7.2 Detonations and explosions - Treccani · 2019. 3. 29. · VOLUME V / INSTRUMENTS 431 7.2.1...

431VOLUME V / INSTRUMENTS

7.2.1 Introduction

From an operative point of view, we can define anexplosion as a release of energy into the atmosphere in asmall enough volume and in a short enough time togenerate a pressure wave of finite amplitude travellingaway from the explosion source, which can be heard.Clearly, also different definitions can be given, but theaforementioned one has the advantage of including all theexplosions that are of some interest for industrial processplant safety. This definition considers only explosions inair, which are the only explosions of some interest in thiscontext. The energy released can be stored, before beingreleased, in various forms, such as: nuclear, chemical,pressure, etc.

The pressure wave generated by the explosion is themain phenomenon to focus on when damage to people orgoods are concerned. However, it is worthwhile stressingthat the main damage, as far as accidental explosions in aprocess plant are concerned, does not stem from the pressurewave itself, but from some indirect effect generated by thepressure wave. In particular, the main injuries to people areusually caused by fragments that may form missiles, and bystructure collapse. Moreover, the damage caused to theprocess units by the blast wave are not usually very large, butlarge enough to cause a dispersion into the environment ofthe chemical stored in the unit. When such a chemical istoxic or flammable, much more serious accidents can betriggered, such as large fires or the dispersion of large cloudsof toxic chemicals into the atmosphere, through what isusually called a domino effect.

A diagram of the accidents involving explosions in aprocess plant is shown in Fig. 1. From such a diagram, wecan see that any major accident in the process industryoriginates from a loss of containment from a process unit(e.g. tank, reactor, apparatus, pipe, etc.), leading to a releaseof chemicals in the environment.

This loss of containment can be catastrophic followingthe collapse of an apparatus. When the chemical in theapparatus is a compressed gas, it will expand from thebursting pressure of the apparatus up to the atmosphericvalue. Since the characteristic time of such an expansion isvery short, the energy released from the expansion

generates an explosion. The gas released will form a cloudthat will be dispersed and diluted into the atmosphere bymixing with air. When no ignition sources are present (oreven when the gas is not flammable), no more energy canbe released and, consequently, no more explosions canfollow. However, when the gas is flammable and ignitionsources are present close to the release point, the smallamount of flammable gas mixed with air catches fire,incorporating the gas not yet mixed with air (that is, withlarger concentration than the upper flammability limit) ina diffusive flame. In this case, the characteristic time forthe combustion energy release is determined by the fueland air transport rate towards the flame, and it isconsequently too large to generate any explosion. Whenthe ignition is delayed, the cloud of flammable gas canmix with air, thus creating a large amount of fuel-airmixture with concentration inside the flammability range.In this case, following the ignition, the characteristic timefor the combustion energy release is determined by thevelocity of the flame that propagates into the flammablemixture, and it can be short enough to generate anexplosion, usually called UVCE (Unconfined VapourCloud Explosion).

When the collapsing apparatus also contains some liquidat room temperature, we can have two different situationsdepending on the value of the normal boiling point (Teb) ofthe liquid compared to the room temperature.

When the liquid temperature, which is equal to theambient one, is lower than its normal boiling temperature,then the liquid is subcooled; so the characteristic time for itsevaporation, and consequently for the energy release relatedto the expansion from the liquid to the vapour volume, isquite long and no explosion can occur. The vapours from theslow evaporation of the liquid pool will be dispersed in theatmosphere, which could lead to the aforementionedphenomena.

When the liquid temperature is higher than its normalboiling temperature, then the liquid is overheated (andconsequently in a non equilibrium condition) and thecharacteristic time for its complete or partial evaporation isquite short; such a fast evaporation can cause a flash or anexplosion called BLEVE (Boiling Liquid Expanding VapourExplosion). The vapours from the fast liquid evaporation

7.2

Detonations and explosions

will be dispersed in the atmosphere and they could lead tothe aforementioned phenomena.

A process fluid can also be released through a non-catastrophic failure of the apparatus, such as a loss from aflange or from a pipe breakage. In this case, the releasedchemical, that can be either gas, liquid or two-phase mixture,will be dispersed into the atmosphere (maybe afterevaporation) and may result in the aforementionedphenomena.

The collapse of the apparatus can also be caused by aninternal explosion. In this case, apart from theaforementioned phenomena, the blast wave will travel frominside to outside the apparatus. Such an explosion can becaused by various situations, such as the explosivedecomposition of an unstable compound, the deflagration ofa flammable mixture of gas or dust, an exothermic chemicalreaction out of control (i.e. a runaway reaction).

Below, the main characteristics of the various kinds ofexplosion, as well as the models available that estimate theconsequences of different explosions will be discussed.

7.2.2 Decomposition of unstablecompounds: ideal explosions

The more common explosions are related to thedecomposition of unstable chemical compounds, such as thecondensed phase explosives.

A chemical compound is said to be thermodynamicallyunstable when the free energy change of its decompositionreaction is negative, DG°

R(T)�0. Since DG°R is the Gibbs free

energy change related to the complete transformation, in thereference state, of a stoichiometric amount of reactants in theproducts, a negative value of DG°

R means that the products of

the decomposition reactions have a lower free energy thanthe original compound. This means that the equilibriumconstant of the decomposition reaction is larger than unity,K(T)�1, from the well known relationK�� iai

vi�exp(�DG°R�RT ), where ai is the activity of the

species i, ni is the stoichiometric coefficient of the samespecies (taken as positive for the reaction products andnegative for the reactants), R is the ideal gas constant and Tthe temperature. In practice, from these relations it followsthat a system kept at constant temperature and initiallycontaining all the species (reactants and products) with unityactivity, evolves spontaneously towards the products. Inother words, at the given temperature, the consideredproducts are more stable than the reactants.

Quite often, unstable compounds contain some oxygenatoms in the molecule, with the consequent formation ofsome oxides (which are usually very stable species, inparticular CO2) among the decomposition products. Atypical example refers to TNT (TriNitroToluene, one of themost common explosives), whose chemical formula isC7H5O6N3.

Not all the thermodynamically unstable compounds canexplode. To do that, firstly it is necessary to provide thecompound with an amount of energy (such as a collision, aspark, a hot surface, etc.) large enough to start thedecomposition reaction; secondly, the decompositionreaction must be fast and exothermic enough, i.e. it mustrelease enough energy in a short time.

Although the explosive potential of a compound mustalways be verified experimentally, there are somepreliminary analytical criteria that can determine, at least asa first approximation, the explosive potential of a compound.A first criterion is based on the analysis of the functionalgroups which are present in the molecule of the compound.

COMBUSTION AND DETONATION

432 ENCYCLOPAEDIA OF HYDROCARBONS

unitfailure

containingalso liquid

T�Teb

BLEVE

T�Teb

containingonly vapour

gasexpansion

atmosphericdispersion

liquidevaporation

notflammable

no

yes

flammable

ignition

immediate

delayedUVCE

internal explosion:.gas/dustdeflagration.decomposition.runaway

loss ofmaterial

explosion

explosion

explosion

Fig. 1. Flow diagram for explosion generationfollowing an industrialprocess plant accident.

The presence of some of the groups in Table 1 is a clearindicator that the compound may be instable. Moreanalytical criteria require the evaluation of some parameters,basically related to the decomposition and combustionenthalpies as well as the stoichiometry of the completedecomposition or combustion reaction. The mostwell-known of such criteria are used in the software CHETAH(CHEmical Termodynamic And Hazard evaluation, originallydeveloped by ASTM, American Society for Testing andMaterials, Seaton et al., 1974) that evaluates thedecomposition of a compound formed by carbon, hydrogen,oxygen and nitrogen to give CO2, H2O, N2, CH4, C, H2 andO2 into the proportions leading to the largest decomposition(or combustion) enthalpy. The lower the decompositionenthalpy (i.e. the larger in absolute value while negative,since it is related to an exothermic reaction), the larger theexplosive potential, because more energy is released by thedecomposition reaction. Moerover, the closer thedecomposition and combustion enthalpies are, the larger theexplosive potential is, since this means that the largest partof the oxygen required for the complete oxidation is alreadycontained in the molecule of the compound. Similarinformation is provided by the oxygen balance that indicatesthe amount of oxygen (in grams) required for the completeoxidation of 100g of the compound. The closer to zero thisindicator is, the larger the explosive potential is; in fact,similar to the combustion and decomposition enthalpies, thismeans that the largest part of the oxygen required for thecomplete oxidation is already contained in the molecule ofthe compound. For each indicator, some empiricalsemi-quantitative scales are defined giving the explosivepotential of the compound. The criteria used in the CHETAHsoftware are summarized as an example in Table 2.

The decomposition of an unstable compound leads to theinstantaneous transformation of the solid compound into itshigh temperature decomposition products that expand in avery short time. Such an expansion is the easiest way togenerate a blast wave (which is, as previously mentioned, theessential feature of an explosion) and can be described bythe movement of a plug accelerating inside a cylinder. Thepressure waves created by the accelerating plug propagateinto the atmosphere in front of the plug at the speed of soundwhich is, for an ideal gas, equal to c��

1233333333333

gRT�M333

, where g is theratio of the specific heat at constant pressure and volume,and M is the molecular weight. The pressure wavesgenerated later propagate into an atmosphere at higherpressure, since the previous pressure waves have alreadytravelled through it, and at higher temperature because thecompression of the atmosphere is adiabatic. It follows thatthe pressure waves generated later travel at a higher speedthan those generated previously and consequentlysuperimpose them. The sharp change in all the statevariables (e.g. pressure and temperature that change sharplyacross the pressure wave) becomes larger and larger, andfinally results in a shock wave propagating from theexplosion source. These explosions can be defined idealsince the explosion source can be regarded as a point, andthe energy release from such a point can be regarded asinstantaneous – in other words, the energy is released in anegligible volume and over a negligible time.

The shock wave generated by an ideal explosionpropagates from the explosion epicentre at an increasingspeed up to a limit value. The qualitative behaviour ofthe pressure vs. time at a given distance from theexplosion epicentre is shown in Fig. 2: pressure risesalmost instantaneously up to a maximum value, P°, and

DETONATIONS AND EXPLOSIONS


Table 1. Functional groups which are typical of unstable compounds

C

C

C

CN2

C

C Me

XC

C ON

ON

C NO2

OC NO2

OC

C NN C

CC NN O

CC NN S

CC NN N

R

NN N N

C O MeN

ON N

N NO2

HZ�N�

OHZ�N�

N2��C

N Me

Ar Me X

N[ ]��Me

Hg HgX�N�

C C

OC O H

OC O C

O O Me

O O O

O X

N3

O

O

N

X Ar Me

N X

then falls down gradually as the shock goes away. In thisphase, the overpressure can be also negative, i.e.absolute pressure can become lower than theatmospheric value. These negative overpressure valuesare usually quite small and rarely exceed 30 kPa. Inorder to estimate the consequences on people andstructures, shape and intensity of a blast wave are usuallycharacterized through several parameters; among them,the maximum side-on overpressure and thecorresponding positive impulse. The side-onoverpressure is defined as the overpressure (with respectto the atmospheric value) occurring at the side of astructure hit by the blast wave, as shown by the P° valuein Fig. 2. The local behaviour of the side-on overpressurevs. time t during the positive overpressure phase isusually represented through the general relation

[1]

where a is a decay parameter and the time starts when theshock wave of finite amplitude P° arrives.

The positive overpressure impulse, iP, is defined as

[2]

As the blast wave travels away from the explosionepicentre, side-on overpressure decreases.

Damage to people and structures due to the blast wavedepend on several factors, characteristic of both the shockwave and the structure or person hit. In practice, fourthreshold values can be used to estimate the expecteddamage to structures. The complete destruction of structuresthat cannot be repaired is expected for side-on overpressurelarger than 83 kPa. Heavy damage to the structures, that is,partial destruction of the structure due to the collapse ofsome structural element, is expected for side-onoverpressure between 83 and 35 kPa. Moderate damage tothe structures involving heavy but repairable damage tosome component (such as walls collapsing) is expected forside-on overpressure between 35 and 17 kPa. Minor damage,such as shattering of windows and doors or cracks in walls,and roofing and tiling, is expected for side-on overpressurebetween 17 and 3.5 kPa.

7.2.3 Physical explosions

The term physical explosions means explosions that do notinvolve any chemical reaction in the energy releasemechanism, such as those occuring due to the rapidexpansion of a compressed gas or the vapour produced by aRapid Phase Transition (RPT). The shock wave formationmechanism can always be depicted as an accelerating plug,similar to the aforementioned ideal explosions. The maindifference is that both the volume from which the energy isreleased and the time necessary for its release are no longerpractically zero as for ideal explosions. For instance, whena pressurized vessel containing only gas collapses, the gasrapidly expands from the bursting pressure to theatmospheric one generating an explosion. Not all theenergy stored in the compressed gas is used to generate theshock wave. A significant amount of this energy is

i Pdt P tP

t

D

D

= = − − −( )

∫0

2

1 11°

α ααexp

P P tt

ttD D

= −

−

° 1 exp α



pres

sure

timetD

P°

0

Fig. 2. Overpressure vs. time at a given distance from the ideal explosion epicentre.

Table 2. Semi-quantitative scales used by CHETAH software for definingthe explosive potential of a CxHyOz compound with n atoms

Indicator

Explosive potential

low medium high

∆hD ��1.25 �2.92-�1.25 ��2.92

∆ ∆

∆

h h

hC D

D

− ��20.93��1.25

�0��1.25

�12.55-�20.93��2.92

��20.93�2.92-�1.25

��12.55��2.92

16 2 2100

x y zM+ −( )/ �240-�160 �160-�80 120-240 �80-120

102

∆h Mn

D( ) �30 30-110 �110

dissipated through various phenomena, such as plasticdeformation before a vessel bursts or kinetic energytransferred to the fragments. As a rough estimation, about30% of the energy stored in the compressed gas istransferred into the shock wave; about 40% provideskinetic energy to the fragments that can consequently bepropelled over great distances, while the remaining 30% isdissipated.

When the vessel that collapses also contains a liquidphase, we can distinguish two different situations, as afunction of the normal boiling point value compared to theroom temeprature, as previously discussed. When liquidtemperature (which is assumed equal to the ambient one) islower than its normal boiling point, the liquid is subcooledand it has no tendency to flash (the fast evaporation of anoverheated liquid is usually referred to as a flash). However,when liquid temperature is higher than its normal boilingpoint, the liquid is overheated (i.e. it is in a non-equilibriumcondition) and it has a strong tendency to flash. This leads tomore vapour, whose expansion usually results in a minorexplosion (since the characteristic time of a flash is notusually short enough), apart from when the liquid conditionstrigger the BLEVE phenomenon. In this case, thecharacteristic time of the phase transition is very short (ofthe order of some ms) and a serious explosion can occur. Theclassic BLEVE theory explains the increase of theevaporation rate as a consequence of the homogeneousnucleation triggering, i.e. the formation of bubbles in thefluid bulk without any nucleation centre that are usuallyprovided by the vessel surface roughness. Homogeneousnucleation leads to a much faster flash and, consequently, toa much stronger shock wave. Moreover, when the vesseldoes not collapse completely but simply breaks (as is theusual case), resulting in fast depressurization, the rapidevaporation due to the homogeneous nucleation does notallow the vapours to exit without pressurizing the vessel,which consequently collapse propelling their fragmentsaround.

Homogeneous nucleation can only happen when theliquid is sufficiently overheated. It is not enough for theliquid temperature to be higher than its normal boilingpoint, but it must also be higher than another limit value,which characterizes each compound. As a consequence, inorder to have a BLEVE, as a first necessary condition theliquid must be overheated, i.e. rapidly lead to a nonequilibrium condition characterized by a temperaturevalue much higher than its normal boiling point. The shorttime for overheating the liquid is a necessary condition inorder to avoid a slow evaporation of the liquid. In thiscase, the liquid would reach equilibrium conditions notinstantly (i.e. with a short characteristic time necessary tocause an explosion) but gradually (i.e. with a longcharacteristic time not suitable for generating anexplosion). The fast overheating of a liquid can be due to afast depressurization of a gas liquefied by compression(e.g. following the collapse of an LPG, LiquifiedPetroleum Gas, tank), but also following the contact of alow-volatile hot liquid with a more volatile cold one.Typical examples are the accidental contact of water withmelting metals or salts, or of LNG (Liquified Natural Gas)with water. For the first two cases, water is the morevolatile cold liquid that can vaporize producing anexplosion, while in the latter case it is LNG.

The second necessary condition to have a BLEVE is tooverheat the liquid enough, i.e. the liquid temperature mustbe higher than the threshold value for the homogeneousnucleation triggering. This means that when two liquidscome into contact, the temperature of the hotter liquid mustbe higher than the critical temperature for homogeneousnucleation triggering of the more volatile cold liquid. In thecase of fast depressurization of a liquefied gas, itstemperature must be higher than its critical temperature forhomogeneous nucleation triggering when depressurizationstarts. The critical temperature value for homogeneousnucleation triggering, TSL, as a function of pressure can beestimated for hydrocarbons as TSL�TC (0.11 PR�0.89)where TC is the critical temperature in K and PR the reducedpressure, i.e. the ratio of pressure to critical pressure. Atatmospheric pressure, the previous relation is wellapproximated by TSL�0.895 TC. For hydrocarbon mixtures,the value of TSL can be approximated by a weighted averageon the mole fractions of the pure species critical temperaturevalues for homogeneous nucleation triggering.

The second condition, i.e. sufficient overheating, isshown in Fig. 3 for an accident involving the heating of avessel containing a liquefied gas. In the figure, the vapourpressure and the threshold overheating for triggering thehomogeneous nucleation are shown. Let’s assume that theliquefied gas in the vessel is characterized by point 1 whichlies on the vapour pressure line since the liquid is inequilibrium with its vapour at room temperature, TA, at apressure value equal to its vapour pressure at thattemperature, PV(TA). In this case, an accidental heating of thevessel can lead the liquid conditions (represented by a pointon the vapour pressure line, since vapour-liquid equilibriumconditions prevail inside the vessel) up to point 2. If thevessel collapses under these conditions, the pressuresuddenly drops to the atmospheric value, PA, represented bypoint 3. Liquid is then in a non-equilibrium condition sinceits temperature is higher than its normal boiling point, andconsequently it will flash. The homogeneous nucleationphenomena cannot be triggered since the liquid overheatingis not enough. However, if the vessel collapses when theliquid temperature (and pressure) has reached point 4, therapid depressurization leads the liquid to point 5, fartherthan the homogeneous nucleation boundary. In this case, aBLEVE is possible.



pres

sure

temperature

vapourpressure line

overheatingline

criticalpoint

TA

12

4

3 5PA

PV(TA)

Fig. 3. Vapour pressure and critical overheating for homogeneous nucleation lines.

7.2.4 Gas or dust explosions

Gas or dust explosions (note that when the diameter of theparticles is small enough, dusts behave like gas) areexplosions involving a combustion chemical reaction in theenergy release mechanism, such as Unconfined VapourCloud Explosions (UVCE) or confined deflagrations insidebuildings, pipelines, or process units.

In order for a cloud of vapours to explode, a largeamount of flammable vapours must be dispersed into theatmosphere before finding an ignition source. In this case,the vapour cloud can create a significant amount of fuel-airmixture with concentration inside the flammability range.The reason why it is necessary that a significant amount offuel should be inside the flammability range (making itimpossible for small flammable gas releases, say lower than1,000 kg, apart from very reactive compounds such ashydrogen, acetylene or ethylene oxide to explode, as well asfor clouds ignited immediately after the release) can beexplained by considering how the flame propagation inside aflammable mixture (in the following, such a flame will alsobe referred to as a combustion wave) can generate a pressurewave. Such a mechanism is similar again to an acceleratingplug. Let us consider the analogy between a pipe filled witha flammable mixture ignited at one end and the plugmovement previously discussed, as shown in Fig. 4. When thepipe is open on both the sides, the burned gases (that are at amuch higher temperature than the unburned ones andconsequently have a much lower density) expand and exitfrom the pipe end close to the ignition point. The flame (i.e.the combustion wave) propagates inside the still unburnedgases with a velocity which is characteristic of thisflammable mixture. However, when the pipe is closed at theend near the ignition point, the expanding burned gasescannot exit the pipe from this end and, consequently, theymove towards the unburned gases pushing them. The burnedgas expansion acts as a plug pushing the unburned gases andgenerates a pressure wave. A cloud of flammable vapoursignited at the centre acts similarly, with the only differenceof the hemispherical geometry of the accelerating plug.

In order for the pressure wave to reach significantvalues, it is necessary that the combustion wave acceleratessignificantly so as to allow for the formation of a shockwave as a consequence of the superimposition of thepressure waves generated by an accelerating plug, aspreviously discussed. In other words, since the volume in

which the energy is released is not negligible, it is necessarythat the characteristic time of the energy release is shortenough.

When the unburned mixture is motionless, the flamefront velocity compared to the unburned gases not onlydepends on the physical and chemical characteristics of theflammable mixture, but also the composition, ambientpressure and room temperature. Flame front propagationinside a flammable mixture is, in fact, determined by theheat and radical species transport from the high temperatureburned gases to the low temperature unburned ones, asshown in Fig. 5. When the flame propagates into a motionlessmixture, the mass and heat transport phenomena aredetermined by the effective transport coefficients (mass andheat diffusivity) according to the Fick and Fourier laws.Consequently, the flame propagates with a well definedspeed that is usually of the order of 0.5 m/s for ahydrocarbon-air mixture slightly richer in fuel compared tothe stoichiometry. Some of these velocity values, usuallyreferred to as burning velocity, are summarized in Table 3,from which it is evident that the more relevant exceptions tothe aforementioned typical value refer to very reactivecompounds, such as hydrogen or acetylene. These velocityvalues, as previously mentioned, are relative to the unburnedgases in front of the flame. For motionless gases, suchvelocity values correspond to the flame front velocity inrelation to a fixed observer; this situation is depicted in thediagram of Fig. 4, referring to a pipe with both ends open. Inthis case, the unburned gases do not move and the flamepropagates with a velocity, in relation to an observer unitedwith the pipe, equal to the burning velocity.

When the pipe is closed at one end, or the ignition sourceis inside the cloud, the burned gases cannot discharge andconsequently they push the unburned gases away from theflame front. Flame propagates with a velocity compared tounburned gases, which is always equal to the burningvelocity, but since the unburned gases move in relation to afixed observer, the flame front velocity in relation to such anobserver is larger than the burning velocity. This velocity iscalled flame speed. If the fluid dynamic conditions of theflammable mixture in front of the flame are laminar, thevelocity of the flame only depends on the physical andchemical characteristics of the flammable mixture and iscalled laminar flame velocity. The burning velocity, S0, andthe laminar flame velocity, Sf , are related to each other bythe expansion factor, E, defined as the ratio of the burned to



piston

vp vpw

vpw

burned gasescan discharge

burned gases cannotdischarge:

they act as a piston vcw

vcw

v�0

v�0

v�0

pressurewave

combustionwave

combustionwave

burned

burned

unburned

unburned

pressurewave

Fig. 4. Combustion wavepropagation in a duct (vp is the velocity of the piston;vpw is the velocity of the pressure wave;vcw is the velocity of the combustion wave).

the unburned volume, E�V2�V1, which is roughly equal to 8for hydrocarbon – air mixtures:

[3]

This relation holds true as far as the flammable mixtureis initially still, fluid dynamic conditions are laminar, flamesurface is regular and burned gases are trapped inside theflame front.

Even in laminar conditions, the velocity of the flamefront can be larger than that predicted by the previous

relation due to the presence of flame wrinkling that increasesthe flame front surface, generated by both flame frontinstabilities and by the flame front passing over someobstacles. Since the burned gas production rate isproportional to the flame front area, the flame speedincreases and can be evaluated as:

[4]

where Af is the real flame front area and A0 the regularshaped flame front area (e.g, planar or hemispherical).

In turbulent conditions (with reference to unburnedgases), the velocity of the flame front not only depends onthe physical and chemical characteristics of the flammablemixture, but also on the fluid dynamics that defines theturbulence level of the flammable mixture where the flamepropagates. In the case of flames that propagate, therebypushing the unburned gases in front of them, increasing theunburned gases velocity inevitably leads to an increase intheir turbulence level. The presence of obstacles in theunburned flow creates wakes that in turn further increaseturbulence. The effect of the unburned turbulent flow istwofold: the larger vortexes induce a flame wrinkling, thusincreasing the flame front area and, consequently, leading toa flame velocity increase through the aforementionedmechanism, while the smaller vortexes induce an increase ofthe mass and heat transport rates from the flame to theunburned gases. The value of the effective heat and masstransfer coefficients (turbulent mass and heat diffusivity) isin fact much larger than the analogous molecular value. Thisincrease of the mass and heat transport rate results in asimilar increase of the flame velocity, which in turn leads toa further increase of the unburned gases velocity, theirturbulence level and, through a feedback mechanism, afurther increase of the flame front acceleration and, finally,the generation of larger pressure waves.

In conclusion, as previously mentioned, in order for agas or dust explosion to generate significant overpressure, asubstantial flame front acceleration is required. For this tohappen, on one side there must be some obstacles to increasethe unburned turbulence level, and on the other side, thecloud must be large enough so that the mechanisms forflame front acceleration can become effective. If the flamefront acceleration becomes really strong, the shock wavegenerated can heat the unburned gases so much that it ignitesthem. In this case, the combustion wave, through theaccelerating plug mechanism, no longer generates thepressure wave propagating in front of it, but it is the shockwave that ignites the flammable unburned mixturegenerating the combustion wave. The energy released by thecombustion allows the shock wave to sustain itself.

These two phenomena are markedly different. In the firstcase (combustion wave that generates the pressure wave), thetwo waves (the pressure and the combustion one) areseparated and the combustion wave velocity is subsonic (ofthe order of some m/s) compared to the unburned gasconditions in front of the flame. The overpressure valuesacross the pressure wave are quite small (lower than a fewbar, 1 bar being the maximum for UVCE), and pressure anddensity reduces through the combustion wave. Thisphenomenon is called deflagration and the relatedphenomenology is illustrated in Fig. 6. Also the pressure peakshape experienced by a target at a given distance from the

S ESAAf

f=0

0

S ESf =0



conc

entr

atio

n, te

mpe

ratu

recold

reactantregion

reactants

temperatureT0

Ti

Tf

intermediates

preheatingregion

reactionregion

productregion

visibleflameregion

Fig. 5. Diagram of a flame front propagating in a flammable mixture.

Table 3. Maximum burning velocity,S0, and maximum laminar flame velocity,

Sf, for some flammable gases

Fuel S0 (m s�1) Sf (m s�1)

Acetylene 1.58 14.2

Benzene 0.62 4.9

Butane 0.50 3.7

Butene 0.57 4.3

Ciclohexane 0.52 4.1

Heptane 0.52 4.0

Hexane 0.52 4.0

Ethane 0.53 4.0

Ethene 0.83 6.5

Hydrogen 3.50 28.0

Methane 0.45 3.5

Pentane 0.52 4.0

Propane 0.52 4.0

Propene 0.66 5.1

vapour cloud centre is different from that previouslydiscussed for ideal explosions and illustrated in fig. 2. Inparticular, the pressure gradient is less sharp, as shown inFig. 7.

In the second case (pressure wave that generates thecombustion wave), the two waves (the pressure and thecombustion one) are coupled and their velocity is supersonic(in the order of some thousands of m/s) compared to theunburned gas conditions in front of the flame. Theoverpressure values across the pressure wave are quite high(a few tens of bar), and pressure and density increasethrough the coupled pressure and combustion wave. Thisphenomenon is called detonation and the relatedphenomenology is illustrated in Fig. 8. Detonations of vapourclouds are not common and they require, for common fuels,the ignition by a detonator, i.e. by the blast wave generatedby a condensed phase explosion.

The difference between the two phenomena(deflagration and detonation) will be discussed furtherbelow. It is interesting to note that the existence of twodifferent kinds of combustion wave can be easily deducedfrom the mass and energy balance equations across acombustion wave using a reference system united with theflame front, as shown in figure Fig. 9. Here, a onedimensional flat combustion wave moving through aconstant section duct is illustrated. With reference to acoordinate system united with the duct, the combustionwave moves towards the right (that is, into the stillunburned gases) at a constant velocity equal to u1;whereas in the case of a reference system united with theflame front, the flame front itself is still and the unburnedgases move towards it with a velocity equal to u1, Usingsuch a reference system and labelling it with the subscript:

1 the unburned gases, and the subscript: 2 the burnedgases, the steady state mass and momentum balanceequations become:

[5]

[6]

These equations give

[7]

[8]

From relation [7] (usually referred to as the Rayleighline, since it represents the equation of a straight line on theplane P2 vs. 1�r2 as will be discussed in section 7.2.5), thefollowing inequality can be deduced:

[9]

from which we can see that pressure and density change inthe same direction through the flame front. From relation[8], it follows that when the pressure increases, velocitydecreases. A first kind of combustion wave is thuscharacterized by an increase of both pressure and densitythrough the flame front: these compression waves arecalled detonations. In this case, the burned gas velocity inrelation to the flame front is lower than the unburned gases.The second kind of combustion wave is characterized by adecrease of both pressure and density through the flamefront: these are expansion waves called deflagrations. Theburned gas velocity in relation to the flame front, in thiscase, is larger than the unburned gases. As previouslydiscussed, in the case of detonations, combustion andshock waves are coupled, and through them pressureincreases. In the case of deflagrations, the two waves(combustion and shock waves) are uncoupled: pressureincreases through the shock wave, whereas it decreases

P P u2 1

2 11 1

22 1 0−

−= ( ) >

r rr r r

P Pu u

u2 1

2 11 1

−−

= −r

P P u2 1

1 21 1

2

1 1−−

= ( )r r

r

P u P u2 2 22

1 1 12+ = +r r

r r2 2 1 1u u=



vpwvcw v�0

combustionwave

burnedunburned

pressurewave

Fig. 6. Conceptual scheme of a deflagration.

vpw v�0

combustionwave

burnedunburned

pressurewave

Fig. 8. Conceptual diagram of a detonation.

pres

sure

timetD

P°

0

Fig. 7. Overpressure vs. time at a given distance from the epicentre of an unconfined vapour cloud explosion (UVCE).

burned gasP2T2u2M2

unburned gasP1T1u1M1

flamefront

reference system fixedwith regard to the flame front

Fig. 9. Diagram of a combustion wave according to an observerfixed with regard to the flame front (M, Mach number).

through the combustion wave. The pressure decreasethrough the flame front is usually very small: the ratio ofthe burned to unburned gas pressure for hydrocarbons-airmixtures is usually close to 0.98.

Transition from deflagration to detonation (DDT,Deflagration to Detonation Transition) requires a very highacceleration of the flame front that is not usually reached infree field explosions (UVCE). On the other hand, thisphenomenon can arise in duct explosions, due to the largeturbulence generated by the unburned flow in ducts.

The mechanism leading to the pressure rise when aflammable gas or dust mixture is ignited in a confinedenvironment is completely different from that previouslydiscussed for unconfined explosions. For unconfinedexplosions, the combustion proceeds at almost constantpressure. Due to the higher temperature of the burned gasescompared to the unburned ones (about 8 times higher forstoichiometric hydrocarbon-air mixtures), the burned gasvolume increases (about 8 times for stoichiometrichydrocarbon-air mixtures, from the ideal gas law) and apressure wave can be generated through the acceleratingplug mechanism.

When the deflagration is completely confined, thecombustion proceeds at constant volume and the temperatureincrease inevitably results in a pressure increase (about 8times for stoichiometric hydrocarbon-air mixtures, from theideal gas law). In this case, the characteristic time of thephenomenon (i.e. the flame front velocity) does not play anyrole in the maximum pressure value, which is determined bythe constant volume constraint.

If the plant unit where deflagration occurs is not fittedwith emergency release systems (explosion panels or rupturedisks), the pressure increase can lead to the collapse of theunit with consequences similar to those previouslydiscussed. On the contrary, if the explosion panels arecorrectly designed and the unit is designed to sustain thedeflagration, the unit will not collapse and the burned andunburned gases will be discharged through the explosionpanels. Obviously, rupture disks or explosion panels areeffective (when correctly designed) only for protecting anapparatus from deflagrations since in a detonation the shockwave travels at supersonic speed; it consequently reaches thevessel walls before the information that pressure in thevessel is increasing, which propagates at the speed of sound,can reach the emergency system.

7.2.5 Hugoniot relation

The mass [5] and momentum [6] balance equations can becomplemented by the steady state energy balance equation(referring to the reference system illustrated in Fig. 9):

[10]

and by an equation of state for the burned gases, that forideal gas is

[11]

where r is the molar density. In these relations, M is themolecular weight, CP is the specific heat at constantpressure, while q is the massive enthalpy difference betweenunburned and burned gases, i.e. the heat released. If the

thermodynamic conditions of the unburned gases (T1, P1 andr1) are known, the unknown variables involved in the fourequations [5], [6], [10] and [11] are the burned gas statevariables (T2, P2 and r2) and the burned and unburned gasvelocity in relation to the combustion wave (u1 and u2). Intotal, there are 5 unknown variables in 4 equations and thesystem cannot be solved without giving the value of oneunknown.

The four equations in five unknown ones can be recast inone equation with two unknown ones, P2 and r2 with theRankine-Hugoniot relation:

[12]

The curve representing this relation for a given value of q ona P2 vs. 1/r2 diagram (or, since the values of P1 and r1 aregiven, on a P2/P1 vs. r1/r2 diagram) is called the Hugoniotcurve and is shown in Fig. 10.

In this figure, two curves are shown: the first one forq�0 refers to adiabatic shock waves (Rankine-Hugoniotcurve), while the second one for q�0 belongs to a familycharacterized by different q values (Hugoniot curves).Hugoniot curves are the locus of all the possible values of P2and 1/r2 for given values of P1 and 1/r1 and a given value ofq. The point (P1, 1/r1), characteristic of unburned gases, isusually called the origin of the Hugoniot diagram and it isindicated in the figure with the symbol O. From this point,all the Rayleigh lines [7] pass, which represent a conditionthat must be fulfilled by the burned gases state variables.The burned gases state will consequently be represented bythe crossing of one Hugoniot curve with one Rayleigh line.The figure shows two Rayleigh lines that are tangent to theHugoniot curve shown on the diagram. The two tangentpoints are usually called the Chapman-Juoguet (C-J) points,upper (labelled U in the figure) and lower (labelled L in thefigure) for that tangent to the upper or the lower branch ofthe Hugoniot curve. The intersections of the Hugoniot curvewith the horizontal and vertical straight lines through theorigin identify two other important points, labelled X and Yin the figure. These four points identify five significantregions on the Hugoniot curve, labelled in the figure withthe Roman numbers from I to V. In the following, only someproperties of the different branches of Hugoniot curves (ofsome practical relevance) will be discussed; moremathematical details can be found in the relevant literature.

The Hugoniot curve represents all the possible solutionsof the Hugoniot relation; however, not all of these solutionshave a physical meaning. In fact, on the V branch it isrequired that P2�P1 and 1/r2�1/r1, which violates theconstraint enforced by the Rayleigh line [7]. This regionrepresents solutions without any physical meaning. BranchesI and II represent compression waves where P2�P1: theseare detonation waves. Branches III and IV representexpansion waves where P2�P1: these are deflagrationwaves. It can be proved that in both the C-J points, the Machnumber (defined as the ratio of the velocity and the soundvelocity) in the burned gas conditions is equal to one, M2�1,i.e. the burned gas velocity compared to the combustionwave is equal to the sound speed in the burned gasconditions, u2�c2. Combining relations [7] and [8] results in:

[13]u u u2 1

2 11 11 1

−−

=r r

r

γγ −

−

− −( ) +

=1

12

1 12

2

1

12 1

1 2

P P P Pr r r r

qq

P RT M2 2

= r

C T u C T u qP P2 22

1 121

212

+ = + +



Since 1/r2�1/r1 on branches I and II, related todetonations, it follows that u2�u1, In a reference systemunited with the duct where the detonation wave propagates(shown in Fig. 11 for the same combustion wave in Fig. 9,where the positive directions for velocities are alsoindicated), the unburned still gas velocity is equal to zero,while that of the burned gases is equal to v2�vPW�u2, wherevPW�u1 is the detonation wave velocity compared to areference system united with the duct. From these relations itfollows that v2�vPW�u2�u1�u2�0; this means that theburned gases behind the detonation wave move in the samedirection as the detonation wave. Depending on the value ofv2 compared to vPW, the burned gases may or may not reachthe detonation wave.

At the C-J point u2�c2, and thereforevPW�u2�v2�c2�v2�c2: the detonation wave at this pointtravels at supersonic velocity. Moreover, although the burnedgases travel in the same direction as the detonation wave,they cannot reach it since vPW�v2.

Branch I is called the strong detonations branch wherepressure and density are higher than in the C-J point. Passingthrough the detonation wave, the gas velocity compared tothe detonation wave decreases drastically from supersonicvalues to subsonic ones (it can be proved that M2�1). Inpractice, it is quite difficult to generate a strong detonationwave even in controlled laboratory conditions and,consequently, this branch is of no significant practicalinterest.

Branch II is the weak detonation one where pressure anddensity of the burned gases are lower than that of the C-Jpoint. Passing through the detonation wave, the gas velocitycompared to the detonation wave decreases but still remainssupersonic (it can be proved that M2�1). In practice, it isquite difficult to generate a weak detonation wave since itrequires very reactive gases and, consequently, this branch isalso of limited practical interest.

Point X, which is characteristic of constant volumedetonations (1/r2�1/r1), would require a detonation wavevelocity (with respect to a fixed reference system) equal toinfinity, as deduced from the Rayleigh line together with theprevious relations. Since such a velocity cannot be achieved,point X also represents a meaningless physical system state.

In conclusion, most of the detonation waves lie on theC-J point, while branches I and II represent meaningless

regions from a practical point of view. This allows the valueu2�c2 to be assigned to one of the five variables of theproblem. The system of 4 equations [5], [6], [10] and [11]can therefore be solved giving the values of the 4 variables:T2, P2, r2 and u1. It should be stressed that in this case, theproblem can be solved without any information on thechemical reaction rates that transform, in the flame frontfollowing the detonation wave, unburned gases into burnedones providing the energy required by the detonation wave tobe sustained. This is a consequence of the supersonic speedof the detonation wave, which cannot obtain informationfrom the flame front that follows it.

Branch IV is known as strong deflagration. Whilepassing through the deflagration wave, the unburned gasvelocity compared to the deflagration wave must drasticallyaccelerate from subsonic to supersonic values (it can beproved that M2�1). It can also be proved that this is notpossible in constant section ducts, and therefore a strongdeflagration wave (including the C-J point) has never beenobserved in practice.

Branch III is known as weak deflagration. In this region,P1�P2�PL and 1/r1�1/r2�1/rL. The burned gas velocitycompared to the deflagration wave increases, u2�u1, but it isstill subsonic. In this case, using a reference system unitedwith the duct and following the same procedure used for thedetonation waves, seeing as the unburned gases are motionlessand the velocity of the deflagration wave is equal to vPW�u1,it can be proved that v2�vPW�u2�u1�u2�0; this means thatthe burned gases behind the deflagration wave move in the



P2

l/r2l/r1

P1O

X

YHugoniot curve, q�0

Rankine-Hugoniot curve, q�0

branch IV, strong deflagrations: M1�1 and M2�1

branch III, weak deflagrations: M1�1 and M2�1

branch II, weak detonations: M1�1 and M2�1

branch I, strong detonations: M1�1 and M2�1

branch V, not possible

Rayleigh straight lineunburned gas condition

lower Chapman-Jouguet point: M2�1

upper Chapman-Jouguet point: M2�1

M2��

M2�0

L

U

Fig. 10. Rankine-Hugoniotdiagram.

burned gasP2T2v2

unburned gasP1T1

v1�0

detonationwave

reference system fixedwith regard to the duct

vPW

Fig. 11. Diagram of a detonation wave according to an observerfixed with regard to the duct in which detonation propagates.

opposite direction compared to the deflagration wave itself,which is another important difference between deflagrationand detonation waves. Deflagration waves lying in region IIIare often observed in practice. However, in this case, there isno condition like the C-J one for detonation waves that allowsthe last degree of freedom in equations [5], [6], [10] and [11]to be saturated and therefore to solve the problem withoutproviding any information on the flame front. The solution tothe problem (or, in other words, the definition of the slope ofthe Rayleigh line for a given system) requires a detaileddescription of the flame front, which is the goal of severalmodels developed for predicting the burning velocity. Athorough discussion of such models is outside the aim of thistopic and can be found in the dedicated literature.

Point Y, which is characteristic of the constant pressuredeflagrations (P2�P1), requires a nil deflagration wavevelocity (compared to a fixed reference system), asdeducible from the Rayleigh line together with the previousrelations, and it is consequently meaningless from a practicalpoint of view.

7.2.6 Estimation of overpressurecaused by an explosion

An explosion can be in a steady state or not; a steady state(or self-sustained) explosion requires the deflagration ordetonation wave to travel inside a flammable mixture, whosecombustion is able to sustain the combustion wave motionmaking the phenomenon stationary. When the combustionwave reaches the flammable mixture boundary, the shockwave propagates in the atmosphere and gradually dissipatesits energy. Similarly, the shock wave generated by the pointrelease of a given amount of energy is transitory, and theoverpressure generated decreases by increasing the distancefrom the energy source point. It is therefore important toknow the shock wave velocity (and thus its location after agiven time) as well as how the overpressure graduallydecreases as the shock wave moves away from the epicentre.

Rankine-Hugoniot relations [7], [8] and [10], togetherwith some relations for the ideal gas as well as the massbalance [5], can be shown in the following equivalentrelations:

[14]

[15]

These relations show how the shock wave intensity (i.e.the value of the shock wave overpressure, P2) increases withM1, i.e. with the shock wave velocity. In the limit conditionfor which M1�

�1 , we obtain P2��P1, i.e. the pressure wave

vanishes. In the limit condition for which M1��1 (andconsequently the shock wave overpressure is much higherthan the atmospheric value), the following approximaterelations can be deduced:

[16]

[17]

from which we can see that for supersonic shock waves, theratio of the densities before and after the shock wave is ofthe order of unity, while that of the pressure values is muchlarger than unity.

Through order of magnitude considerations on thevarious energy terms, it is possible to derive an importantrelation between the energy released by an ideal explosion,q, and the distance, R°(t), where the shock wave generatedby such a release of energy arrives after a given time t. In thefollowing relations, the superscript ° means shock waveconditions, subscript 1 means still atmosphere in front of theshock wave, and subscript 2, gas behind the shock wave.After a given time t, the energy released by the explosionwill be of the same order of magnitude of the energy of theshock wave computed as internal and kinetic energy of thegases behind the shock wave. In other words, the explosionenergy is used to heat and to move the still atmospherethrough which the shock wave propagates. The volumebehind the shock wave is of the order of V2�R°3, andtherefore the mass of gas behind the shock wave (which hasbeen heated and moved by the shock wave itself) is of theorder of m2�r1R°3. The fluid velocity behind the shockwave is of the same order of magnitude as the radial velocityof the shock wave, that is of the order of v2�dR°�dt�R°�tand consequently the kinetic energy Ek of the moving gas isof the order of Ek�m2v2

2�r1R°5�t2. The specific internalenergy eu of an ideal gas is of the order ofeu�CVT�CVP�(rR)�P�r(g�1)�P�r and therefore theinternal energy per unit volume of the gases behind theshock wave, also using relation [17] and the equation givingthe sound speed for an ideal gas, is of the order ofr2eu2�P2�P1M1

2�P1v2PW �c1

2�P1v2PW �(P1�r1)�

�r1v2PW�r1(dR°�dt)2�r1(R°�t)2. The internal energy Eu

behind the shock wave is therefore of the order ofEu�r1(R°�t)2R°3�r1R°5�t2, which is of the same order ofmagnitude as the kinetic energy. The explosion energy mustbe of the same order of magnitude, that is

[18]

and therefore

where k is a constant of unity order. This means that after agiven time, t the shock wave has reached a distance equal to

[19]

This relation provides two items of information: how thevariables change behind the shock wave and a functionalrelation between the distance travelled by the shock wave,the overpressure and the explosion energy. The first item ofinformation (known as the Sedov-Taylor solution of theshock wave) can be obtained by solving the details of theflow field behind the shock wave; this can be done bysolving the partial differential equations of the radial fluidflow, which are made up of the continuity equation, theEulero equation in spherical coordinates and the energyconservation equation for an adiabatic flow, that isP�rg�cost (assuming that the wave motion has a sphericalsymmetry and that the atmosphere behaves like an ideal gaswith g�cost). These equations can be integrated once theboundary conditions are known, which is not a trivial taskfor explosion problems. However, it is possible to assume

R qk

t° =

r1

1 52 5

//

q k R t= r1

5 2° /

q R t≈ r1

5 2° /

PP

M2

1

122

1=

+γγ

r

r1

2

2

1

11

= = −+

uu

γγ

PP

M2

1

122

111

=+

− −+

γγ

γγ

r

r1

2

2

1 12

11

21

1= = −+

++

uu M

γγ γ



that after a given time, all the details concerning the energyrelease, provided it can generate a shock wave, becomeirrelevant. This implies that the shape of the pressure,density and velocity profiles does not depend on time, andthe solution of the previous equation system is self-similar.From a mathematical point of view, this means that thepressure, density and velocity trends behind the shock wavecan be computed as a characteristic value (e.g, that on theshock wave) times a universal function of a dimensionlesscoordinate x�r�R°(t). In order to compute the variableprofiles behind the shock wave, further information isrequired: the shock wave position as a function of time,R°(t). More precisely, knowing the functional behaviourgiven by equation [19], R°�t2/5, suffices. The differentialequations system that derives from it is very complex, butcan be integrated analytically.

From a practical point of view, the second item ofinformation that can be deduced from relation [19] is moreimportant, i.e. the functional relation between the distancetravelled by the shock wave, the overpressure and theexplosion energy. Inserting the time relation that derivesfrom P2�r1(R°�t)2 into equation [19], it follows:

[20]

or, in other words,

[21]

where the more common symbol P° has been used toindicate the overpressure P2 on the shock wave. This relationshows how the pressure values at a given distance from thedetonation of a given amount of condensed explosive mustlie on a straight line on a log-log plot that showsoverpressure vs. the ratio R°�3��

333

q , R° being the distance fromthe epicentre and q the amount of energy released by theexplosion (or, equivalently, the mass of explosive). Thisjustifies the Hopkinson-Cranz scaling rule used toextrapolate experimental measurements concerningcondensed phase explosion to larger scales. This scaling lawstates that the explosion of different amounts of the sameexplosive, the same geometry and in the same atmospherewill produce similar shock waves at the same scaleddistance. By similar shock waves, we mean two shock waveswith the same overpressure but with an impulse scaledthrough a suitable factor. The most used scaling factor is thatproposed by Sachs, stating that the overpressure scaled bythe ambient pressure,

33

P�P°�P1, and the dimensionlessimpulse as,

33

I�(iPc1)�(q1/3P1

2/3), are a universal function ofthe scaled distance,

33

R�(R°P11/3)�(q1/3). As far as the

overpressure vs. distance trend is concerned, this scaling lawfollows the qualitative behaviour predicted by equation [21].

Changes in atmospheric pressure values (and,consequently, also in the sound speed in the atmosphere, c1)are of minor relevance; thus, simpler relations are often usedin practice to scale impulse and distance in which only thedependence on the amount of energy released appears, suchas33

I�iP�q1/3 and33

R�R°/q1/3. Experimental data referring to overpressure generated by

condensed phase explosions lie in a straight line on a log-logplot as required by this scaling law, as shown in Fig. 12 forTNT. For a given explosive, it obviously does not matter if

the amount of energy released, q, or the amount of explosive,W are used in the scaling laws, since these two values areproportional through the decomposition energy, which isequal to 4,437-4,765 kJ/kg for TNT.

Explosion effects from condensed phase explosionsThese explosions are very similar to ideal explosions. A

simple and effective method to foresee the effects of thiskind of explosion assumes that different condensed phaseexplosions releasing the same amount of energy willproduce similar effects. This allows the explosion of anequivalent amount of any condensed phase explosive to beconverted, from the standpoint of released energy, into TNTwith the simple relation:

[22]

Therefore, the equivalence ratio between the amount ofTNT and a given compound, a, is simply given by the ratiobetween the decomposition enthalpies of the twocompounds:

[23]

The expected consequences can be estimated using theequivalent amount of TNT together with the informationshown in the diagrams of Fig. 12.

The main uncertainty factor regarding the decompositionof unstable compounds accidentally accumulated in aprocess plant is the estimation of the amount of unstablecompound that participates in the shock wave formation: thetotal amount present does not always decompose so quicklyas to fully contribute to the explosion.

Explosion effects from unconfined vapour cloudexplosions (UVCE)

Estimating the overpressure generated by a UVCErequires the knowledge of some key parameters; amongthem, the mass of flammable gas involved in the explosionand the ignition point. To estimate the mass of flammablegas involved in the explosion, atmospheric dispersionmodels can be used which give the amount of flammablegas in the cloud with concentration inside theflammability range. A short cut method assumes thatabout 10% of the mass released lies in the flammableregion; the effect of such an approximation is obviouslyquite different depending on the meteorological andrelease conditions. On the other hand, although theinfluence of the ignition point can be relevant, it is notusually considered in the simulation models since it ispractically unpredictable.

TNT equivalent model. This method approximates thegas cloud deflagration effects to those of an ideal TNTdetonation. As previously discussed, the two phenomena aremarkedly different. On the other hand, how to estimate theeffects of the detonation of a given amount of TNT is wellknown and supported by much experimental evidence;consequently, the use of this simple approach as a firstapproximation is quite common.

The application of this method to UVCE requires theevaluation of the amount of TNT which is equivalent, fromthe amount of energy released standpoint, to the mass of gas

α = =W

Wh

hTNT

compound

D compound

D TNT

∆∆

,

,

W Wh

hTNT compoundD compound

D TNT

=∆

∆,

,

ln constantP Rq

° = − °

+31 3

ln/

R qk

RP

° °≈

r

r

1

1 5

1

2

1 22 5

/ //

tthat is R P

q°3

2 = constant



in the cloud. This estimate can be made with a relationsimilar to equation [23]:

[24]

In this case, the equivalence ratio between the amount ofTNT and flammable gas, a, is no longer equal but simplyproportional, through the parameter e (i.e. an explosionefficiency), to the ratio of the gas combustion enthalpy to theTNT decomposition enthalpy. The ratio between the energyreleased by burning 1 kg of flammable gas and that releasedby the decomposition of 1 kg of TNT is equal to about 10 formany hydrocarbons.

The uncertainties related to the use of this method arenot so much linked to the amount of flammable gas in thecloud, but mainly to the estimation of the parameter ε; thissummarizes all the differences between an ideal TNT andUVCE explosion, and can only be estimated by analyzingthe consequences of past real accidents. For such accidents,the amount of gas inside the flammability range when thecloud was ignited is not usually known, and consequently theexplosion efficiency is usually estimated based on the wholemass of gas released. Unfortunately, the efficiency value isaffected by many parameters which are characteristic of thesingle accident (meteorological or release conditions, etc.),resulting in a wide variability of the explosion efficiency.This is shown in Fig. 13 which gives the values of theequivalence ratio, a, estimated assuming an enthalpy ratioequal to 10.

It should be noted that the central value of thedistribution is equal to about 0.3 (an explosion efficiencyequal to e�0.03), even if some higher values haveoccasionally been recorded. However, 97% of the data show avalue lower than 1. Instead of considering the whole mass ofgas released, let’s look at an atmospheric dispersioncomputation carried out to estimate the real amount of gasinside the flammability range. From such an amount, you canobtain an equivalent amount of TNT multiplying it by 3instead of 0.3 on the basis of the rough assumption that about10% of the gas released is inside the flammability range.

This method cannot be used close to centre of theexplosive cloud since it would result in unrealistic highoverpressure values. A common modification limits themaximum overpressure value for a UVCE to 1 bar andconsequently modifies the TNT curve as shown in Fig. 14,where the amount of explosive refers to the gas released.Assuming there is an explosion efficiency equal to 10%and a ratio between the gas and TNT enthalpies equal to10, the amount of gas released and equivalent TNT areobviously the same (in other words, the equivalence factor,a, is equal to 1); whereas if there is a lower efficiency (say3%, as shown in the figure), the curve will be moved tothe left.

Wiekema model. This method assumes that theunconfined vapour cloud deflagration is similar to anexpanding hemispherical plug moving at a given averagevelocity which is defined, with reference to Fig. 15, as theratio of the burned gas cloud radius to the deflagration time.High average deflagration velocities characterize highreactive compounds or compounds whose flame velocity isvery sensitive to turbulence induced accelerations (forinstance, ethylene oxide), while the opposite is true for lowvalues of the average deflagration velocity (for instance,methane).

Solving the mathematical model simulating anaccelerating hemispherical plug with different averagevelocities, various correlations (one for each averagevelocity) can be obtained between the relative dimensionlessoverpressure (compared to the atmospheric value,[P°�PA]�1) and the dimensionless ratio of the distance to thecubic root of energy in the cloud, R�L0�R�3��

1233333333333

V1Dhc�PA

333333

. In thisrelation, V1 is the gas volume which is initially inside theflammability range, while DhC�3.5 106 J/m3 is an averagevalue of the energy released by the combustion of one cubicmetre of a stoichiometric hydrocarbon-air mixture.According to the previous discussion concerning idealexplosions, Fig. 15 gives an almost linear behaviour on alog-log plot of the overpressure vs. the ratio of the distanceto the cubic root of the explosion energy, which isrepresented by V1DhC.

α ε= =WW

hh

TNT

gas

C gas

D TNT

∆∆

,

,



A B

0.01 0.1 1

1

210 102

10�2

102

103

10�1

101

1

1

10�1

10�2

10�3

10�4

10�5

10�6

0.01 0.1 1 10 10210�2

102

103

10�1

101

P_

R_

(m/kg1/3) R_

(m/kg1/3)

P_

I_ I

_

tA_

tA_

tD_ tD

_

1

1

10�1

10�2

10�3

10�4

10�5

10�6

_P

_P

_ _I, tA, tD

_ _I, tA, tD

Fig. 12. Scaledoverpressure,

33

P�P°/P1,scaled impulse,

3

I�iP /W1/3

(Pas/kg1/3), scaled arrival timeof the shock wave,

3

tA�tA/W1/3

(s/kg1/3), and scaled positivephase duration of the shockwave,

3

tD�tD /W1/3 (s/kg1/3),vs. scaled distance,33

R�R°/W1/3 (m/kg1/3),for TNT explosions. A, freeatmospheric explosions(spherical symmetry); B, ground explosions(hemispherical symmetry)(adapted from Lees, 1996).

This method gives four curves delimiting three ranges,each of which is characteristic of a given range of averagedeflagration velocity. The main novelty introduced by thismethod compared to the equivalent TNT one is theclassification of several flammable compounds as a functionof their reactivity and tendency to accelerate their flamefront into three reactivity categories: low, medium, and high.This method associates a range of overpressure as a functionof the distance (in Fig. 15) with each reactivity class.

In other words, this method recognizes that the energy ofthe cloud is not the only relevant parameter that defines theoverpressure generated by a UVCE, since the flame frontacceleration is the real mechanism able to generate a shock

wave. From a different point of view, this method solves theproblem beforehand of choosing the explosion efficiency inthe equivalent TNT method, which is the parameter thatsummarizes all the differences between a TNT detonationand a gas cloud deflagration. More precisely, it reduces thevariability range of such a parameter. The method actuallygives a range of overpressure for a given distance for eachreactivity class, and thus for each flammable compound. Thewidth of such a range represents the influence of thepresence of some obstacles, and consequently of theturbulence induced acceleration of the flame front, on theoverpressure value. The lower value refers to deflagrations ina low congested area, while the upper one to deflagrations inhigh congested regions. This results in a value of the ratiobetween the distances estimated with the upper and lowerlimits equal to about 2-3. Such an uncertainty is quite similarto that found in the choice of the efficiency value with theequivalent TNT method. A classification of somecompounds as a function of their reactivity for this method issummarized in Table 4.

Multi-Energy model. This method further develops theidea of the Wiekema method for which a flammable gascloud can result in a UVCE only when it is ignited in asufficiently congested area so that the flame front velocitycan significantly increase. As a consequence, not onlydoesn’t all the released gas participate in the explosion, butneither does all the gas inside the flammability range. Onlythe flammable mixture inside congested areas (or in highturbulence regions, such as those close to high velocityreleases of a flammable gas) contributes to a UVCE. Thestrength of the explosion depends on the kind of gas as wellas the degree of congestion of the area. It follows that the



num

ber

of a

ccid

ents

0.01 0.05 0.1

30%

0.5 1.0 5 10WTNT/Wgas

Fig. 13. Number of the accidents vs. the ratio of equivalent TNT amount to the released mass of flammable gas that generated the UVCE (adapted from Giesbiecht, 1988).

over

pres

sure

(ba

r)

maximum overpressureinside the cloud

cloudradius

a�30%

base line for TNT

1086

43

2

1.00.80.6

0.40.3

0.2

0.10.080.06

0.040.03

0.02

0.01

scaled distance (m/t1/3)10 20 30 50 70 100 200 500300 1,000

Fig. 14. Overpressure curve for a UVCE as a function of the scaled distance (adapted from Lees, 1996).

(P°/

PA

)�1

10�2

10�1

100

101

R/L0

unburned

burned

R1

R2

100 101

highm

edium

low

Fig. 15. Diagram for Wiekema method (adapted from Lees, 1996).

gas cloud usually results in more than one explosion, locatedin the various congested area of the plant, whose effectsmust be considered separately.

The effect of various explosions can be estimatedthrough a similar approach to Wiekema, using 10 differentaverage deflagration velocities, as shown in Fig. 16. The fulllines refer to detonations, whereas the dashed lines refer todeflagrations. The parameter values on both the axes aredimensionless as in the previous Wiekema method.

Although this approach tries to represent the UVCEphenomena more precisely, it also requires an arbitrarychoice of some parameters that have, in the final analysis,the same physical meaning as the explosion efficiency in theequivalent TNT method. In particular, it is necessary todefine which regions of a plant should be consideredcongested or not, at which distance two congested areasgenerate two separate explosions (25 m is a common butarbitrary suggestion) and finally the explosion class, rangingfrom 1 to 10.

While the identification of the congested area can beintuitive, the choice of the explosion class is moreproblematic. The value of 10 gives results similar to theequivalent TNT method (obviously limited to the gas presentin a given congested region) with an equivalence factorequal to 20% (which is close to the 30% value referred to themass of gas inside the flammable region previouslydiscussed for the equivalent TNT method). The value equalto 7 seems to be reasonable for many practical situations;values of 6 and 7 provide similar results if the overpressureis lower than 0.1 bar and the maximum pressure is equal to1 bar. For non congested areas and a still atmosphere, a valueequal to 1 seems adequate, whereas for non congested areasand a moving atmosphere, a value equal to 3 may be moreadequate.

CFD models. The availability of more powerfulcomputers as well as more efficient numerical methodsenables us to deal with the problem of solving theNavier-Stokes equations for turbulent flows in complexgeometries within the context of CFD (ComputationalFluid Dynamics).

Theoretically, these models could be considered reallypredictive since they solve the local balance equations ofmass, momentum and energy with suitable boundaryconditions defining the problem to be solved. On the otherhand, the main problem of CFD models (apart from thelarge amount of resources required, both in terms ofoperator experience and CPU performance, which limitsthe application of this approach to relatively simpleproblems) is the representation of the interaction betweenturbulence and combustion chemical reactions. This is animportant research field where no consolidated andgeneral results have been obtained yet. The variousapproaches developed are often limited to a single kind ofproblem and the predictions’ reliability of this approachneeds to be assessed by comparing it with experimentaldata. An agreement with a given set of experimental datais often obtained by introducing one or more adjustableparameters, whose value is defined by the experimentaldata in consideration. This makes the model no longercompletely predictive and limits its use tosemi-quantitatively predicting the influence of the changein some parameters (such as the geometric ones) on theoverpressure generated by the explosion.

Physical explosion effects As previously discussed, this definition summarizes

all the explosions generated by the rapid expansion of aliquid or vapour compound without involving anychemical reaction in the formation of the shock wave,such as the combustion reactions involved in UVCE. Achemical reaction could be involved during a vesselpressurization, such as in the case of a confineddeflagration able to increase the internal pressure up tovessel failure. Typically, physical explosions follow avessel collapse, with the consequent release into theatmosphere of both mass and internal energy previouslycontained in the vessel. The difference between theinternal energy of the compound in the vessel and in theatmosphere represents the maximum energy amountavailable for generating the explosion.

The shock wave is generated by the partialtransformation of the internal fluid energy into mechanicalenergy. The fluid internal energy available for the formationof a shock wave depends on the thermodynamic state of thefluid, which in turn depends on the kind of fluid and on thebursting conditions. The estimation of the internal energy ofthe fluid when the vessel collapses can be done in differentways depending on the assumed scenario: ideal or real gasexpansion, liquid evaporation (flash or BLEVE), confineddeflagration, etc.

As previously discussed for UVCE, the morecommon methods are variations of the equivalent TNTmethod. Also in this case, the predictions are lessaccurate in the region close to the vessel (up to about10-20 vessel diameters), whereas they are reasonable atlarger distances.

The use of the equivalent TNT method (or othersimilar methods, providing some corrections for theregion near the vessel) requires an estimation of the



Table 4. Classification of some compoundsaccording to the Wiekema method

Reactivity Compounds

Low

ammonia1,3-dichloropropeneepichlorhydrine methanecarbon monoxidetetra ethyl lead

Medium

acetaldehydeacetonitrileformic acidacrylonitrile1,3-butadienen-butane1-butenevinylchloridediethylaminedimethylamineethaneethenepropanepropene

Highethylene oxideacetylene

energy used for generating the explosion. Not all theavailable energy can be used for generating the shockwave: part of it is transformed into fragment kineticenergy, that can be thrown at large distances(frequently, the major hazard in the case of vessel collapseis due to the possible domino effects that canbe triggered); part of it can be used to strain the vesselbefore failure; and part of it is dissipated.The last part of the internal energy can be used to expandthe fluid and, consequently, to form a shock wavewhose characteristics depend on the expansion velocity:the larger the phenomenon velocity, the more similarthe characteristics of the generated shock waveare to those of an ideal explosion. It is difficultto quantify the amount of energy used to generatethe shock wave. Brittle fractures of a vessel give largervalues than ductile ones. A conservative approach(reasonable since the overpressure values generatedby a vessel collapse are not usually so large as to producedisastrous effects at large distances) needs to considerthat all the available energy is used to generatethe shock wave.

If the vessel content can be assimilated to an ideal gas,the difference of internal energy between the initial statebefore vessel collapse and the final one at ambientconditions can be computed as:

[25]

where the subscript 0 refers to the initial conditions in thevessel before failure, PA is the ambient pressure, V thevolume and g the specific heats ratio.

When the vessel also contains some liquid, flash orBLEVE possibility should be checked by comparing roomtemperature value with the normal boiling point and thecritical temperature for homogeneous nucleation. If theliquid cannot flash, there is only some vapour expansion. Ifit can produce a BLEVE, the expansion of the evaporatedliquid also contributes to the shock wave, whereas if itflashes, the expansion of the evaporated liquid cancontribute (not necessarily) to the shock wave. In this case, aconservative approach is to also consider the evaporatedliquid for computing the available energy.

In all the cases, the internal energy change can becomputed using a thermodynamic state diagram for the fluidbeing tested or performing an adiabatic flash computation.From a state diagram, the values of the specific enthalpy, h,and specific volume, v, at the bursting vessel temperatureand pressure are readily available. Following an isoentropicline, the same diagram gives the values of these variables atambient pressure. The specific internal energy in the twostates can then be computed as eu�h�Pv. When the finalstate involves a mixture of saturated liquid and vapour, thespecific state variables can be computed as mmix�(1�x)mL�x mV. In this relation, m is a specific variable, L and Vrefer to saturated liquid and vapour, respectively, while x isthe vapour quality, which is usually provided by the samestate diagram.

Bibliography

Fannelöp T.K. (1994) Fluid mechanics for industrial safety andenvironmental protection, Amsterdam, Elsevier.

Glassman I. (1996) Combustion, San Diego (CA), Academic Press.

Kuo K.K. (2005) Principles of combustion, New York, John Wiley.

References

Giesbiecht H. (1988) Evaluation of vapor cloud explosions by damageanalysis, «Journal of Hazardous Materials», 17, 247-257.

Lees F.P. (1996) Loss prevention in the process industries. Hazardidentification, assessment and control, Boston (MA), Butterworth-Heinemann, 3v.

Seaton W.H. et al. (1974) CHETAH. The ASTM chemicalthermodynamic and energy release potential evaluation program,Philadelphia (PA), American Society for Testing and Materials,Data Series 51.

List of symbols

A0 smooth surface area of the flame frontAf real surface area of the flame frontai i-th species activity c sound speedCP constant pressure specific heatE expansion factoreu specific internal energyh specific enthalpyI3

scaled impulseq

P P VA≈−( )

−0 0

01γ



(P°/

PA

)�1

0.001

0.002

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

2

5

1010

9

8

7

6

5

4

3

2

1

R/L0

0.1 0.2 0.5 1 2 5 10 20 50 100

Fig. 16. Diagrams to value parameters of an explosion with the Multi-Energy method (adapted from Lees, 1996).

iP positive impulseK equilibrium constantm variableM Mach numberM molecular weightn number of atomsn parameter defining the symmetryP pressureP° maximum overpressurePC critical pressurePR reduced pressurePV vapour pressureq combustion heatr spatial coordinateR ideal gas constantR distanceR443

scaled distanceRj distance travelled by the shock wave at time tjS0 burning velocitySf flame velocityT thermodynamic temperaturet timetA shock wave arrival time TC critical temperaturetD positive impulse durationTSL critical temperature for triggering the homogeneous

nucleationu velocity in relation to the flame frontv velocity in relation to a fixed observerV volume v specific volume W amount of explosivex vapour quality

Greek lettersa decay parameter a equivalence ratio of the amount of explosive to that

of TNT g ratio of specific heats at constant pressure and

volumeDGR

° free Gibbs energy change of reaction DhC combustion enthalpyDhD decomposition enthalpye explosion efficiencyni stoichiometric coefficient of species ix dimensionless coordinater density

Superscripts° of the shock wave

Subscripts1 unburned gases or atmosphere in front

of the shock wave2 burned gases or atmosphere behind the shock wave o of the vessel before collapseA ambientCW combustion waveL lower C-J pointP plugPW pressure waveU upper C-J point

Renato Rota

Dipartimento di Chimica, Materialie Ingegneria chimica ‘Giulio Natta’

Politecnico di MilanoMilano, Italy



7.2 Detonations and explosions - Treccani · 2019. 3. 29. · VOLUME V / INSTRUMENTS 431 7.2.1...

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