Burning Velocity of Propane–Air Mixtures from Pressure–Time

Article pubs.acs.org/EF

Burning Velocity of Propane−Air Mixtures from Pressure−Time Records during Explosions in a Closed Spherical Vessel Domnina Razus,*,† Venera Brinzea,† Maria Mitu,† Codina Movileanu,† and Dumitru Oancea‡ †

“Ilie Murgulescu” Institute of Physical Chemistry, 202 Splaiul Independentei, 060021 Bucharest, Romania Department of Physical Chemistry, University of Bucharest, 4-12 Regina Elisabeta Boulevard, 030018 Bucharest, Romania

‡

ABSTRACT: The flame propagation during the deflagration of the propane−air mixtures with variable initial concentration, pressure, and temperature ([C3H8] = 2.60−6.20 vol %, p0 = 30−200 kPa, and T0 = 298−433 K) in a spherical closed vessel with central ignition was monitored by means of pressure measurements. Using an improved relationship for the burnt mass fraction, the burning velocities were calculated from pressure−time records over an extended duration of spherical propagation. A very good agreement was found between the burning velocities of atmospheric laminar C3H8/air flames and literature data. Computed burning velocities, obtained from numerical modeling of one-dimensional laminar flames using two detailed chemical kinetic schemes (Warnatz mechanism and GRI Mech 3.0, respectively) are discussed in comparison to experimental burning velocities. The measured burning velocities are correlated with pressure and temperature using a power law; the baric and thermal exponents, ν and μ, lie within the usual range characteristic to alkane−air laminar flames.

1. INTRODUCTION The laminar burning velocity is a fundamental property of flammable gaseous mixtures characterizing the fuel reactivity in the presence of an oxidizer. It is a most useful parameter for a wide range of applications: design of explosion vessels and active suppression and/or relief devices for confined explosions, mitigation of damaging effects of explosions, optimization of internal combustion (IC) engines. In basic studies, the laminar burning velocity is used for validating chemical reaction mechanisms and for modeling turbulent combustion. Therefore, accurate determination of the laminar burning velocity is of great interest. Measurements of the laminar burning velocity are made on either steady flames (e.g., conical or flat flames anchored on a burner, counterflow twin flames, and jet-wall stagnation flames) or unsteady flames, propagating under isobaric (in an extensible bubble) or isochoric (in a closed vessel or combustion bomb) conditions. The method based on the study of unsteady flame propagation in a closed spherical vessel with central ignition is widely used1−21 because it offers some major advantages in comparison to other experimental methods: (i) the normal burning velocity can be determined within an extended range of pressure and temperature, and (ii) flames propagating in non-atmospheric conditions (especially at high pressures and temperatures, where normal burners cannot be used) can be studied. Because of these features, the bomb method delivers relevant results at elevated pressures and temperatures, useful for characterizing the flame propagation in chemical reactors and IC engines.1−3 Another important advantage stems from the possibility to determine the normal burning velocity from data of a single experiment, after making suitable corrections for flame stretch and curvature in the early stage of combustion and for heat losses from the flame front in the late stage of combustion. In most cases, synchronous records of the pressure and flame radius were made, offering a complete image of flame propagation through the transient values of the flame radius rb, space velocity Ss, and normal burning velocity Su.3−10 Sometimes, © 2012 American Chemical Society

only pressure−time records were obtained, so that additional information was necessary to test the validity of results, e.g., flame radius measurements by means of ionization probes or thin thermocouples.11−15 The main drawback of the closed vessel technique is the difficulty to find exact solutions to equations describing the flame movement in confined conditions, when strong gradients appear on both sides of the flame front (especially in the burned gas, behind the flame front). A number of contributions used numerical methods to solve the differential equations describing the specific volume and energy conservation during flame propagation under variable temperature and pressure of the unburnt gaseous mixture. The first approaches used a single burnt gas zone model,2,12,16 where the bomb contains the burnt and unburnt gas, separated by a flame front of negligible thickness. Further studies used the multi-zone approach, where the gas inside the bomb is divided into multiple zones (20, ..., 200). The zones were formed so that they consist of equal gas masses or they have equal radii. For each gas zone (shell), the authors assumed that the unburnt gas undergoes an isobaric combustion, and accordingly, in every moment of flame propagation, a different position of the chemical equilibrium is reached within the burned gas, corresponding to the instantaneous values of pressure and temperature ahead of the flame front.17,18 In many studies of combustion in closed vessels, the flame radius and the normal burning velocity were derived by means of the burnt mass fraction n, using the basic equations proposed by Fiock and Marvin22 and Manton, Lewis and von Elbe23

r b = R c[1 − (1 − n)π−1/γu]1/3

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Received: October 12, 2011 Revised: December 22, 2011 Published: January 2, 2012 901

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R 3 ⎛ 1 ⎞1/γu dn Su = c 2 ⎜ ⎟ dt 3r b ⎝ π ⎠

Article

hemispheric heads made from stainless steel that are bolted together to make a 10 cm inner diameter sphere. The chamber is fitted with ports for filling and evacuating the chamber, spark electrodes, and an ionization probe. The homemade ionization probe was formed by two thin stainless-steel electrodes. A distance of 0.4−0.5 mm was set between their tips. The electrodes were mounted in series with a 100 kΩ to 1 MΩ potentiometer, and the circuit was connected to a 3 V direct-current (DC) power supply. The voltage drop across the resistor was recorded and stored by means of the data acquisition system. The ignition was made with inductive−capacitive sparks produced between stainless-steel electrodes (1 mm diameter, round tips); the spark gap with a constant width of 3.0 mm was located in the geometrical center of the vessel. The sparks were generated by means of a capacitor discharge ignition system fed from a DC variable power supply between 200 and 400 V. A capacitor with C between 0.1 and 1 μF is discharged through a thyristor across a standard automotive induction coil. Spark energies were adjusted to a minimum value, between 1 and 5 mJ, to avoid the turbulence produced by an excessive energy input at initiation. The combustion vessel was electrically heated; its temperature was adjusted by ±1 °C using a AEM 1RT96 controller and monitored by a K-type thermocouple. The piezoelectric pressure transducer was mounted in a special adapter, maintained at 25 ± 0.1 °C by a water jacket. The pressure variation during explosions was recorded with a piezoelectric pressure transducer (Kistler 601A), connected to a charge amplifier (Kistler 5001SN) and an acquisition data system Tektronix TestLab 2505, at 5000 signals/s. An ionization probe mounted in equatorial position immersed at 0.5 cm far from the wall allowed for the detection of the moment when the flame reached a radius R = 4.5 cm. A vacuum and gas feed line, tight at pressures between 50 Pa and 500 kPa, connected the combustion vessel with the gas cylinders containing fuel and air, the metallic cylinder for mixture storage, and a vacuum pump. The fuel−air gaseous mixtures were obtained in metallic cylinders by the partial pressure method and used 24 h after mixing the components, at a total pressure of 500 kPa. Before each test, the combustion vessel was evacuated down to 50 Pa; the explosive mixture was admitted and allowed 15 min to become quiescent and thermally equilibrated. Gas chromatography (GC) analyses [GC Carlo Erba 2307, with a thermal conductivity detector (TCD), 2 m Porapak Q, 40 °C, H2 as the carrier, and 45 cm3/min] of several test propane− air mixtures revealed no change of the initial composition as a result of preheating, in the range of 333−450 K, even using longer heating periods (30 min). The studied propane−air mixtures had a propane concentration between 2.60 and 6.20 vol % at various initial temperatures between 298 and 433 K and various initial pressures between 30 and 200 kPa. Propane (99.99%) (SIAD Italy) was used without further purification. Other details were previously given.34,36,37

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where rb is the instantaneous flame radius (at pressure p), Rc is the radius of the explosion vessel, γu is the adiabatic compression coefficient of the unburnt gas, and π is the relative pressure, defined as π = p/p0, with p being the transient pressure (at time t) and p0 being the initial pressure. Various equations for the burnt mass fraction were used,23−27 in conjunction with simplifying assumptions meant to offer the possibility to obtain an analytical solution for the normal burning velocity. Recent contributions15,28−30 use numerical methods for solving the differential equations obtained using more accurate formulations of the burnt mass fraction. Even in such approaches, simplifying assumptions were made, e.g., neglecting the heat losses toward the end of the process,3,30 the flame front thickness,13 or the stretch effect, even in the initial stage of the process.17,18 The present paper examines a procedure for normal burning velocity calculation from transient pressure records in a closed spherical vessel by means of a simple model for the burnt mass fraction.24,27 The validity of this model was tested earlier27 on literature results reported by Rallis et al.31 referring to two acetylene−air mixtures, when a fair agreement was found between calculated and measured flame radii and calculated and measured burning velocities, respectively. The model was used for the determination of the normal burning velocities of propylene−air and propylene−air−inert mixtures,32,33 when only pressure records were made; the burning velocities agreed well with the literature data obtained by other experimental techniques. In our recent experiments made in a spherical vessel with central ignition on propane−air mixtures at variable initial concentrations, pressures, and temperatures, p(t) data were acquired by means of a better instrumentation, at 5000 signals/s. In a first stage, the pressure−time records were evaluated only in the early stage of combustion (Δp ≤ p0), when the coefficients of cubic law and the normal burning velocities referring to the initial moment of combustion were computed,34 according to a recent simplified model.35 However, a lot of information is lost when examined data are restricted only within the early stage of combustion. These p(t) data, supplemented by additional measurements on propane−air mixtures at higher initial pressures (within 130 and 200 kPa), are used now to compute the burning velocities over an extended duration of flame propagation, using the improved relationship for the burnt mass fraction.27 The main objective of the present paper is to report these recently determined values of normal burning velocities of propane−air mixtures for a wide initial concentration range of C3H8 in air (2.60−6.20 vol %) and to examine their correlations with the temperature and pressure (p0 within 30− 200 kPa and T0 within 298−433 K). The burning velocities from experimental data will be compared to burning velocities obtained from a detailed numerical modeling of one-dimensional (1D) propane−air laminar flames propagating under the same initial concentrations, pressures, and temperatures. Both data sets will be critically examined against burning velocities obtained from other experimental techniques.

3. DATA EVALUATION Using the assumptions widely used for modeling the spherical flames (the ignition source is of negligible dimension; the flame propagates spherically throughout the process; the flame front thickness is negligible in comparison to the flame radius; the heat transfer from the flame front to electrodes and to the wall of the vessel is negligible; both the unburnt and burnt gas are ideal gases; etc.), O’Donovan and Rallis derived the following form of the burned mass fraction n:24 n=

2. EXPERIMENTAL SECTION

θ(π − π1 − 1/γu) πe − θπ1 − 1/γu

(3)

where the relative parameters are π = p/p0, πe = pe/p0, and θ = (T̅ f,e)/(T̅ f,p), with T̅ f,p being the average burned gas temperature

Experiments were made in a spherical explosion vessel with radius Rc = 5 cm, tight up to a static pressure of 4 MPa. The vessel consists of two 902

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at time t and pressure p and T̅ f,e being the average end burned gas temperature. Equation 3 is considered valid for the whole duration of the process until the moment when the flame comes into contact with the wall and the heat losses from the burned gas to the explosion vessel become important. The main problem is a correct estimation of the temperature profile within the burned gas, in various moments of the unsteady process and at its end. According to a previous study,27 the transient relative burnt gas temperature θ varies as a function of the transient pressure π according to a power law, similar in form to the adiabatic compression law:

θ=

⎛ πe ⎞1 − 1/γ* ⎜ ⎟ ⎝π⎠

molecular databases by Sandia National Laboratories, according to the international standard (format for CHEMKIN).

5. RESULTS AND DISCUSSION 5.1. Measured Burning Velocities. Typical plots of burnt mass fraction, flame radius, and burning velocity variations against pressure during a single experiment are given in Figures 1 and 2. A

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The initial relative temperature of the burned gas is θ0 = Tf,V/Tf,p, and the end relative temperature of the burned gas is θe = 1, where Tf,p is the adiabatic flame temperature of isobaric combustion at p = p0 and Tf,V is the adiabatic flame temperature of isochoric combustion at p = p0. The parameter γ* is a composite parameter, calculated as:

γ*=

ln πe ⎛π ⎞ ln⎜ e ⎟ ⎝ θ0 ⎠

Figure 1. Burned mass fraction variation during the explosion of the stoichiometric propane−air mixture ([C3H8] = 4.02 vol %) at p0 = 101.3 kPa and T0 = 298 K. (5)

Because the original paper27 is seldom accessible, the significance of γ* and the derivation of eqs 4 and 5 are given in Appendix A, together with the computing algorithm in Appendix B. The data from the early stage of explosion propagation (within the interval p0, ..., 1.5p0) were left aside, to avoid the disturbing effects of flame stretch and curvature on normal burning velocity during this stage. Data evaluation was carried out up to the inflection point of p(t) records, when usually the heat losses toward the vessel become significant.

4. COMPUTING PROGRAMS The adiabatic flame temperatures of isobaric and isochoric combustion for propane−air mixtures at various initial temperatures within 300−550 K and various initial pressures within 50−300 kPa were calculated with the 0D COSILAB package.38 The program is based on a general algorithm meant to compute the equilibrium composition of products for any fuel−oxidizer gaseous mixture using the thermodynamic criterion of chemical equilibrium: the minimum of free Gibbs energy, at a constant temperature and pressure, or the minimum of free Helmholtz energy, at a constant temperature and volume. A total of 53 compounds were considered as combustion products. The kinetic modeling of 1D, premixed, laminar, free propane−air flames was made with packages INSFLA, developed by Warnatz and co-workers,39,40 and COSILAB (version 3.0.3), developed by Rogg and Peters.38 INSFLA was run using the mechanism CH4−C4 (53 chemical species and 592 elementary reactions) with updated values of rate coefficients for the rate-limiting reactions in propane−air oxidation, as reported by Heghes.41 COSILAB was run using the GRI mechanism, version 3.0 (53 chemical species and 325 elementary reactions). The runs were performed for isobaric combustion of propane−air mixtures at various initial pressures within 50 and 300 kPa and various initial temperatures within 300−550 K. The input data were taken from thermodynamic and

Figure 2. Flame radius and burning velocity during the explosion of the stoichiometric propane−air mixture ([C3H8] = 4.02 vol %) at p0 = 101.3 kPa and T0 = 298 K.

linear correlation n = f(p) was found for all data sets. Such a correlation was typical for the burnt mass fraction computed by the earlier model by Manton, Lewis and von Elbe23

nLvE =

p − p0 pe − p0

(6)

which can be easily derived from the present eq 3 of burnt mass fraction n, by assuming γu = γb = 1, that is, by neglecting the temperature gradient in the burnt gas. In Figure 1 we also plotted nLvE calculated for the examined data set. In the early stage of flame propagation, n and nLvE are quite close; they differ by 1.2− 1.5% in the late stage of the process. A linear correlation n = f(p) was also found for the more accurate relationship of the burnt mass fraction recently derived by Luijten and co-workers30 for CH4−air flames propagating in 903

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For data of any single experiment, the pressure influence on the burning velocity was modeled by a simple power law

a spherical vessel. This behavior shows that corrections brought to the burnt mass fraction by taking into account various disturbing aspects represent less than a few percentages from the classical burnt mass fraction, nLvE. In Figure 3, the dependency of the burning velocity upon the unburnt gas temperature Tu, calculated by means of the

Su = Su,0 πε

(7)

where Su,0 is the burning velocity at the initial moment of combustion (p0 and T0) and ε is a composite parameter named the “thermokinetic coefficient”.14 The thermokinetic coefficient is closely bound to μ and ν, the thermal and the baric exponents of the burning velocity, respectively, used to model the pressure and temperature influence on burning velocities by a simple power law

⎛ 1⎞ ε = ν + μ⎜⎜1 − ⎟⎟ γu ⎠ ⎝

(8)

⎛ T ⎞μ ⎛ p ⎞ Su = Su,ref ⎜ ⎟ ⎜⎜ ⎟⎟ ⎝ Tref ⎠ ⎝ pref ⎠

ν

(9)

when the law of adiabatic compression is considered valid in the unburnt gas zone.

⎛ p ⎞1 − 1/γu Tu = Tref ⎜⎜ ⎟⎟ ⎝ pref ⎠

Figure 3. Burning velocity versus unburnt gas temperature during the explosion of the stoichiometric propane−air mixture ([C3H8] = 4.02 vol %) at p0 = 101.3 kPa and T0 = 298 K.

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For all propane−air mixtures under studied conditions, the parameters Su,0 and ε were determined by a nonlinear regression analysis, according to eq 7. Relevant results are given in Table 1. The standard deviations of Su,0 and ε are

adiabatic compression law, is plotted. In the early stage of propagation, when 1.5p0 ≤ p ≤ 2p0, data are scattered in both curves from Figures 2 and 3; at later stages, the burning velocity has a monotonous variation against p or Tu. Similar plots were obtained for all examined systems. Following a literature suggestion,6,18 some of our measurements were made at initial pressures and temperatures chosen along the isentropes starting at 30 kPa and 298 K. According to this procedure, burning velocities could be obtained for a pressure range from 30 to 830 kPa and a temperature range from 298 to 650 K. The isentrope Su(Tu) shown in Figure 3 could thus be extended toward higher pressures and temperatures by choosing the adequate sets of data among measurements. A typical plot is given in Figure 4, where

Table 1. Best Fit Parameters for the Burning Velocity Dependency upon Pressure during Single Explosions of Propane−Air Mixtures in a Spherical Vessel [C3H8] (vol %)

p0 (kPa) 50

3.60 100

50 4.20 100

T0 (K)

Su,0 (cm s−1)

298 373 408 298 373 408 298 363 423 298 363 423

49.93 68.55 86.20 35.52 48.50 56.08 54.72 76.36 97.83 41.30 54.13 67.97

± ± ± ± ± ± ± ± ± ± ± ±

0.05 0.04 0.11 0.04 0.04 0.04 0.05 0.07 0.09 0.03 0.07 0.08

ε 0.2509 0.2379 0.2748 0.2036 0.2435 0.2628 0.2465 0.2556 0.2850 0.1809 0.2227 0.2513

± ± ± ± ± ± ± ± ± ± ± ±

rn2 0.0021 0.0010 0.0027 0.0011 0.0008 0.0008 0.0015 0.0017 0.0020 0.0006 0.0012 0.0012

0.983 0.990 0.971 0.990 0.992 0.995 0.981 0.984 0.985 0.992 0.986 0.988

between 0.3−1.5% and 0.3−1.0%, respectively. Within examined ranges, the exponents ε exhibit a dependency upon the pressure and temperature. For all mixtures, ε increases with the temperature and decreases with the pressure. The burning velocities of the propane−air mixture with a variable propane concentration at ambient initial conditions are shown as a function of the equivalence ratio φ in Figure 5, together with the best fit of data, in the form of a second-order polynomial

Figure 4. Normal burning velocities of the stoichiometric propane−air mixture during explosions at various initial pressures and temperatures, along several isentropes.

Su,0 = −(118.32 ± 8.86) + (285.04 ± 16.12)φ − (128.51 ± 7.07)φ2

three partly overlapping runs are shown. A good overlapping, especially in the early stage of runs, is a correctness proof for the proposed model.

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with a determination coefficient of 0.977. 904

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The burning velocities have a peak value of 41.3 cm/s around φ = 1.1 and fall off for lean and rich mixtures. In Figure 5, the

Figure 7. Burning velocity versus pressure during the explosion of the stoichiometric propane−air mixture at p0 = 101.3 kPa and various initial temperatures.

initial temperatures are given. From such data, the burning velocity variation as a function of the initial concentration and temperature was plotted in Figures 8 and 9.

Figure 5. Normal burning velocities of propane−air mixtures at 298 K and 101.3 kPa. Spherical bomb method: (•) present data; (□) Babkin et al.,14 and (△) Huzayyn et al.15 Counter-flow twin-flame method: (◇) Vagelopoulos et al.42

burning velocities from the literature are also plotted. A very good agreement was found between our data and literature values, especially with those obtained by means of the spherical bomb technique. The reference values of the peak laminar burning velocity at ambient initial temperature and pressure are scattered within 34 and 47 cm/s: 47 cm/s43 at φ = 1.08, 46.4 cm/s44 at φ = 1.10, 43 cm/s,19,45 41.0 cm/s,46 and 40.5 cm/s47,48 at φ = 1.1, 39.7 cm/s15 at φ = 1.1, 39.3 cm/s6 at φ = 1.08, 38.7 cm/s49 at φ = 1.06, 38.1 cm/s14 at φ = 1.0, and 34.2 cm/s13 at φ = 1.1. Our data range well within these results. Burning velocities of the stoichiometric mixture computed from single experiments at ambient initial temperature and various initial pressures are given in Figure 6. Similar diagrams

Figure 8. Burning velocities extrapolated at the initial moment, Su,0, versus propane concentration, at ambient initial pressure and various initial temperatures.

Figure 6. Burning velocity versus pressure during the explosion of the stoichiometric propane−air mixture at T0 = 298 K and various initial pressures.

Figure 9. Burning velocities extrapolated at the initial moment, Su,0, versus initial temperature at atmospheric pressure and various [C3H8]: (1) 2.80, (2) 3.15, (3) 4.08, and (4) 4.20 vol %.

were obtained for all examined mixtures and initial temperatures. At a constant initial temperature, the burning velocity decreases when the pressure increases, for all mixtures. Another set of isentropes is shown in Figure 7, where burning velocities computed from single experiments with the stoichiometric mixture at ambient initial pressure and various

Examination of Su,0, the burning velocities extrapolated at the initial moment of each explosion, in correlation with the initial temperature (at constant pressure) and the initial pressure (at constant temperature) was made by means of the simple 905

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velocities have close values in the domain of rich propane− air mixtures; for the other examined compositions, the computed burning velocities are higher at all examined initial temperatures. When the initial pressure and/or initial temperature are increased, the GRI mechanism, version 3.0, succeeds qualitatively in simulating the evolution of the laminar burning velocity. However, calculations globally overestimate experimental results.

power law equations derived from eq 9. Representative values of μ and ν obtained by this procedure are given in Tables 2 and 3, Table 2. Thermal Coefficients of Burning Velocities at Ambient Initial Pressure [C3H8] (vol %)

μ (from present data)

μ (from Su evaluated by means of cubic law coefficients)34

3.61 4.02 5.06

1.450 ± 0.196 1.276 ± 0.073 1.433 ± 0.076

1.588 ± 0.079 1.625 ± 0.044 1.582 ± 0.064

6. CONCLUSION Examination of transient values of pressure during explosions in a closed spherical vessel affords calculation of transient values for several deflagration parameters: the burnt mass fraction, the flame radius, and the burning velocity. In the present paper, the calculation was made with an improved model for the burnt mass fraction valid for an extended duration of spherical propagation, up to the moment when energy losses become important. Propane−air mixtures with variable initial concentration, pressure, and temperature were studied in a spherical bomb with central ignition. The burning velocities obtained with this method agree well with burning velocities obtained from other experimental techniques. The burning velocities of propane−air were examined in correlation with pressure and temperature, using either data obtained in a single experiment or data obtained in several experiments (when burning velocities extrapolated to initial conditions were acquired). In both cases, a power law was chosen to describe the correlation of burning velocities with the pressure and temperature. The thermal and baric exponents of burning velocities for propane−air have values within the usual range of variation, characteristic of hydrocarbon−air mixtures. The burning velocities obtained from a detailed numerical modeling of 1D propane−air laminar flames based on the GRI mechanism, version 3.0, and propagating under the same initial concentrations, pressures, and temperatures follow the same trend of variation as the burning velocities from experiments. A better agreement between experimental and computed burning velocities is observed when the Warnatz mechanism is used for modeling the propane−air flames.

Table 3. Baric Coefficients of Burning Velocities at Ambient Initial Temperature [C3H8] (vol %)

−ν (from present data)

−ν (from Su evaluated by means of cubic law coefficients)34

3.61 4.02 5.06

0.151 ± 0.002 0.163 ± 0.009 0.198 ± 0.003

0.168 ± 0.002 0.226 ± 0.004 0.215 ± 0.002

together with thermal and baric coefficients previously reported,34 obtained from burning velocities determined by analysis of the early stage of explosion in the spherical vessel. Another possibility to calculate the baric coefficient ν of burning velocities is to use the thermokinetic coefficients ε and the thermal coefficients μ determined with eq 8, when only experiments at variable initial temperature are carried out. When reliable experiments at variable initial pressure can be performed, the thermal coefficients μ may be calculated from ν and ε, according to a similar procedure. The thermal and baric coefficients determined in this way have close values to those determined from burning velocities extrapolated at the initial moment. The present baric coefficient for the stoichiometric propane− air mixture at T0 = 298 K ranges well within literature data: ν = −0.17,6 −0.13,13 and −0.25.49 The thermal coefficient μ of the stoichiometric propane−air mixture at p0 = 101.3 kPa is lower in comparison to literature values (2.13,6 1.83,49 and 1.6613) but still within the usual range of variation. 5.2. Burning Velocities from Kinetic Modeling. The burning velocities at ambient initial conditions obtained from kinetic modeling with the packages COSILAB and INSFLA are plotted in Figure 10 together with burning velocities from

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APPENDIX A The transient values of the burnt mass fraction were calculated with eq A1 derived after examination of the mass and volume balance and state equations of unburned and burned gas for the initial, final (end), and intermediate moment of combustion (time t and pressure p)24 n=

θ(π − π1 − 1/γu) πe − θπ1 − 1/γu

(A1)

with n, θ, π, πe, and γu as defined in the text. The key problem in using this equation is the correct estimation of the transient average burned gas temperature at time t, T̅ f,p, and the average end burned gas temperature, T̅ f,e. Many authors assumed that, in every moment of flame propagation, a different position of the chemical equilibrium is reached within the burned gas, corresponding to the instantaneous values of pressure and temperature ahead of the flame front. In our previous paper,27 the assumption that chemical equilibrium is reached within the burned gas in each intermediate moment of combustion, until its end, was maintained. In this approach, the burned gas temperature reached immediately after

Figure 10. Measured and calculated burning velocities of C3H8−air flames at ambient initial temperature and pressure.

experimental measurements, computed by means of the burnt mass fraction. The computed and experimental burning 906

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Table B1 πe θ0 γu

relative end pressure initial relative burned gas temperature adiabatic coefficient of unburned gas

πe = pe/p0, where pe is the ideal (adiabatic) end pressure, obtained from equilibrium calculations θ0 = Tf,V/Tf,p, where Tf,V is the flame temperature of isochoric combustion and Tf,p is the flame temperature of isobaric combustion, obtained from equilibrium calculations γu = cp̅ /cV̅ , where cp̅ is the average heat capacity of unburned gas at constant pressure and cV̅ is the average heat capacity of unburned gas at constant volume

ignition is Tf,p, the adiabatic flame temperature of isobaric combustion at p = p0, considering that the energy input of the ignition source is negligible. At the end of closed vessel combustion, the average temperature of the burned gas is T̅ f,e, equal to Tf,V, the adiabatic flame temperature of isochoric combustion at p = p0. Under these conditions, the relative burned gas temperature θ varies between the initial value θ0 = Tf,V/Tf,p and the end value θe = 1. It was assumed27 that, between the initial (θ0) and end (1.0) value, θ varies monotonically as a function of the transient pressure p. Among several tested equations, the most adequate was the power law: θ = aπb, obtained from the model of adiabatic compression, leading to

θ=

⎛ πe ⎞1 − 1/γ* ⎜ ⎟ ⎝π⎠

where a and b are pressure and time corrections, respectively, meant to eliminate the signal shift of the pressure transducer and the possible delays in signal recording.35 Their values were determined from the nonlinear regression of Δp versus t, over a restricted number of points from every experiment, satisfying the condition

p0 ≤ p ≤ 1.5p0

Determination of the Position of the Inflection Point on p(t) Curves (Upper Limit of the Pressure Range for Data Evaluation)

At the end of explosion in the spherical vessel, various phenomena (preferential ascendent rise of the burnt gas, followed by a premature cooling at the wall, and deviations from sphericity because of the asymmetry of the initiation spark) determine deviations from the ideal evolution of the process, revealed by the appearance of an inflection point on Δp(t) curves before the maximum explosion pressure is reached. After the corrections described above in step B1 were made, the p(t) data were read and the first derivative dp/dt(t) was calculated by means of the Savitzky−Golay method, using a 10% smoothing level. The peak of the first derivative designates the inflection point pinf on the p(t) curve, assumed to reveal the moment when the heat losses from the flame front to the vessel become non-negligible. The smoothing procedure also delivered smoothed p(t) data, necessary for the calculation of the burned mass fraction, flame radius, space velocity, and normal burning velocity.

(A2)

with γ* as the adiabatic coefficient of the burned gas, considered a composite parameter, calculated from the limiting conditions (θ0 = Tf,V/Tf,p and θe = 1) as

γ*=

ln πe ⎛π ⎞ ln⎜ e ⎟ ⎝ θ0 ⎠

(A3)

The initial relative temperature θ0 and the end relative pressure πe are readily obtained from equilibrium calculations. Using γ* = γb as an adjustable parameter depending upon the transient pressure was considered a good approximation, removing errors associated with the use of γb,0 (the adiabatic coefficient of burnt gas at temperature Tb,p0, in the initial stage of combustion) or γb,e [the adiabatic coefficient of burnt gas at temperature Tb,e, in the last (end) stage of combustion]. In the former case, the adiabatic compression law would be a good approximation for the early stage of flame propagation but increasingly inadequate toward the end of the process. In the latter case, the reverse would be true.

Calculation of the Burnt Mass Fraction, n

In this step, only data satisfying the initial and end restrictions (p between 1.5p0 and pinf) were examined. Several parameters characteristic of the unburned gas are also necessary for this step (Table B1). Using the transient p(t) data, transient values of π, γ*, θ, and n were obtained.

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Calculation of the Flame Radius, rb, and Normal Burning Velocity, Su

APPENDIX B: COMPUTING PROTOCOL Because p(t) records are affected by different recording and reading errors, a special procedure is recommended for calculation of the normal burning velocities, according to the following steps.

The time derivative of the burned mass fraction, dn/dt(t), was analytically calculated in each moment, after finding the best fit equation n(t) using a nonlinear least-square procedure. The flame radius rb and normal burning velocity Su were calculated with eqs 1 and 2 (see the text). For each data set, the results are examined to check that the flame radius rb calculated at the inflection point pinf verifies the restriction rb ≤ 0.9Rc.

Calculation of Time and Pressure Corrections

For each experiment, a certain delay of time exists in the signal measurement, because of the unavoidable variation of the trigger level of the signal and the poor reproducibility of electric sparks (geometry of the spark, total energy of the spark, amount of energy transferred to the gas, position of the spark in the very centre of the bomb, etc.). Therefore, the usual correlation valid for the early stage of explosion propagating in a spherical vessel is50

Δp = kt 3

Calculation of Su versus p Correlation, for Data from a Single Experiment

This correlation was written as a power law

⎛ p ⎞ε Su = Su,0⎜⎜ ⎟⎟ ⎝ p0 ⎠

(B1)

(B4)

with ε being a composite coefficient, named the thermokinetic coefficient.14 Su,0 and ε were calculated by a nonlinear least-square method.

which was rewritten as

Δp = a + k(t − b)3

(B3)

(B2) 907

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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ACKNOWLEDGMENTS This work was supported by CNCSIS−UEFISCSU, project number PNII−IDEI code 458/2008. The authors gratefully thank Prof. U. Maas (Institut für Technische Verbrennung, Karlsruhe, Germany) and Dr. D. Markus [Physikalisch-Technische Bundesanstalt (PTB), Braunschweig, Germany] for the permission to run the program INSFLA and for the provided assistance.

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NOMENCLATURE k = cubic law coefficient (kPa s−3) n = burnt mass fraction p = pressure (kPa) r, R = radius (cm) S = burning velocity (cm s−1) t = time (s) T = temperature (K)

Greek Letters

γ = adiabatic compression coefficient θ = relative temperature π = relative pressure ε = thermokinetic coefficient μ = thermal coefficient of burning velocities ν = baric coefficient of burning velocities Subscripts

b = burned gas c = combustion vessel e = end (final) state of the burned gas f = flame inf = inflection point of a diagram u = unburned gas

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