ments and the rupture probability of the target were investigated,with Target A chosen as the target. The results are shown in Figs. 16and 17. These figures indicate that the impact probability of thefragments and the rupture probability of the target depend verylittle on the kind of explosion.

5. Conclusions

A parametric approach to the domino effect for structuralfragments was investigated:

� The intervals of the number of fragments from BLEVE, ME, CE,and RR were all [1e9], with numbers of fragments from 1 to 4accounting for larger proportions in each event. The numbersof fragments and their observed probabilities were different fordifferent accidental events. More specific probabilistic modelsof the number of fragments generated by horizontal cylindricalvessel accidents of BLEVE, ME, CE, and RR types were devel-oped using the maximum entropy principle based on historicalaccident data. All the theoretical results from the four proba-bility density functions were compared to observed data, andthe number of fragments was found to follow discrete expo-nential distributions in the interval [1, 9]. Furthermore, a morereasonable correlation between the number of fragments andthe source vessel volume for spherical vessel explosions wasrecommended, as well as a synthesis of the other features.

� For the ground distributions of fragments generated byexplosions of a horizontal cylindrical vessel and a spherical

vessel, the complete model predicted that the fragments wouldimpact in shorter distances than predicted by the simplifiedmodel, the absence of fragment rotation caused the fragmentsto impact in shorter distances, wind shear and turbulencecaused the fragments to impact in shorter distances than withno wind, and the ground distributions of fragments with windshear were almost the same as those with turbulence. Theprobabilistic confidence intervals became narrower withincreasing number of simulations, and the accuracy of theresults increased due to the larger number of samples corre-sponding to the larger number of simulations.

� As for the probability of fragment impact, the probability ofimpact with fragment rotation was smaller than with nofragment rotation, and there was almost no difference betweenthe rupture probabilities of the target with and without frag-ment rotation. The probability of impact with wind shear orturbulence was greater than with no wind, the probabilities ofimpact with wind shear were almost the same as those withturbulence, and there was almost no difference between therupture probabilities of the target with no wind, with windshear, and with turbulence. 104 simulations provided enoughaccuracy for the probability of impact and the rupture proba-bility of the target. The probabilistic confidence intervalbecame narrower with increasing number of simulations, andthe accuracy of the probability of impact and the ruptureprobability of the target increased due to the larger number ofsamples corresponding to the larger number of simulations.

� The probability of impact increased with target volume,a larger target surface area was found to correspond to theprojection area in the direction from the target to the source,and a larger upper area increased the probability of impact. Theprobability of fragment impact increased with degree of fillingand did not depend on the kind of explosion; the ruptureprobability was little affected by the target volume, increasedslowly with degree of filling, and depended very little on thekind of explosion.

Acknowledgments

The financial support of National Natural Science Foundation ofChina (No. 71001051, 50904037), the “eleventh five-year” nationaltechnology support plan of China (2007BAK22B04), the AcademicFund for Young Teachers of Nanjing University of Technology (No.39714005), the Discipline Fund of Nanjing University of Technology,Research and Innovation Plan for Graduates of Colleges and Univer-sities in Jiangsu Province (CX09B_142Z), and the Shanghai LeadingAcademic Discipline Project (B506) are gratefully acknowledged.

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