prediction of liquid flammabilimit at low pressure

7/28/2019 Prediction of Liquid Flammabilimit at Low Pressure

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Chemical Engineering Science 56 (2001) 38293843www.elsevier.nl/locate/ces

Prediction of ammability limits at reduced pressures

J. Arnaldos , J. Casal, E. Planas-Cuchi

Centre dEstudis del Risc Tecnologic (CERTEC), Institut dEstudis Catalans-Universitat Politecnica de Catalunya,

Diagonal 647, 08028-Barcelona, Catalonia, Spain

Received 27 March 2000; received in revised form 11 October 2000; accepted 1 March 2001

Abstract

This work presents a method to estimate the variation of ammability limits as a function of the pressure of fuelair mixtures atconditions of reduced pressure, for CmHnOx type fuels. The accuracy of the method has been tested through the comparison withexperimental values of methaneair, ethaneair, propaneair and butaneair mixtures drawn from the bibliography; the agreement

between the predicted values and the experimental data is very good. ? 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Flammability limits; Pressure; Modelling; Fuel; Explosions; Safety

1. Introduction

Knowledge of ammability limits of ammable gas=airmixtures or ammable gas=air=inert gas mixtures is ex-

tremely important for safety in certain plants and in theperformance of a range of operations (for example, dry-ing of solids, loading and unloading tanks). Thus, thevalues of both upper (UFL) and lower ammability lim-its (LFL) are available for most gases and vapours mixedwith air at atmospheric pressure and room temperature(usually 298 K). The best study published in this eld is

probably that by Zabetakis (1965), with a set of valuesobtained experimentally with the apparatus developed atthe US Bureau of Mines by Coward and Jones (1952).

Furthermore, for the most common substances,ammability diagrams of ammable gas=air=inert gas are

also available in the literature. Again, these diagrams aregiven for the most common conditions, i.e., 298 K andatmospheric pressure.

Very few data are available on ammability limits inconditions other than the aforementioned. However, in anumber of processes mixtures of ammable gas and air,with or without an inert gas, can exist in dierent con-ditions, such as relatively high temperatures or reduced

pressure. For example, vacuum is used to increase thedrying rate when low temperatures are required. In these

Corresponding author. Tel.: +34-93-401-6675;fax: +34-93-401-7150.

E-mail address: [email protected] (J. Arnaldos).

cases, if the moisture is a ammable liquid (as oftenhappens in the ne chemicals industry) a ammable gasmixture can be created, with the consequent risk of de-agration. Variation of the vacuum has been suggested

as a means of moving the operation outside the amma-bility limits (it is known that if pressure is sucientlydecreased, ammable mixtures will not sustain combus-tion). Unfortunately, as stated before, data on these limitsare very scarce.

A method enabling the prediction of ammability lim-its as a function of pressure and temperature would there-fore be of great interest. The purpose of this paper is todiscuss a new mathematical model for the estimation ofUFL and LFL at reduced pressures. The model is basedon that proposed by Lihou (1993), which has been con-siderably modied to allow more complete calculations.

2. Variation of ammability limits as a function oftemperature

As temperature increases, the range of ammabilitylimits widens (see Fig. 1). Thus, a mixture, which is notammable at room temperature, can become ammable iftemperature increases. The value of LFL decreases whentemperature increases, as less combustion energy will berequired to spread the ame (Drysdale, 1985):

LFLTLFL298

= 1 T 298Tlim

298

: (1)

0009-2509/01/$ - see front matter? 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 (0 1 ) 0 0 0 9 0 - 2


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3830 J. Arnaldos et al. / Chemical Engineering Science 56 (2001) 38293843

Fig. 1. Variation of ammability limits as a function of temperature.

Taking Tlim = 1573 K (minimum temperature whichmust be reached to allow ame propagation) (Zabetakis,1965):

LFLT =LFL298(1 7:8 104(T 298)): (2)This expression is similar to the empirical one proposed

by Zabetakis, Lambiris, and Scott (1959) on the basis ofthe Burguess and Wheeler law:

LFLT =LFL298(1 7:21 104(T 298)): (3)UFL follows a similar relationship (Drysdale, 1985):

UFLT = UFL298(1 + 7:21

104(T

298)): (4)

According to this expression, a limiting temperatureshould also exist for UFL. It is not possible to calcu-late the adiabatic ame temperature for UFLas can bedone for LFLsince the combustion products cannot be

predetermined (they will be a complex mixture of prod-ucts with diering degrees of oxidation). However, Stull(1977) demonstrated theoretically that the ame temper-ature at UFL is approximately the same as that at LFL.

3. Variation of ammability limits as a function ofpressure

As pressure increases above atmospheric pressure,ammability limits enlarge their domain. In fact, LFLdecreases only very slightly as pressure increases, butUFL increases very signicantly (Zabetakis et al., 1959;Vanderstraeten et al., 1997; Goethals et al., 1999).

Below atmospheric pressure, as pressure decreases (atconstant temperature) the values of the two limits ap-

proach each other and the gap between them is narroweduntil a certain pressure is reached (for methane, for exam-

ple, 125 mm Hg at 293 K) at which they have the samevalue (Coward & Jones, 1952; Stull, 1977); at lower pres-sures, the ame cannot propagate through the mixture.

According to Stull (1977), this is due to the fact that theconcentration of gas is too low to sustain combustion.In a more recent paper, Lihou (1993) presented a math-ematical model for the calculation of ammability lim-its at reduced pressure from their values at atmospheric

pressure and room temperature. This model allows the

calculation of the minimum pressure at which the amecan propagate through the mixture, although it does notallow the prediction of the evolution of UFL and LFL asa function of pressure (Table 1).

The model proposed by Lihou considers that whena small spherical volume of a ammable gas mixturereaches a certain temperature and starts to react, then thehot reaction products expand and the expansion absorbs

part of the combustion energy. Thus, the products un-dergo a certain cooling. If the temperature of burnt gasesat the ammable mixture (unburnt gases) interface isstill higher than the minimum or limiting temperature for

ame propagation in the ammable mixture, the reactionwill continue towards the mixture. But if the temperatureat the interface is lower than the limiting temperature,then the reaction cannot propagate. A balance betweenheat generation and heat absorption will indicate whetherthe limiting temperature has been reached or not:

radiative heat loss =

combustion heat sensible heat gain:The heat released in the incomplete combustion of 1 kmolof ammable vapour can be calculated as

Qc = fH: (5)

The sensible heat gain required to bring the temperatureof the mixture from ambient temperature (298 K) up tothe ame temperature is

Qs =MrCp(T 298); (6)where Cp, the specic heat of the reacting mixture (con-sidered equal to that of air at average temperature, Tm),can be calculated as

Cp =4:18

28:8(6:713 + 0:04697 102Tm + 0:1147

105T2m

0:4696

109T3m) (7)

and r is

r=(M 28:8)y + 28:8

My: (8)

The radiant heat emitted in time dt can be determined,for a spherical volume, by the StefanBoltzmann law:

dQr = D2T4 dt: (9)

It may be assumed that the ammable mixture surround-ing the ame front has an absorptivity of zero. Fur-thermore, the value of ame emissivity () has beenconsidered to be unity: a luminous ame is made of in-candescent particles (=0:95, Siegel and Howell (1992))


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J. Arnaldos et al. / Chemical Engineering Science 56 (2001) 38293843 3831

Table 1Limiting pressures below which the ame does not propagate through the mixture (Lihou, 1993)

Methane Propane Heptane Methanol Toluene Acetone Acetic Acid THF

Pressure (bar) 0.168 0.137 0.101 0.106 0.129 0.126 0.131 0.115

and gases with low emissivity; therefore, supposing thatthe ame is a grey body it will always have values of 0:95. Dierent authors (Eisenberg, Lynch, & Breeding,1975; Raj & Kalekar, 1974) have used values of 1for the ame in radiation models for hydrocarbon res.

The spherical volume emitting radiation changes withtime. The velocity at which its diameter increases can beexpressed as (Lihou, 1993)

dD

dt= 2V: (10)

From Eqs. (9) and (10) the following expression is ob-tained:

dQr =D2T4

2VdD: (11)

Integrating between D = 0 and D = S gives

Qr =

S0

T4

2VD2 dD =

T4S3

6V: (12)

For 1 kmol of ammable vapour the nal diameter of theradiant shell can be calculated from (Stull, 1977)

S=D

ET

298

1=3: (13)

By substituting the StefanBoltzmann constant and Eq.(13) into Eq. (12) the following expression is obtained:

Qr =

D3

6

1000E

298

T

1000

5 56:7

V

: (14)

In this expression, the rst term on the right-hand sidecorresponds to the volume of 1 kmol of ammable vapourmixed with air at 298 K and a pressure P, with a molefraction y of vapour in the mixture. By applying the idealgas law, this volume can be expressed as

v =24:77

yP: (15)

Finally, by substituting this relationship into Eq. (14), thefollowing expression is obtained, which gives the heatradiated from the combustion of 1 kmol of ammablevapour:

Qr =4713E

yPV

T

1000

5: (16)

Now, by substituting Eqs. (5), (6) and (16) into the heatbalance:

4713E

yPV

T

1000

5= fH 1000MrCp

T

1000 0:298

:

(17)

In the model it has been assumed that the ame speedand the burning velocity have the same value, as it isvery dicult to establish experimentally any dierence

between these two parameters (Drysdale, 1985). Then,the burning velocity can be expressed as (Lihou, 1993)

V =K

De: (18)

K is a constant lightly dependent on the molecular struc-ture of the fuel. However, the introduction of its exactvalue would extraordinarily complicate the model. There-fore, an average value for all fuels has been taken; in

this way, the knowledge of the value ofK is not required(see Eqs. (20) and (21)) and the proposed model is muchsimpler.

The diusivity can be calculated from the followingexpression for a mixture of a vapour or gas and air (Reid,Prausnitz, & Poling, 1987):

De = 0:0043T1=3

P(air + )2

1

28:8+

1

M: (19)

To avoid the use of diusivities, Lihou suggested thatthe burning velocity for each substance, V, can be ob-tained from that corresponding to another one, Vref, al-ready known,

V =K

De

Vref =K

Deref

V =

De

DerefVref: (20)

Thus, substituting De and Deref according to Eq. (19) intoEq. (20),

V =

(air + ref)

(air + )

Mref(M + 28:8)

M(Mref + 28:8)

1=4Vref: (21)

The value of can be obtained (Reid et al., 1987) fromthe atomic volumes of the dierent atoms forming the gasmolecule. For air, vair = 29:9 103 m3 kmol1; there-fore, air = 0:31 m kmol

1=3. As an example, the value of

for a generic fuel CmHnOx is calculated as

= 3

mvC + nvH +xvO; (22)

where vC = 0:0148 m3 atom1; vH = 0:0037 m

3 atom1

and vO = 0:0074 m3 atom1.

The burning velocity thus obtained corresponds to at-mospheric pressure. At reduced pressure, burning veloc-ity is slightly higher (Spalding, 1979):

Vreduced pressure = V

1

P

0:1: (23)

The burning velocity for any concentration (Vy) can alsobe expressed as a function of burning velocity for the


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stoichiometric mixture, Vs, and of mole fraction ofammable vapour for the mixture, y:

Vy = Vs

ay

s

2+ b

ys

+ c

: (24)

By combining Eqs. (23) and (24) we can obtain an ex-

pression for the burning velocity as a function of pressureand concentration:

Vy = Vs

ay

s

2+ b

ys

+ c

1

P

0:1: (25)

Substituting this expression into Eq. (17) we obtain

4713E

yP0:9Vs(a(y=s)2 + b(y=s) + c)

T

1000

5

=fH 1000MrCp

T

1000 0:298

: (26)

3.1. Minimal pressure below which the ammabilitylimits do not exist

The concentration at which the two limits concur (themixture then becoming non-ammable) can essentially

be considered as equal to the stoichiometric concentra-tion (y = s). Substituting the ame temperature corre-sponding to UFL (which gives a more conservative valuefor pressure) into Eq. (26), the minimal pressure belowwhich the mixture cannot be ammable is (Lihou, 1993)as follows:

Pmin =4713Es(Tu=1000)

5

sVs(fsH 1000MrsCpu ((Tu=1000) 0:298))

1:11:

(27)

4. Variation of ammability limits at reduced pressures

Although the method described above provides usefulinformation, it would be of great interest to know thecomplete variation of ammability limits as a function of

pressure, at low pressures. As stated above, when pressuredecreases LFL increases gradually and UFL decreases,until both concur when the concentration of the mixture,y, is equal to the stoichiometric concentration, s. Belowa new method is presented enabling the prediction of thisvariation.

For LFL:

P=

4713E(T=1000)

5

y(a(y=s)2 + b(y=s) + c)Vs(fH 1000MrCp ((T=1000) 0:298))

1:11: (28)

For UFL:

P=

4713Eu(Tu=1000)

5

y(au(y=s)2 + bu(y=s) + cu)Vs(fuH

1000MrCpu ((Tu=1000)

0:298))

1:11

: (29)

By giving several values to P, from atmospheric pressureto minimal pressure (Pmin), this implicit equation can besolved with respect to concentration y. Thus, the variationof both limits as a function of pressure can be obtained.

4.1. Calculation of parameters a;b;c, at LFL and UFL

The method proposed by Lihou allows prediction ofburning velocity for a given compound from the burningvelocity of another substance (which will be referred to

below as the reference substance and will be indicatedby subscript ref). As burning velocity varies with the con-centration of the mixture, a value can be obtained for the

burning velocity for a given concentration. Lihou, for thereference substance, provided a value of burning veloc-ity (Vref) for the ammability limits (supposing that at298 Kand 1 atm it has the same value for both limits) andanother one for the stoichiometric concentration (Vs

ref

).Thus, according to Eq. (21), these two velocities (V andVs) can be directly obtained for a given substance.

In order to obtain the burning velocity for concentra-tions other than these two, Eq. (24) must be applied.Therefore, parameters a;b;c, must be known (a; b andc for concentrations below the stoichiometric concentra-tion and au; bu and cu for concentrations above it).

There are three unknown parameters in Eq. (24); aswe know three values of the burning velocity for threedierent concentrations, then three equations can be setup to obtain these three values. As previously mentioned,the method proposed by Lihou provides two of them, the

third one must be determined. The burning velocity as afunction of concentration has been published for methaneand propane (Stull, 1977), the two reference substanceschosen by Lihou. Thus, another value can be obtainedfor each range of concentrations. The values y = 0:8s andy = 1:2s were selected.

Then, the aforementioned parameters can be deter-mined by solving the following two equation systems:

1 = a + b + c;

V0:8s

Vs = a0:82

+ b0:8 + c;

V

Vs= a

s

2+ b

s

+ c; (30)


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1 = au + bu + cu;

V1:2s

Vs= au1:2

2 + bu1:2 + cu;

V

Vs= au

u

s 2

+ bu u

s + cu: (31)According to this, the burning velocities for each amma-

bility limit, as a function of pressure and concentration,are

V = Vs

a

ys

2+ b

ys

+ c

1

P

0:1; (32)

Vu = Vs

au

us

2+ bu

us

+ cu

1

P

0:1: (33)

5. Calculation of ame temperature

To solve the heat balance at dierent pressures (thusobtaining the ammability limits at each pressure), theame temperatures T and Tu corresponding to LFL andUFL at 298 K and 1 atm must be known. To nd thetemperature corresponding to LFL it is necessary to solveEq. (28) forP=1 and y = ; and to nd the temperaturefor UFL, Eq. (29) must be solved for P= 1 and y = u.Both equations are implicit with respect to temperature.

As stated by Lihou (1993), the ame temperature atPmin is equal to Tu; therefore, Tu does not change withpressure at vacuum conditions, while the ame temper-ature at LFL changes from T (at atmospheric pressure)to Tu (at Pmin). This inuence of pressure must be takeninto account to obtain accurate results close to real data.

T can be expressed as a function of both Tu and con-centration (which changes with pressure) through the fol-lowing expression:

T = Tu

ys

0:2: (34)

The error involved when using this expression to calculateLFL ranges between 5.2% and 2:4%, with an averagevalue of1:7% (lower than the experimental error whichcan be introduced when determining Tu). Fig. 2 showsthe deviation of the values obtained with Eq. (34) withrespect to the experimental data.

6. General methodology for CmHnOx type fuels

In the following paragraphs a general procedure is pre-sented to determine the variation of ammability limits

Fig. 2. Deviation of T as calculated from Eq. (34) with respect toexperimental data (at 298 K and 1 atm).

as a function of pressure for any CmHnOx type gas fuel(see Fig. 3).

The data required (available in the literature for mostsubstances) are summarised in Table 2.

6.1. Calculation of ame temperature at LFL and UFL

Calculation of the mass ratio of reaction mixture toammable vapour at LFL and UFL at 298 Kand 1 atm,

from Eq. (8):

r =(M 28:8) + 28:8

M; (35)

ru =(M 28:8)u + 28:8

Mu: (36)

Calculation of the fraction of ammable vapour burntat LFL and UFL, at 298 K and 1 atm:Of course, for LFL and the stoichiometric concentra-tion, as there is an excess of air (for the stoichiometricconcentration it is considered that there is 5% excessair) this fraction will be 1,

f = 1; (37)

fu =0:8(1 y)

y(4m + n 2x) : (38)

Calculation of the mole fraction of reaction productswith respect to reactants for the concentrations corre-sponding to LFL and UFL at 298 K and 1 atm:

E = 1 + 0:25n + 0:5x; (39)

Eu =(4:2 0:2u)m + (1:26 0:26u)n (1:68 + 0:32u)x 0:84(1 u)

(4m + n

2x)(1:05

0:05u)

: (40)


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Fig. 3. General methodology for CmHnOx type fuels.


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Table 2Data required for the new methodology

Denition Nomenclature

Number of carbon atoms in the molecule m

Number of hydrogen atoms in the molecule nNumber of oxygen atoms in the molecule x

Molecular weight of ammable vapour M (kg kmol1)Molar heat of combustion H (kJ kmol1)Lower ammability limit at 298 K and 1 atm

Stoichiometric concentration ammable vapour=aira s = 1=[1 + (4:762m + 1:191n 2:381x)1:05]Upper ammability limit at 298 K and 1 atm u

Molecular weight of the reference substance Mref (kg kmol1)

Burning velocity at the limits for the reference Vref (m s1)

Burning velocity at the stoichiometric conc. for the reference Vsref (m s1)

Burning velocity for the reference at y = 0:8s V0:8sref (m s1)

Burning velocity for the reference at y = 1:2s V1:2sref (m s1)

Cube root of molar volume of the reference ref (m kmol1=3)

aSupposing an air excess of 5%.

Calculation of burning velocities at LFL and UFL fromthe value of the burning velocity of the reference sub-stance at LFL and UFL:

V = Vref(air + ref)

(air + )

Mref(M + 28:8)

M(Mref + 28:8)

1=4: (41)

Heat balance as a function of temperature (taking intoaccount that specic heat also depends on tempera-ture):

1=4713E

T

1000

5

V

fH1000Mr 4:18

28:8

6:713+0:0469710

2 (T

+298)

2

+0:114710

5 (T

+298)

220:469610

9 (T

+298)

23 T

10000:298

;(42)

1=4713Eu

Tu

1000

5uV

fuH1000Mru

4:1828:8

6:713 + 0:04697102

(Tu+298)

2

+0:1147105

(Tu+298)

2

2 0:4696 109

(Tu+298)

2

3Tu

1000

0:298

:(43)

These two expressions allow the calculation of T andTu, respectively.

6.2. Calculation of ammability limits as a function ofpressure

With the calculated values of temperatures T and Tu,

the specic heats for both limits are calculated fromEq. (7). Then the burning velocitiesrequired to de-termine parameters a; b and care obtained as follows:

Vs = Vsref(air + ref)

(air + )

Mref(M + 28:8)

M(Mref + 28:8)

1=4; (44)

V0:8s = V0:8sref(air + ref)

(air + )

Mref(M + 28:8)

M(Mref + 28:8)

1=4

; (45)

V1:2s = V1:2sref(air + ref)

(air + )

Mref(M + 28:8)M(Mref + 28:8)

1=4

: (46)

Now, by solving the equation systems (30) and (31) theparameters a; b and c can be obtained.

The next step is the determination of the parameters r,f and E as a function ofy:

r=(M 28:8)y + 28:8

My; (47)

f = 1;

fs = 1;

fu =0:8(1 y)

y(4m + n 2y) ; (48)

E = 1 y + 0:25ny + 0:5xy;Es =

5m + 1:5n 2x5m + 1:25n 2:5x + 1 ;

Eu =(4:2 0:2y)m + (1:26 0:26y)n (1:68 + 0:32y)x 0:84(1 y)

(4m + n 2x)(1:05 0:05y) : (49)

Although, when determining the entire range of variationof ammability limits as a function of pressure the min-imal pressure value is also obtained, this value can also

be calculated previously by substituting the values givenby Eqs. (47) (49) for the stoichiometric concentrationinto Eq. (27).


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Table 3Burning velocity and cube root of molar volume for the reference substances

Reference Mref V

refVs

refV0:8s

refV1:2s

refref

(kg kmol1) (m s1) (m s1) (m s1) (m s1) (m kmol1=3)

Methane 16 0.08 0.33 0.231 0.309 0.3093Propane 44 0.06 0.35 0.29 0.327 0.4198

Table 4Data for the fuels studied

Combustible m n x M H s u

(kg kmol1) (kJ kmol1) (m kmol1=3)

Methane 1 4 0 16 802,703 0.3093 0.06a 0.091 0.132a

Ethane 2 6 0 30 1,560,700 0.3727 0.0285 0.054 0.131Propane 3 8 0 44 2,042,043 0.4198 0.024 0.0385 0.095Butane 4 10 0 58 2,877,600 0.4582 0.018 0.0298 0.0835Heptane 7 16 0 100 4,503,508 0.546 0.011 0.0179 0.067Methanol 1 4 1 32 665,239 0.3332 0.067 0.118 0.36Toluene 7 8 0 92 3,773,679 0.5107 0.012 0.0217 0.07Acetone 3 6 1 58 1,690,139 0.4198 0.0215 0.0476 0.13

Acetic acid 2 4 2 60 832,930 0.3996 0.054 0.0909 0.16THF 4 8 1 72 2,510,590 0.4582 0.02 0.0351 0.118

aThese values, given by Lihou, do not correspond to the reference temperature of 298 K but to a temperature of 293 K. Thus, the referencetemperature in the equations given in the text (298 K) must be changed to 293 K when the limits of methane are calculated.

Table 5Parameters r; f and E for the dierent fuels at ammability limits

Combustible r ru f fu E Eu

Methane 29.2 12.84 1 0.658 1 1Propane 27.618 7.235 1 0.381 1.024 1.036Heptane 26.894 5.011 1 0.253 1.033 1.051Methanol 13.53 2.6 1 0.237 1.0335 1.0434Toluene 26.77 5.159 1 0.295 1.012 1.02

Acetone 23.62 4.326 1 0.335 1.0215 1.0438Acetic acid 9.41 3.52 1 0.525 1.054 1.0846THF 20.6 3.99 1 0.272 1.03 1.0482

Table 6Parameters r; f and E for the dierent fuels, as a function of concentration

Combustible r fua E Eu

Methane (1:8=y 0:8) 0:1(1 y)=y 1 1Propane (0:345 + 0:655=y) 0:04(1 y)=y (1 + y) (1:092 0:092y)=(1:05 0:05y)Heptane (0:712 + 0:288=y) 0:0182(1 y)=y (1 3y) (1:107 0:107y)=(1:05 0:05y)Methanol (0:1 + 0:9=y) 0:1333(1 y)=y (1 + 0:5y) (1:12 0:12y)=(1:05 0:05y)Toluene (0:686 + 0:313=y) 0:0222(1 y)=y (1 + y) (1:073 0:073y)=(1:05 0:05y)Acetone (0:503 + 0:497=y) 0:05(1 y)=y (1 + y) (1:1025 0:1025y)=(1:05 0:05y)

Acetic acid (0:52 + 0:48=y) 0:1(1 y)=y (1 + y) (1:155 0:155y)=(1:05 0:05y)THF (0:6 + 0:4=y) 0:0364(1 y)=y (1 + 1:5y) (1:107 0:107y)=(1:05 0:05y)

af is in all cases 1, as explained above.

Table 7Burning velocity and parameters a; b and c (reference: propane)

Combustible V Vs V0:8s V1:2s a b c au bu cu(m s1) (m s1) (m s1) (m s1)

Methane 0.08 0.47 0.39 0.44 11:07 20.77 8:71 6:361 13.674 6:313Heptane 0.048 0.28 0.232 0.262 7:04 13.53 5:49 0.0068 0:335 1.328Methanol 0.0705 0.411 0.341 0.384 4:615 9.162 3:547 0:0455 0:22 1.265Toluene 0.05 0.294 0.243 0.274 4:04 8.127 3:087 0:0256 0:2636 1.289Acetone 0.0585 0.341 0.283 0.319 1:893 4.263 1:37 0:104 0.0917 1.195Acetic acid 0.06 0.35 0.29 0.327 5:75 11.21 4:46 1:375 2.705 0:33THF 0.0547 0.319 0.264 0.298 4:66 9.243 3:583 0:0144 0:288 1.303


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Table 8Flame temperature at ammability limits (reference: propane)

Combustible T Tu(K) (K)

Methane 878.4 968.4

Heptane 802.3 883.0Methanol 835.5 886.1Toluene 793.0 899.9Acetone 767.1 909.8Acetic acid 808.6 893.9THF 824.3 916.3

Table 9Specic heat at ammability limits and minimal pressure (reference:propane)

Combustible Cp Cpu Pmin(kJ kg1 K

1) (kJ kg1 K

1) (bar)

Methane 1.058 1.066 0.1314Heptane 1.051 1.059 0.125Methanol 1.053 1.058 0.0863Toluene 1.05 1.06 0.126Acetone 1.047 1.06 0.1202Acetic acid 1.05 1.059 0.122THF 1.052 1.06 0.119

To determine the complete range of variation ofammability limits the values from Eqs. (47)(49) (asa function of y) and the values of ame temperature (inthe case of LFL with the correction of Eq. (34)) must besubstituted into Eqs. (28) and (29). Thus, two implicitequations of y are obtained, which can be solved byattributing values to the pressure in the range between

P= 1 and P=Pmin.Lihou proposed two reference substances: methane and

propane. In order to be in a position to compare our resultswith Lihous, ammability limits as a function of pres-sure were calculated for the same fuels proposed by him(methane, propane, heptane, methanol, toluene, acetone,acetic acid and tetrahydrofuran (THF)), using methaneas a reference in one case and propane in the other. Theresults obtained are described below.

7. Results obtained for dierent fuels

The data corresponding to the reference substancesproposed by Lihou, which are used in the following cal-culations, can be seen in Table 3. The data required forthe dierent fuels studied are summarised in Table 4.

To determine the ame temperature at the ammabil-ity limits, a set of parameters (which do not depend onthe reference substance) must be obtained (see Table 5).And to establish the general equation of the heat balanceas a function of pressure and concentration, we need todetermine the expressions for r, f and E (these parame-

ters do not depend on the reference substance) as a func-tion ofy for LFL and UFL (see Table 6).

7.1. Determination of the ammability curves using areference substance

For the case of propane as a reference substance, thevalues of burning velocity and parameters a; b and c have

been summarised, for the various fuels selected, in Ta-ble 7. Two implicit equations (Eqs. (42) and (43)) weresolved, yielding the results shown in Table 8. With thesetemperature values, specic heats can be determined (seeTable 9) at ammability limits, as well as the minimal

pressure at which the mixture becomes non-ammable.The results nally obtained (i.e., the ammability limitsas a function of concentration and pressure) are plottedin Fig. 4.

The procedure was also applied to the same fuels takingmethane as a reference substance. The values obtainedfor the various parameters are summarised in Tables 10,11 and 12. The results corresponding to the variation ofammability limits as a function of pressure are plottedin Fig. 5.

The results obtained for a given fuel are very similarwith the two reference substances, and show the sametrend with only slightly dierent values. Fig. 6 showsthe variation of ammability limits for heptane usingmethane and propane, respectively, as references. LFLvaries very slightly when pressure decreases, this varia-tion increasing somewhat near minimal pressure. How-ever, UFL shows a signicant variation over the wholerange of pressures studied. It can also be observed thatthe values obtained when propane is used as a refer-ence substance are always below those obtained whenusing methane; the zone corresponding to ammabilityconditions is therefore larger when propane is used as areference substance, the minimal pressure being lower.Minimal pressure was lower for all the fuels studied when

propane was used as a reference (Fig. 7).The same trend was found for all the fuels studied.

Therefore, while no experimental values are available, theuse of propane as a reference substance is a sound option,

as it will always provide more conservative results.Flame emissivity was used to study the sensibility of

the method. The variation ofPmin as a function of emissiv-ity was analysed for the dierent hydrocarbons studied,using propane as the reference substance. The data cor-responding to ve representative compounds have been

plotted in Fig. 8. These data show that the ame emis-sivity has a practically negligible inuence on the valueof Pmin, even in the case of low luminosity ames with 0:5 (Siegel & Howell, 1992); the deviation of thevalue of Pmin at = 1 with respect to the value obtainedat = 0:5 was lower than 2% for most hydrocarbons, and4.4% in the worst case (methanol).


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Fig. 4. Variation of ammability limits for dierent fuelair mixtures, as a function of pressure, taking propane as a reference substance.


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Table 10Burning velocity and parameters a; b and c (reference: methane)

Combustible V Vs V0:8s V1:2s a b c au bu cu(m s1) (m s1) (m s1) (m s1)

Propane 0.06 0.25 0.172 0.23 0:0744 1.6964 0:622 0:0906 0:2028 1.2933

Heptane 0.0477 0.197 0.138 0.184

2:5743 6.1337

2:5594 0.0214

0:377 1.3557Methanol 0.07 0.288 0.2019 0.27 1:141 3.5493 1:4083 0:0298 0:2464 1.2762Toluene 0.05 0.206 0.1442 0.1929 0:7718 2.8883 1:1164 0:0119 0:2907 1.3027Acetone 0.058 0.2394 0.1676 0.2242 0.3389 0.8895 0:2284 0:0791 0:143 1.2221Acetic acid 0.059 0.242 0.1696 0.2269 1:785 4.7081 1:923 1:2205 2.3737 0:1531THF 0.054 0.2237 0.1566 0.2095 1:1486 3.5675 1:4189 0:0014 0:3149 1.3163

Table 11Flame temperature at ammability limits (reference: methane)

Combustible T Tu(K) (K)

Propane 796 921

Heptane 802 882Methanol 835 885Toluene 793 900Acetone 766 908.5Acetic acid 806 891THF 822.5 914

Table 12Specic heat at ammability limits and minimal pressure (reference:methane)

Combustible Cp Cpu Pmin(kJ kg1 K

1) (kJ kg1 K

1) (bar)

Propane 1.0503 1.0622 0.185

Heptane 1.0508 1.0585 0.184Methanol 1.054 1.0588 0.127Toluene 1.05 1.0602 0.186Acetone 1.0474 1.061 0.176Acetic acid 1.0512 1.0594 0.18THF 1.0527 1.0616 0.174

In order to test the new method, the results calculatedwere compared to the experimental data available in theliterature. In Fig. 9 the results predicted by the method(using as substances of reference propane, in the case ofthe methane, and methane for the other gases) are plotted

together with the experimental results published by Stull(1977) for methane at 298 K, and published by Latteand Delbourgo (1952) for ethane, propane and butane at298 K. The calculated values t very well with the exper-imental ones; a slight dierence is observed in Pmin val-ues, the calculated values being somewhat conservativenear this pressure.

8. Conclusions

The method proposed by Lihou (1993) for calculatingthe minimum pressure at which a fuelair mixture be-

comes non-ammable has been considerably improvedand extended to determine the variation of ammabilitylimits as a function of pressure, at reduced pressure con-ditions.

The application of this new methodology has demon-

strated that ame temperature at UFL is not a functionof pressure, while at LFL it varies with pressure. A cor-relation is proposed to calculate the ame temperature atLFL as a function of ame temperature at UFL and of

pressure.

A new general methodology has been developed forCmHnOx type fuels to obtain the ammability limits as afunction of pressure. The comparison of the results ob-tained from the method with experimental values takenfrom literature has shown that using propane as the ref-erence substance leads to more conservative results (and,therefore, safer operating conditions) than if methane is

taken as a reference substance.The proposed method has produced results that closelymatch the experimental values available on the variationof ammability limits as a function of reduced pressures.

Notation

a;b;c parameters in Eq. (24)

Cp specic heat of reactant mixture, consid-ered equal to that of air at the average tem-

perature (Tm) of the mixture, kJ kg1

K1

Cpu specic heat of air at the average tempera-ture between Tu and 298 K, kJ kg

1 K1

Cp specic heat of air at the average tempera-ture between T and 298 K, kJ kg

1 K1

De diusivity, m2 s1

D diameter of spherical volume of reactantemitting radiation, m

E molar ratio of reaction products to reac-tants, dimensionless

Es molar ratio of reaction products to reactantswhen the concentration of the mixture isstoichiometric, dimensionless


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Fig. 5. Variation of ammability limits for dierent fuelair mixtures, as a function of pressure, taking methane as a reference substance.


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Fig. 6. Comparison of the results obtained for heptane using the tworeference substances.

Fig. 7. Minimal pressure for the various fuels and for the two referencesubstances (propane and methane).

Fig. 8. Minimal pressure as a function of ame emissivity.

f fraction of ammable vapour burnt, dimen-sionless

H molar heat of combustion, kJ kmol1

K proportionality constant, s0:5

M molecular weight of ammable vapour,kg kmol1

m number of carbon atoms in the molecule ofthe substance of interest

n number of hydrogen atoms in the moleculeof the substance of interest

P pressure, barPmin pressure below which the mixture cannot

be ammable, barQc combustion heat. Heat released in the com-

bustion of 1 kmol of ammable vapour, kJQr radiative heat loss for a spherical volume,

kJQs sensible heat gain required to increase the

temperature of the mixture up to the ametemperature, kJ

r mass ratio of reaction mixture to ammablevapour, dimensionless

s mole fraction of ammable vapour in thestoichiometric mixture, dimensionless

S nal diameter of the radiant spherical vol-ume per kmol of ammable vapour, m

T ame temperature for a given mixture, KTm average temperature of the mixture, Tm =

(T + 298)=2, KTlim minimum temperature which must be

reached to allow ame propagation, Ku mole fraction of ammable vapour in the

UFL mixture, dimensionlessV burning velocity, m s1

V burning velocity at the ammability limits,m s1

V1:2s burning velocity for a concentration y =1:2s; m s1

V0:8s burning velocity for a concentration y =0:8s; m s1

Vy burning velocity for any concentrationy; m s1

v molar volume, m3 kmol1

x number of oxygen atoms in the moleculey mole fraction of ammable vapour in any

mixture, dimensionless

Greek letters

emissivity, dimensionless cube root of molar volume at boiling point,

m kmol1=3

StefanBoltzmann constant (= 56:7 1012 kW m2 K4)

mole fraction of ammable vapour in theLFL mixture, dimensionless


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Fig. 9. Comparison of the ammability limits predicted by the new method and the experimental ones at 298 K. (a) Methane: reference propane:(b) propane: reference methane, (c) ethane: reference methane, (d) butane: reference methane.

Subscripts=superscripts

indicates LFL conditions (as a superindex

it also indicates the same conditions)u indicates UFL conditions (as a superindexit also indicates the same conditions)

s indicates stoichiometric conditions (as asuperindex it also indicates the same con-ditions)

ref corresponds to the reference substanceair corresponds to the air

References

Coward, H. F., & Jones, G. W. (1952). Limits of ammability ofgases and vapors, Bureau of Mines Bulletin No. 503, US.

Drysdale, D. D. (1985). An introduction to re dynamics. Chichester:Wiley.

Eisenberg, N. A., Lynch, C. J., & Breeding, R. J. (1975). Vulnerabilitymodel: A simulation system for assessing damage resulting frommarine spills. Report CG-D-136-75 (Enviro control). Rockville,MD: National Technical Information Service.

Goethals, M., Vanderstraeten, B., Berghmans, J., De Smet, G.,Vliegen, S., & Vant Oost, E. (1999). Experimental study of theammability limits of toluene=air mixtures at elevated pressureand temperature. Journal of Hazardous Materials, A70, 93104.

Latte, P., & Delbourgo, R. (1952). Fourth Symposium

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Raj, P. K., & Kalekar, J. H. (1974). Assessment models in supportof the hazard assessment handbook (USCG Rep. CG-D-65-74).Cambridge: Arthur D. Little.


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Reid, R. C., Prausnitz, J. M., & Poling, B. E. (1987). The propertiesof gases and liquids (4th ed.). Boston: Mc Graw-Hill.

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