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Afterburning Aspects in an Internal TNT Explosion

Article  in  International Journal of Protective Structures · March 2013

DOI: 10.1260/2041-4196.4.1.97

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Afterburning Aspects in an Internal TNTExplosionby

I. Edri, V.R. Feldgun, Y.S. Karinski and D.Z. Yankelevsky

Reprinted from

International Journal of Protective Structures

Volume 4 · Number 1 · March 2013

Multi-Science PublishingISSN 2041-4196

Afterburning Aspects in an Internal TNTExplosion

I. Edri1, V.R. Feldgun2*, Y.S. Karinski2 and D.Z. Yankelevsky3

1Israel Defense Forces, Combat Engineering Corps, FortificationsBranch

2PhD, Senior Researcher, National Building Research Institute, Technion,Haifa, ISRAEL

3Professor, Faculty of Civil & Environmental Engineering, NationalBuilding Research Institute, Technion, Haifa, ISRAEL

[Received date 21 Oct 2012; Accepted date 20 Jan 2013]

AABBSSTTRRAACCTTThe afterburning is a complex chemical process which stems from thereaction of the detonation products with the oxygen in the air whenappropriate conditions exist. TNT is a very fuel-rich explosive as indicatedby the large negative oxygen balance value of −74%. It means that thereis not enough oxygen in its initial chemical compound and extra oxygenis needed to make the afterburning energy release possible. This articledescribes in details the calculation process for evaluating the amount ofenergy release in a confined TNT explosion. Moreover, partialafterburning energy release is also calculated for cases of oxygendeficiency. Commonly, numerical simulations take into account only thedetonation energy in blast pressure analysis and it is responsible forunder prediction of blast pressures in a confined explosion. Accountingfor the afterburning energy as well considerably improves thepredictions and yields pressures that are in good correspondence withmeasured data. The calculation time however increases by an order ofmagnitude. An afterburning coefficient was defined as the relationbetween the total energy released and the detonation energy. Thiscoefficient was found useful for correcting numerical simulation resultsof TNT confined explosion which take into account only the detonationenergy. This correction can be achieved by multiplying the pressure-timehistory with the afterburning coefficient. In addition, an analytic method,based on thermodynamic rules, was developed for calculating the gaspressure resulted by TNT confined explosion. This unique method takesinto account the variation of the total energy released and the heatcapacity ratio depending on the ratio between the charge weightdivided by the confined air volume. The gas pressure obtained using thismethod was shown to be in good agreement with experimental resultsthat are published in the literature.

KKeeyy wwoorrddss:: Confined Explosion, Wave Reflection, Afterburning,Detonation Products, Oxygen Balance

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 97

*Corresponding author. E-mail address: aefeldgo@tx.technion.ac.il

11.. IINNTTRROODDUUCCTTIIOONN1.1. INTERNAL EXPLOSIONConfined explosions may occur due to various reasons and their effects may be extremelysevere and may lead to serious damage to structural elements and even to structuralcollapse (Griffiths et al, 1968). The confined explosion causes more damage than asimilar external free-air explosion, and this damage depends on the geometricalparameters of the space where the explosion occurs (geometrical dimensions, the chargelocation, the openings’ size and location, etc.) as well as on the charge characteristics. Theanalysis of an internal explosion in a confined space aims at both assessment of the pressuresacting on the inner space boundary and analysis of the boundaries structural elementsresponse to predict their response and assess the induced damage. The damages occurshortly after the shock waves arrival at the structural components and may lead to furthercollapse of the entire structure (Dorn et al, 1996; Luccioni et al, 2006). Such an analysismay be performed either without coupling of the pressure prediction and the structuraldynamic response (Yankelevsky et al, 2009; Feldgun et al, 2011) or with coupling(Chock, Kapania, 2001; Vaidogas, 2003; Corneliu, 2004). The first stage of the analysisof this general problem is the prediction of the contact pressures acting on the inner sidesof the surrounding structural elements (walls, ceiling, floor, dividing partitions etc.)depending on the charge properties and location, the confined space geometry, thepresence of openings in the elements, their size and location, etc. The basis of the analysisis the problem of the shock wave interaction with the boundaries of the confined space.This is a well known problem that has been studied in many books (Baker et al, 1983;Kinney, 1962; Bangash, 1993), handbooks (A718300, 1974; Ben-Dor, 2000), reports(Michael, Swisdak, 1975) and manuals (TM-5-1300, 1990, TM-5-855-1, 1986; AASTP-1, 2006; UFC-3-340-02, 2008).

In papers that were written by the authors, a full scale experimental study waspresented (Edri et al, 2011; Savir et al, 2009), in which TNT charges were detonated atthe center of a cubicle room with rigid boundaries and a limited venting in its roof. Anumerical model validation and numerical study (Feldgun et al, 2011) was presentedaiming at understanding the blast pressures and a new insight regarding the pressuredistribution on the walls as well as the pressure attenuation has been gained. Furthermore,investigation of the influence of TNT afterburning in a confined explosion scenario waspresented (Edri et al, 2012) including a comparison to the full-scale experimental data toquantify and evaluate the results. This article widens the scope of internal blast in termsof partial afterburning in case of oxygen deficiency, correction of numerical simulationsfor accounting the afterburning energy release and developing a thermodynamic modelfor the prediction of the gas pressure.

1.2. AFTERBURNING PHENOMENAThe afterburning is a complex chemical process which stems from the reaction of thedetonation products with the oxygen in the air when appropriate conditions exist. Theprimary source of energy release comes during the detonation process, which has an energyrelease timescale on the order of microseconds. On the other hand, afterburning is a late-timecombustion process which occurs on a timescale on the order of milliseconds. Highexplosives which are CHNO molecules produce detonation products such as CO2, H2O, N2,O2 (in the case of oxygen-rich explosives), C, CO, H2, and a variety of hydrocarbons (in thecase of fuel-rich explosives).

98 Afterburning Aspects in an Internal TNT Explosion

Afterburning (Dewey, 1964) occurs when species in the detonation products, such ascarbon and carbon monoxide, mix and react with oxidizers in the surrounding atmosphere.Under particular combustion conditions (i.e. oxygen content, temperature and mixing), thesedetonation products have the potential for producing an extra energy release in addition tothe detonation process while reacting with the surrounding oxygen.

When an explosion occurs in the free-field, the temperature decays relatively quickly with thedistance from the detonation products zone, therefore an extra reaction as mentioned above anda resulting energy release is not possible. On the other hand, for explosives that are detonated inconfined conditions, the volume becomes pressurized over a long duration, and the temperaturedecays relatively slowly, thus allowing the conditions necessary for the extra reaction.

The afterburning energy release is dependent not only on the availability of oxygen withinthe confined room and on the temperature field. Afterburning takes place efficiently onlywhen the detonation products are well mixed with the surrounding air. The mixing processis enhanced for charges placed in full confinement, such as a bomb calorimeter or anexplosion chamber. In such an internal blast scenario, the blast waves propagate faster thanthe detonation products which gradually cool down behind the front shock. As the front blastwaves interact with the inner structure boundaries and reflect back towards the detonationproducts, the temperature rises, the detonation products mix with the surrounding oxygenand the extra reaction is enabled thus releasing the additional afterburning energy. Thisadditional energy can contribute to the air blast pressures, resulting in a more severeexplosion effect.

Fig. 1 (Bryan, Steward, 2006) shows a schematic process of the internal energydevelopment. As the reaction proceeds (left to right), the reactants overcome the activationenergy, Ea, and are converted to products with a lower internal energy. The process istypically composed of two steps. The first step is detonation where the reactants areconverted to intermediate products with the oxygen present in the system, releasing someenergy, ∆Hdet. As additional oxygen is introduced, the reaction continues to the second step(afterburning) and additional energy ∆Hab is released. The total excess energy released isknown as the heat of combustion, ∆HC.

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 99

Ea

Hdet

Hab

Hc

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 0.1 0.2 0.3

Intermediate products

Reactants

Products

Ene

rgy,

E (

arb.

)

0.4

Reaction coordinate, (arb.)

0.5 0.6 0.7 0.8 0.9 1

ξ

∆

∆

∆

Figure 1. Internal energy as a function of reaction coordinate

22.. AAFFTTEERRBBUURRNNIINNGG OOFF TTNNTTTNT explosion performance has been experimentally studied in both free-fieldconfigurations (unconfined) and closed vessels (full confinement) (Ornellas, 1982; Zhang,et al, 2007; Kuhl et al, 1998; Wolanski et al, 2000). Other researchers have gained insightinto the afterburning phenomena using numerical (Kuhl et al, 1998; Kuhl. et al, 2003; Ripleyet al, 2006; Togashi et al, 2010) and analytical (Ames et al, 2006) techniques.

2.1. TNT DECOMPOSITIONThe chemical formula for TNT is C7H5N3O6. Its molecular weight is 227.13 [g/mole] and itstypical density is 1630 [kg/m3]. As mentioned before, the explosive detonation processproduces high pressure and temperature products composed of species including CO2, H2O,N2, O2 (in the case of oxygen-rich explosives), C, CO, H2, and a variety of hydrocarbons (inthe case of fuel-rich explosives). Depending on the explosive formulation, the products canbe fuel-rich, fuel-lean, or balanced. A measure of the balance of an explosive can bedetermined by calculating the oxygen balance, OB%, (Cooper, 1996):

(1)

Where AW(O) is the atomic weight of oxygen (16.00 [g/mole]), and MW(TNT) is themolecular weight of TNT. Variables x, y, and z are the number of atoms of carbon, hydrogenand oxygen that are present in the explosive, respectively. TNT is a very fuel-rich explosiveas indicated by the large negative oxygen balance value of −74%. It means that there is notenough oxygen in its initial chemical compound, and extra oxygen is needed to make theafterburning energy release possible.

The TNT detonation product species and concentrations are shown in Table 1. They weredetermined from the calculation of an adiabatic expansion following detonation performedusing the Cheetah 2.0 thermo-equilibrium code (Fried et al, 1998). The Cheetah compositionis compared in Table 1 with the detonation products formed according to Paul W. Cooper

OBAW O

MW TNTz x

y%

( )

( )[ ] = ⋅ ⋅ − −

100 2

2

100 Afterburning Aspects in an Internal TNT Explosion

Table 1. TNT Detonation Products (Fried et al, 1998; Ornellas, 1982;Cooper, 1996)

CompositionCheetah Ornellas Cooper

Component ρ = 1600 [kg/m3] ρ = 1533 [kg/m3] ρ = 1630 [kg/m3]C(s) 3.410 3.65 3.5CO 2.233 1.98 3.5

H2O 1.592 1.60 2.5

N2 1.493 1.32 1.5

CO2 1.086 1.25 —

H2 3.516 × 10−1 4.60 × 10−1 —

CH4 2.656 × 10−1 9.90 × 10−2 —

H3N 1.305 × 10−2 1.60 × 10−1 —

C2H4 1.262 × 10−3 — —

CH2O2 1.048 × 10−3 — —

(Cooper, 1996) and with published values from the experiments conducted by Ornellas(Ornellas, 1982). These were bomb calorimeter experiments involving the detonation ofsmall-scale charges (on the order of grams), in which the resulting composition wasmeasured using mass spectrometry. A slight disagreement between most of the productcompositions from the three sources is shown. The detonation energy release also agreesfairly well between these sources. Cheetah predicts an energy release of 4.495 [MJ/kg-TNT],the experimental detonation energy found by Ornellas (Ornellas, 1982) ranges from 4.409 to4.573 [MJ/kg-TNT] and a value of 4.56 [MJ/kg-TNT] reported by Paul W. Cooper (Cooper,1996).

2.2. AFTERBURNING ENERGY RELEASEThe detonation products which have the potential to react with the surrounding oxygen areC(s), CO, H2 and CH4. The chemical reactions of each of the combustible detonationproducts are shown in Eq. (2).

(2)

Table 2 shows the number of moles of each fuel per one mole of TNT according to theaforementioned three sources and the heat of reaction released in each chemical reaction.

When all the required conditions for afterburning exist (meaning the presence of enoughavailable oxygen, mixing of the detonation products with the oxygen and temperatures levelsabove the ignition temperatures of the reacted fuels), the expression for the afterburningenergy can be written as shown in Eq. (3):

(3)

This expression yields the afterburning energy from summing the energy contribution ofeach fuel ‘i’. While, n(i) is the number of moles per one mole of TNT of each combustiblecomponent (taken from Table 1). ∆Hr(i) [KJ/mole] is the amount of energy released in thefull chemical combustion of each component. In other words, the heat of reaction ∆Hr(i), isequal to the difference between the standard heats of formation of the reaction products andthe standard heats of formation of the reactants. M.WTNT is the molecular weight of TNT andequals to 227.13 [g/mole].

∆∆

En i H i

M Wabr

TNTi

=( ) ⋅ ( )∑ .

C O CO

CO O CO

H O H O

CH O CO

+ →+ →+ →

+ →

2 2

2 2

2 2 2

4 2 2

0 5

0 5

2

.

.

++ 2 2H O

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 101

Table 2. Chemical Reactions in the Afterburning Process

Moles of fuel/moles TNT ∆Hr

Fuel Reaction Cheetah Ornellas Cooper [kJ/mol]C C + O2 ∅ CO2 3.410 3.65 3.5 393.60CO CO + 0.5O2 ∅ CO2 2.233 1.98 3.5 282.80H2 H2 + 0.5O2 ∅ H2O 0.3516 0.46 — 241.80CH4 CH4 + 2O2 ∅ CO2 + 2H2O 0.2656 0.099 — 800.00

Table 3 shows the calculation of the maximum afterburning energy contribution of eachfuel in the detonation products according to different concentrations of the detonationproducts.

The heat of combustion, ∆HC, is calculated by adding the heat of detonation to theafterburning energy, ∆Hab, as shown in Eq. (4):

(4)

It can be seen that despite the fact that each source assumes different concentrations ofdetonation products, a minor dissimilarity is obtained for the maximum total energy, ∆HC.

In the case of insufficient available oxygen around the TNT charge, the detonationproducts will react with the present oxygen and only partial extra energy will be released.The ratio of W/V which represents the limit for full afterburning is 0.0241 [lb/ft3] (Imperialunits are used in this article for comparison to other literary sources. Note: 1 kg = 2.205lb,1 m = 3.281ft). The calculation behind this number was extensively described in a recentpaper of the present authors (Edri et al, 2012). In order to quantify the afterburning energyrelease in a case of oxygen deficiency, an assumption of linear dependency between theoxygen percentage for full afterburning and the mass fraction (µ) of each fuel consumed bythe combustion process has been made. This assumption yields to obtain the followingformula:

(5)

While W[kg] is the charge weight, V[m3] is the volume of the confined space andΓ[m3/kg] is defined as the minimum volume of air needed to full reaction of the detonationproducts from 1 kg TNT and equals to 2.58344 [m3/kg] (Edri et al, 2012). The lower andupper values for W/V were arbitrarily taken as 0.0005 [lb/ft3] and 4.3 [lb/ft3] respectively. Infact, µ is the relative portion of the maximum afterburning energy release.

µ =

≤ ≤

⋅≤ ≤

1 0 0005 0 0241

10 0241 4

, . .

, .

W

V

WV

W

VΓ..3

∆ ∆ ∆H H HC ab= +det

102 Afterburning Aspects in an Internal TNT Explosion

Table 3. Afterburning Energy and Total Energy Release

Energy parameter Cheetah Ornellas Cooper∆Hdet [KJ/kg-TNT] 4,495 4,573 4,560Maximum afterburning C(s) 5,912.67 6,328.81 6,068.72energy release for each CO 2,781.90 2,466.71 4,360.35detonation product H2 374.52 489.99 —[KJ/kg-TNT] CH4 936.04 348.90 —Max. Afterburning energy, ∆Hab [KJ/kg-TNT] 10,005 9,634 10,429Max. total energy, ∆HC [KJ/kg-TNT] 14,500 14,207 14,989

According to this assumption, Eq. (3) can be extended to obtain the afterburning energyrelease in a case of oxygen deficiency. Eq. (6) shows the extended version which yields theafterburning energy in dependency to the charge weight divided by the confined air volume(W/V).

2.3. THE AFTERBURNING COEFFICIENTIt was shown that the addition of afterburning energy can be calculated also when there isnot enough oxygen for full afterburning process. This calculation takes into account thepartial energy released in the chemical reaction for each of the combustible fuels. Fig. 2 andFig. 3 show the energy contribution of each fuel for a wide range of W/V according toCheetah and Cooper fuel concentrations respectively obtained from Eq. (6).

It can be seen that energy contribution of each fuel is constant at values of W/V lower than0.0241 [lb/ft3] which represent the oxygen limit for full afterburning. Beyond this oxygenlimit, the energy contribution of each fuel decrease linearly on logarithmic axes. Summing

(6)∆

∆

E

n i H i

M W

W

V

n iab

r

TNTi

=

( ) ⋅ ( )≤ ≤∑ .

, . .0 0005 0 0241

(( ) ⋅ ( )

⋅ ⋅≤ ≤∑

∆

Γ

H i

WV

M W

W

Vr

TNTi .

, . .0 0241 4 3

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 103

20

10

532

1

0.50.30.2

0.1

0.050.030.02

0.01

0.0050.0030.002

0.0005 0.001 0.002 0.005 0.01 0.02 0.03

W/V (lb/ft3)

0.05 0.1 0.2 0.3 0.5 0.7

Heat of detonation

Oxy

gen

limit

for

full

afte

rbur

ning

1 2 3 4 5

Heat of detonationEnergy release due toburning of CEnergy release due toburning of COEnergy release due toburning of H2Energy release due toburning of CH4Afterburning energy releaseTotal energy release

∆E (

MJ/

kg)

Figure 2. Afterburning energy and total energy VS W/V according toCheetah fuel concentrations

all the afterburning energy contributions in addition to the detonation energy yields the totalenergy released, ∆ETOTAL. Mathematically it can be written:

Dividing the total energy, ∆ETOTAL, by the detonation energy, ∆Edet, yields the afterburningcoefficient (Eq. (8)). Fig. 4 shows graphically the afterburning coefficient, Kab, independence to W/V. It can be seen that Kab is constant at values of W/V lower than 0.0241[lb/ft3] which represent the oxygen limit for full afterburning and asymptotically close to 1.0at high values of W/V.

(8)

Using this coefficient was found useful in correcting numerical simulations which areperformed without taking into account the afterburning energy release. That approach yieldsa tremendous advantage: numerical calculation that take into account the extra afterburningenergy require considerably smaller time steps that yield a considerably longer time of

KE

EabTOTAL=

∆∆ det

(7)∆

∆∆

E

n i H i

M WE

W

V

TOTAL

r

TNTi

=

( ) ⋅ ( )+ ≤∑ .

, .det 0 0005 ≤≤

⋅⋅

( ) ⋅ ( )+∑

0 0241

1

.

. dWV

n i H i

M WEr

TNTiΓ

∆∆ eet , . .0 0241 4 3≤ ≤

W

V

104 Afterburning Aspects in an Internal TNT Explosion

20

1075

32

10.70.5

0.30.2

0.010.070.05

0.030.02

0.0005 0.001 0.002 0.005 0.01 0.02 0.03

W/V (lb/ft3)

0.05 0.01 0.2 0.3 0.50.7

Heat of detonation

Oxy

gen

limit

for

full

afte

rbur

ning

1 2 3 4 5

Heat of detonationEnergy release dueto burning of CEnergy release dueto burning of COAfterburning energyreleaseTotal energy release

∆E (

MJ/

kg)

Figure 3. Afterburning energy and total energy VS W/V according toCooper fuel concentrations

analysis, whereas using the coefficient allows one to use larger time steps and save an orderof magnitude shorter calculation time.

Examples for using this coefficient are shown for two cases. The first case (Keenan,Wager, 1992) refers to NCEL test of a fully confined explosion of 2.12 kg of TNT inside a32.6 m3 cylindrical tank (radius: 1.73 m, height: 3.46 m). The second case (Kuhl A.L. et al,1998) refers to LLNL test of a fully confined explosion 875 g of TNT inside a 16.6 m3

cylindrical tank (radius: 1.17 m, height: 3.87 m). For both cases, numerical simulations havebeen performed by the authors with and without taking into account the afterburning energyrelease.

In Figures 5 and 7 one can observe the dissimilarity in the pressure-time history betweenthe test result and the numerical simulation result obtained from AUTODYN whileafterburning energy had not been added into the JWL EOS (Edri et al, 2012). Figures 6 and 8show a considerable improvement and good agreement between the calculated andexperimental pressure profiles as well as between the accumulated impulses (dashed lines)after adding the afterburning energy into the JWL EOS.

It also shows the very good correspondence of the proposed approach using theafterburning coefficient to correct the numerical simulation which was performed withouttaking into account the afterburning energy release by shifting the pressure-time historyaccording to Eq. (9). The parameter Pg,AUTODYN is the averaged quasi-static gas pressureobtained from AUTODYN after adding the afterburning energy into the JWL EOS.

(9)

The latter approach is considerably more efficient in calculation time and is as accurateas the detailed analysis using the JWL EOS including the afterburning energy, and in very

P t P t P Kshifted AUTODYN g AUTODYN ab( ) ( ) ( ),= + ⋅ −1

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 105

20

10

876

5

4

3

2

10.0005 0.001 0.002

∆E (

MJ/

kg),

Kab

0.005 0.01 0.02 0.03

W/V (lb/ft3)

0.05 0.1 0.2 0.3 0.50.7

Heat of detonation

Total energy release

Afterburning coefficient

Oxy

gen

limit

for

full

afte

rbur

ning

1 2 3 4 5

Heat of detonationKab - afterburning coefficient

Total energy release

Figure 4. Total energy and afterburning coefficient VS W/V

106 Afterburning Aspects in an Internal TNT Explosion

10

9

8

7

6

5

4

3

2

1

0

0 4 8 12 16 20 24 28 32 36 40 44 48 52 58 60

Time (msec)

Pre

ssur

e (b

ar)

Impu

lse

(bar

-mse

c)

64

−1

−2

−3 −75

−50

−25

0

25

50

75

100

125

150

175

200

225

250

LLNL Test

AUTODYN simulation without afterburning

Figure 5. LLNL test, experimental data and simulation withoutafterburning

10

9

8

7

6

5

4

3

2

1

0

0 4 8 12 16 20 24 28 32 36 40 44 48 52 58 60

Time (msec)

Pre

ssur

e (b

ar)

Impu

lse

(bar

-mse

c)

64

−1

−2

−3 −75

−50

−25

0

25

50

75

100

125

150

175

200

225

250LLNL TestAUTODYN simulation with afterburningAUTODYN simulation withoutafterburning shifted by Kab

Figure 6. LLNL test, experimental data, simulation with afterburning andcorrected simulation without afterburning

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 107

10

12

14

16

8

6

4

2

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time (msec)

Pre

ssur

e (b

ar)

Impu

lse

(bar

-mse

c)

32

−2

−4 −30

−15

0

15

30

45

60

75

90

105

120NCEL TestAUTODYN simulation without afterburning

Figure 7. NCEL test, experimental data and simulation withoutafterburning

10

12

14

16

8

6

4

2

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time (msec)

Pre

ssur

e (b

ar)

Impu

lse

(bar

-mse

c)

32

−2

−4 −30

−15

0

15

30

45

60

75

90

105

120NCEL TestAUTODYN simulation with afterburningAUTODYN simulation without afterburningshifted by Kab

Figure 8. NCEL test, experimental data, simulation with afterburning andcorrected simulation without afterburning

good correspondence with measured data. The use of the afterburning coefficient has saved,in these particular cases, 8 hours of computer calculation time. The numerical simulationwithout taking into account the afterburning energy had lasted for 1 hour and the numericalsimulation which had considered the afterburning energy had lasted for 9 hours.

33.. TTHHEERRMMOODDYYNNAAMMIICC MMOODDEELL OOFF TTHHEE GGAASS PPRREESSSSUURREEIn order to predict accurately the residual gas pressure obtained in an enclosure due to aninternal explosion, a thermodynamic model is proposed. This model takes into account thevariation of the total energy released and the specific heat capacity depending on the relationbetween the charge weight divided by the volume of air around the charge. The suggestedmodel is based on the ideal gas EOS which gives the relationship between the volume,pressure, quantity and temperature of the gas mixture in a given state (Eq. (10)).

(10)

Where: P is the pressure of the gas (atmospheres), V is the volume of the gas (liters) n isthe number of moles of the gas, T is the absolute temperature of the gas in Kelvin, and R isthe universal ideal gas constant (R = 0.0821 [atm/mole·K]). In order to obtain the gaspressure for any value of W/V, the total number of moles in the gas mixture and the finaltemperature of the gas mixture must be calculated for any value of W/V.

3.1. TOTAL NUMBER OF MOLESThe chemical reaction describing the detonation process of 1.0 mole of TNT according toCooper (Cooper, 1996) is:

(11)

The chemical reaction describing a full combustion of the carbon and the carbonmonoxide in the afterburning process of 1.0 mole of TNT is:

(12)

It can be seen that all the carbon and carbon monoxide produce carbon dioxide if there isenough oxygen to enable full combustion process (5.25 moles of oxygen per one mole ofTNT, Edri et al, 2012).

The calculation of the total number of moles in the gas mixture is based on these twochemical reactions and on the assumption of linear dependency between the oxygenpercentage for full afterburning and the mass fraction of each fuel consumed by thecombustion process. Moreover, the calculation distinguishes between two ranges of W /V.The first range is for W/V values lower than 0.0241 [lb/ft3] and the second range is for W /Vvalues greater than 0.0241 [lb/ft3]. The value of 0.0241 [lb/ft3] represents a transition pointbetween a full potential afterburning and a partially afterburning energy release. Thecalculation of the total number of moles is given in Fig. 9. Here, W is in [kg], V in [m3] andΓ equals to 2.58344 [m3/kg].

Fig. 10 shows the normalized mole fraction for each of the products in the gas mixturecalculated for any value of W/V in the range of 0.0005 < W/V < 4.3.

C H N O O N H O CO7 5 3 6 2 2 2 25 25 1 5 2 5 7+ → + +. . .

C H N O N H O CO C7 5 3 6 2 21 5 2 5 3 5 3 5→ + + +. . . .

PnRT

V=

108 Afterburning Aspects in an Internal TNT Explosion

3.2. REACTION TEMPERATURESection 2.2 above presents a method that enables to calculate the amount of heat liberated bythe exothermic chemical reaction (∆ETOTAL). In a case of a system that consists only of thematerials that are involved in the reaction, all the heat generated goes into heating theproducts to some higher temperature.

It is well known from thermodynamics that:

(13)

This equation describes the amount of energy required to cause the temperature of n moleswith heat capacity of Cp to rise from T1 to T2.

Where ∆Q is the total heat that is generated by the chemical reaction and equals to∆ETOTAL. ∆W, n is the number of moles of the gas, T1 is the standard temperature 298 K, and

∆Q n C dTpT

T

= ∫1

2

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 109

n(TNT ) =

n(H2O) = 2.5·n(TNT )

n(AIR) = 40.871·V

n(N2) = 1.5·n(TNT ) + 0.79· n(AIR)

Input parameters

n(CO) = 0 n(CO) = 3.5·n(TNT ) −

n(C) = 0

n(O2) = 0.21·n(AIR)−5.25·n(TNT )

n(CO2) = 7· n(TNT )

≤ 0.0241 [ft/lb3]

W

WV

M.W(TNT )

W [kg] ; V [m3]

> 0.0241 [ft/lb3]WV

W

V· Γ

1· 3.5·n(TNT )

n(C) = 3.5·n(TNT ) −

n(O2) = 0

n(TOTAL) = n(N2) + n(H2O) + n(CO) + n(C) + n(CO2) + n(O2)

W

V· Γ

1· 3.5·n(TNT )

n(CO2) =W

V· Γ

1· 7·n(TNT )

Figure 9. The calculation of the total number of moles in the gasmixture

T2 is the adiabatic flame temperature at constant pressure. Cp [cal/mole·K] is the heatcapacity at constant pressure (function of temperature).

Cp values for each product as a function of the temperature were taken from Cooper(Cooper, 1996). The variation of Cp with the temperature is shown in Fig. 11.

110 Afterburning Aspects in an Internal TNT Explosion

0.9

0.75

0.6

0.45

Mol

e fr

actio

n

0.3

0.15

00.0005 0.001 0.002 0.005 0.01 0.02 0.03

W/V (lb/ft3)

0.050.07 0.1 0.2 0.3 0.5 0.7

Oxy

gen

limit

for

full

afte

rbur

ning

1 2 3 4 5

C,COCO2

H2OO2

N2

Figure 10. Normalize mole fraction VS W/V

CP (

cal-(

mol

e-K

)−1 )

18

17

16

15

14

13

12

11

10

9

8

7

6

5

40 500 1000 1500 2000

Temperature (K)

2500 3000 3500 4000 4500 5000

COC

CP CURVESCO2H2O O2

N2

Figure 11. Variation of heat capacity at constant pressure withtemperature

A 6th order polynomial curve fit was determined for each Cp variation as shown in Eq. (14):

(14)

The polynomial coefficients for all products are given in Table 4.In order to solve Eq. (13) for determining the final temperature, T2, the average heat

capacity, Cpm, must be calculated according to Eq. (15):

(15)

T2 can be calculated for each value of W/V by using Eq. (16):

(16)

After calculating the gas temperature at constant pressure, it must be corrected toconstant-volume conditions by using the adiabatic index γ.

In general, γ decreases with increasing temperature and increases with increasing pressure(Cooper, 1996). For the range of temperatures and pressures used in most of these types ofcalculations (25 < T < 4000°C, and 1bar < P < 1Kbar), the value of γ at 15°C and 1 atm canbe considered to be constant through-out the range without introducing a significant error(see Table 5).

∆E W n TOTAL C dTTOTAL pm

T

⋅ = ⋅ ∫( )298

2

Cn TOTAL

n i C ipm pi

= ⋅ ⋅∑1

( )( ) ( )

C i A A T A T A T A T A T A Tp ( ) = + + + + + +0 1 22

33

44

55

66

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 111

Table 4. Cp polynomial coefficients for each product

CO CO2 H2O O2 N2 C(s)A0 7.345865E-21 −1.02396E-20 7.869391E-21 −1.02392E-20 7.947220E-21 5.95

A1 −1.21880E-16 1.919884E-16 −1.40565E-16 1.919884E-16 −1.33464E-16 0

A2 7.734933E-13 −1.46212E-12 9.842467E-13 −1.46212E-12 8.636520E-13 0

A3 −2.28900E-09 5.832060E-09 −3.32726E-09 5.832060E-09 −2.64868E-09 0

A4 2.857517E-06 −1.30941E-05 5.000709E-06 −1.30941E-05 3.616769E-06 0

A5 −5.16443E-05 1.655762E-02 −4.39227E-04 1.655762E-02 −8.02825E-04 0

A6 4.98991520 4.98991520 7.76072270 4.98991520 6.89155729 0

Table 5. Values of γ at 15°C and 1 atm

Detonation product Heat capacity ratio - γCO2 1.304CO 1.404O2 1.401N2 1.404H2O 1.324C (solid) 1

The calculation of γ (Eq. (17)) is similar to that of the average heat capacity, Cpm. For anyvalue of W /V there is a different concentration of each of the components in the gas mixture.Therefore, γ varies regarding the value of W/V.

(17)

The variation of the heat capacity ratio with W/V is graphically shown in Fig. 12. It canbe seen that at low values of W/V the heat capacity ratio equals to 1.4 which is more suitablefor low pressure ranges, while for high values of W /V it is close to 1.25 which is moresuitable for high pressure levels.

The final temperature of the gas mixture (at constant pressure) variation with W /V isshown in Fig. 13. It can be seen that there a maximum value is obtained at W /V = 0.0241[lb/ft3] and the temperature decreases at higer values of W/V.

3.3. GAS PRESSUREAfter calculating the total number of moles in the gas mixture and the temperature of that gasmixture in the final pressurized state inside the confined space, the gas pressure can becalculated by putting these values into the equation of state as shown in Eq. (18):

(18)

Fig. 14 shows the gas pressure variation with W/V obtained from the calculation describedabove together with experimental results of TNT confined explosions published by Weibull

∆Pn TOTAL R T

V=

⋅ ⋅ ⋅( ) 2 γ

γ γ= ⋅ ⋅∑1

n TOTALn i i

i( )( ) ( )

112 Afterburning Aspects in an Internal TNT Explosion

Oxy

gen

limit

for

full

afte

rbur

ning

1.425

1.4

1.375

1.35

1.325

1.3

1.275

1.25543210.50.20.10.050.020.010.0010.0005

W/V (Ib/ft3)

HE

AT

CA

PA

CIT

Y R

AT

IO.γ

Figure 12. Heat capacity ratio

(Weibull, 1968) and the commonly used curve presented in the UFC guide (UFC-3-340-02,2008). It can be seen that a very good agreement is obtained between the test results, the UFCcurve and the thermodynamic model result for the gas pressure through all the range of W/V.

International Journal of Protective Structures – Volume 4 · Number 1 · 2013 113

Oxy

gen

limit

for

full

afte

r bu

rnin

g

W/V (Ib/ft3)

Tem

pera

ture

(k)

0.001 0.01 0.02 0.05 0.20.1 0.5 1 2 3 4 50

500

1000

1500

2000

2500

3000

0.0005

Figure 13. Final temperature of the gas mixture

Oxy

gen

limit

for

full

afte

r bu

rnin

g

W/V (Ib/ft3)

0.0005 0.002 0.005 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 0.7 1 2 3 4 50.001

Gas

pre

ssur

e (p

s)

5

10

2030

50

100

200300

500

1000

20003000

5000

10000

20000Gass pressure modelTNT experiments results

UFC 3-340-02

Figure 14. Comparison between the thermodynamic model and testresults

44.. CCOONNCCLLUUSSIIOONNSSAfterburning occurs when fuel-rich explosive detonation products react with oxygen in thesurrounding atmosphere. Complete combustion of these afterburning fuels will produce anadded energy that is equal to approximately twice the detonation energy in the case of TNT,further contributing to the air blast and resulting in a more severe explosion hazard. Thisafterburning takes place efficiently only when the detonation products are well mixed withthe surrounding air and under appropriate combustion conditions.

Afterburning might be a significant energy component, especially in fuel-rich explosivesas TNT. Ignoring this energy release might cause to underestimate the pressures andimpulses.

The total energy released in a confined explosion of TNT was calculated as a function ofW/V even for values of W/V where only partial afterburning energy is released. Thiscalculation was based on the assumption of linear dependency between the oxygenpercentage for full afterburning and the mass fraction of each fuel consumed by thecombustion process.

An afterburning coefficient has been defined as the relation between the total energyrelease and the detonation energy. By using this coefficient, numerical simulations carriedout without taking into account the afterburning energy can be fixed by a very simpleprocedure.

Tremendous advantage using this coefficient is a long calculation time saving reflected innumerical calculations which take into account the extra energy due to a smaller time step inthe calculation process.

A thermodynamic model was developed regarding the residual gas pressure in a confinedspace. This model provides the gas pressure obtained for each value of W /V (charge weightdivided by the internal air volume where it explodes). This model takes into account thenumber of moles produced in the combustion process and the laws of thermodynamics areused to find the blast temperature and the gas pressure. A method for calculating the adiabaticindex variation with W/V is presented based on the mole fraction of the explosion gases.

AACCKKNNOOWWLLEEDDGGEEMMEENNTTSSThis work was supported by a joint grant from the Centre for Absorption in Science of theMinistry of Immigrant Absorption and the Committee for Planning and Budgeting of theCouncil for Higher Education under the framework of the KAMEA Program.

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116 Afterburning Aspects in an Internal TNT Explosion

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