Criteria for Spontaneous Ignition in Exothermic ... - Science Direct

COMBUSTION A N D F L A M E 75:381-395 (1989)

381

Criteria for Spontaneous Ignition in Exothermic, Autocatalytic Reactions: Chain Branching and Self-Heating in the Oxidation of Cyclohexane in Closed Vessels T. J. SNEE Health and Safety Executive, Research and Laboratory Service Division, Harpur Hill, Buxton, Derbyshire, SKI7 9JN, United Kingdom

and J. F. GRIFFITHS Department of Physical Chemistry, The University, Leeds LS2 9JT, United Kingdom

The combustion of fuel-rich hydrocarbon + air mixtures constitutes an autoignition hazard that is of concern in industrial processes. In the present article we report studies of the combustion of cyclohexane + air, in the molar proportions cCd-In:air = 1:2, using five closed vessels covering a very extensive range of volume (0.2-20 dm 3). We use the minimum ignition temperature measured in each vessel to test scaling rules for the prediction of conditions for criticality. Measurements were also made of the yields of some products, the extent of reaction (and from it the rate), and the accompanying temperature change under both subcritical and supercritical conditions. Characteristic cooling times from hot gases in each vessel were also assessed experimentally. We show that fuel-rich compositions undergo isothermal oxidation that follows a quadratic, autocatalytic rate law. Departures from this due to self-heating of the reactants are also reported Spontaneous ignition occurs as a result of chemical autocatalysis augmented by selfheating. On the basis of the Scmenov parameter for criticality in a corrected form of the condition for thermal ignition, as derived by Melentiev and Todes, the autoignition temperatures in vessels of different size are found to match a linear scaling relationship. This would be expected in systems in which spatially uniform temperatures prevail. Although there is no forced gas motion in the present experiments, the Rayleigh numbers associated with the reactants on the verge of ignition confirm that namrai convection may be playing a significant part in smoothing out temperature distributions and aiding heat loss, especially from the largest vessels. Ab initio calculations of a criterion for criticality reveal the need for measurements of activation energy that are more precise than the limits of accuracy normally encountered in global kinetic studies.

INTRODUCTION This article is concerned with criteria for spontaneous ignition in gaseous hydrocarbon-air mixtures at atmospheric pressure in closed vessels, and how simple scaling rules may be used to assess the potential combustion hazard under industrial conditions. It is, however, also concerned with a Copyright © 1989 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY I0010

particular and not much studied regime in which the rate of oxidation is controlled primarily by quadratic autocatalysis: the reaction accelerates due to (degenerate) chain branching and selfheating, but criticality occurs under conditions that avoid much of the kinetic complexity normally encountered in hydrocarbon combustion [1]. Isothermal quadratic autocatalysis was first in-

0010-2180/891503.50

382 vestigated by Semenov [2]. The additional consequences of self-heating were presented in analytical form by Melentiev and Todes [3-5]. There have been very few experimental investigations. One factor that makes this special realm accessible to us is study under very fuel-rich conditions so that the rate dependence approximates well to zero order with respect to the fuel: the change of rate is controlled solely by temperature and the concentration of oxygen. Secondly, spontaneous ignition is brought about in vessels of sufficiently large size that critical ignition temperatures are very low (Ta(cr) < 530 K). The range exploited is 0.2-20 dm 3, with particular emphasis on volumes above 1 dm 3. The largest of these (20 dm 3) approaches the practical limit for laboratory-based tests without very elaborate facilities being set up; it far exceeds the size normally encountered in studies of spontaneous ignition [6, 7]. This investigation was initiated because of a need to assess potential spontaneous ignition hazards associated with batch processes, especially as used in the resin manufacturing industry. Resins comprise complex mixtures of high-molecularweight hydrocarbons (containing six or more carbon atoms) and including many cyclic compounds. The manufacturing process often involves heating the contents of a closed (or at least restricted access) batch reactor of up to 5 m 3 capacity to temperatures in excess of 300"C ( - 5 7 0 K) at atmospheric pressure: a very fuelrich, vapor-air mixture is generated above the liquid contents in the vessel. As yet there has been no fundamental study directed specifically to establishing criteria from which the potential spontaneous ignition hazard of these types of system may be assessed. The present study draws on fundamental kinetics and thermal factors in order to establish a foundation to interpretation of criticality in reactions augmented by chain branching. Cyclohexane was chosen as a substrate related to the components of resins. It has been the subject of several experimental investigations concerning oxidation and spontaneous ignition [8-10] and a comprehensive review [11]. Spontaneous ignition can be brought about in reactant compositions that fall outside the flamma-

T . J . SNEE and J. F. GRIFFITHS bility limits for normal flame initiation by a spark or other external stimulus. The cyclohexane + air mixtures in the molar proportions 1:2 used in the present study lie far beyond the rich, or upper, ignition limit for cyclohexane (which exists at 8 % fuel in air, by volume). The published value for the "autoignition temperature" of cyclohexane, measured over a wide range of fuel concentrations in a 0.2-dm 3 vessel [12, 13], is 255"-259"C (528532 K). This paper addresses (1) the critical ambient temperature for ignition of cyclohexane vapour (i.e., the "minimum ignition temperature" or autoignition temperature) in vessels of different size, (2) autocatalysis during subcritical oxidation and during the induction period preceding ignition, and (3) calibration of heat loss rates as a basis for the interpretation of the diminution of spontaneous ignition temperature as the vessel size is increased.

FORMAL

KINETIC

BACKGROUND

Isothermal Autoeatalysis

An isothermal quadratic autocatalytic (or chain branching) reaction, to which a small amount of the "autocatalyst" has been added initially, progresses in a closed system according to the rate law

a~ v - - - = 4~(6 + ~o)(1 - O,

dt

(1)

where ~ is the fractional extent of reaction of the primary reactant at time t and Go is the initial fraction of autocatalytic intermediate (x) relative to the initial concentration of reactant (Co). The branching factor (~) is related to the initial concentrations of material and the (second-order) rate constant for the autocatalytic step [~ = k(co + x0)], and has the units of a pseudo-first-order rate constant (time-~). Equation 1 yields a fractional extent of reaction as a function of time of the form ~o [ 1 - e - o * ~o)*t]

=

~o + e- (1+I/o),t

(2)

CRITERIA FOR SPONTANEOUS IGNITION

383 Equations of these kinds were first set out in similar forms by Semenov [2], who established a unifying relationship between experimental results from different systems by scaling through the product ~tnm, where tm~ is the time to maximum rate. As far as studies of hydrocarbon oxidation are concerned, the subsequent emphasis has been placed on the principal test via a universal relationship of the form Otmax = constant [14]. Direct evidence of the graphical kind shown in Fig. lc is not so common.

A

t.

B

Autocatalysis

Accompanied

by Self-Heating

Exothermic reaction may be accompanied by selfheating. When the reaction rate is slow the temperature rise is low. When the rate of reaction is highest the temperature excess is also at a maximum. This can lead to thermal runaway. The extent of self-heating and the conditions at which criticality and ignition may be achieved are as much a function of the rate of heat loss as of the rate of heat evolution and its sensitivity to temperature change. Nonisothermal reaction is expressed in a formal way by the coupled equations for conservation of mass and energy that in the present context may take the form:

'o

t.

c

"o

for mass, 0

~

1

Fig. 1. Evolution of isothermal, quadratic autocatalysisin a closed vessel when a very small but finite amount of the

autocatalytic intermediate is ~d_ed to the initial w.actant. A. Fractionalextentof reactionversustime. B. Rate of fractional conversion versus time. C. Rate of fractional conversion versus extent of conversion.

and a fractional conversion rate as a function of time of the form ¢~0(1 +~0)2e -(l+~O)Ot

d~ --

dt

=

[~0+ e-O+~0)*/] 2

O)

When G0 < ~, Eqs. 1-3 may be represented in the more familiar graphical forms shown in Fig. 1. The principal features are the sigmoid evolution in time and the association of its maximum gradient with 50% consumption of the reactant.

-~t = 4~(~ +/jo)(1 - / j ) ,

(4)

and for energy,

dO HS -~t = k( T ) t ~ 8 - - ~ 0

(5)

if Newtonian heat transfer is assumed. 0 ( = E ( T Ta)/RTa 2) is a dimensionless temperature excess; B is a dimensionless adiabatic temperature rise, and C is a heat capacity per unit volume. S~ V is the surface-to-volume ratio for the vessel and H is a heat-transfer coefficient at the surface. The difficulty now is that there are two independent variables (0, ~) linked via the nonlinear (Arrhenius) dependence of the rate constant k on temperature, which is incorporated in ~. Explicit analytical solutions are no longer possible. As shown by -

384

T . J . SNEE and J. F. GRIFFITHS

Gray and Yang [15], recourse to phase plane analysis and theorems of stability derived in nonlinear dynamics yields deep insights into criteria for criticality in chain branching-thermal reactions. More primitive interpretations of quadratic autocatalysis accompanied by self-heating, addressed earlier by Melentiev and Todes [3], yielded the useful expression for criticality under Newtonian cooling ~bcr= 40cre(-°cr) = 4e -1.

(6)

0 takes the value of unity at criticality. For a reaction of exothermicity Q, activation energy E, and with a preexponential factor A, ~kcr, the critical Semenov parameter [2, 4, 5] is given by VQcoEAe t- Z/Rra(cr))

~b~=

HSR(Ta (cr)) 2

(7)

The contrast between Eq. 6 and the corresponding expression for purely thermal criticality under pseudo-zero-order conditions [4], for which ~bcr = e- 1, arises because the onset of ignition is associated with the fastest' fractional reaction rate. In a simple, deceleratory reaction this occurs initially. When quadratic autocatalysis takes place the fastest fractional rate occurs at ~ = 0.5 (Fig. 1). The maximum rate expressed in terms of pseudo-zeroorder conditions derived from Eq. 1 is then dt / m ~

4 (Co+ Xo),

(8)

where k is defined at the critical ambient temperature Ta(cr). The scaling of criticality through ~b for an autocatalytic reaction under nonisothermal conditions in vessels of different size follows that for Semenov criticality, namely as a linear function of the characteristic dimension (e.g., radius of a sphere). The particular form of this relationship guides the direction of the present experimental study.

EXPERIMENTAL METHODS Minimum ignition temperatures Ignition experiments were carded out using a conical Pyrex flask (0.2 dm 3) and spherical Pyrex

reaction vessels ranging in size from 1 to 20 dm 3. The 0.2-dm 3 conical flask was similar to that specified in the standard methods for determining minimum ignition temperature [12, 13]. The vessels were held in an oil bath that could be controlled thermostatically in the range from 290 K to 570 K to an accuracy of +_ 1 K with variations throughout the volume of the oil of less than 1 K. The temperature of the oil bath was measured using a chromel-alumel thermocouple pair with the reference junction at 273 K. Ignition temperatures were determined by injecting a measured quantity of liquid cyclohexane from a syringe into the reactor and monitoring the temperature at the center of the flask. Thus, for example, a gaseous composition corresponding to the molar proportions cyclohexane:air = 1:2 was achieved by injecting 1 cm 3 C6H12(1) per dm 3 of vessel volume. After injection of fuei the vessels were closed using a lightweight "stopper" consisting of thermal insulation covered with aluminum foil. In the case of the large vessels cyclohexane was injected using a syringe needle passing through the lightweight closure. Air was drawn from a filtered laboratory supply. Vaporization of cyclohexane (b.p. 358 K) occured rapidly when a relatively small quantity was injected into the hot vessel (Ta - 520 K). The time for vaporization was very much less than the total induction time at conditions close to the minimum ignition temperature. Moreover, the evidence for rapid convection (see Discussion) suggests that satisfactory mixing would have been achieved soon after fuel injection. The oxygen concentration-time records from sampling at the center of the flask (see below) showed no indication of slow mixing having a significant influence on the induction times. Liquid fuel injected into the flask would fall to the bottom, vaporize, and expand causing the predominant expulsion of air from the top of the vessel past the loose-fitting stopper. In the experiments to measure oxida~:ion rate (see below) the flask was partially evacuated before injection of fuel. Air was admitted to restore the flask to atmospheric pressure only after vaporization and expansion had taken place. In either circumstances it seems likely that very little or no fuel vapor will have escaped, and certainly not


385

enough to reduce the fuel-rich composition significantly. Gas temperatures were monitored using an internal thermocouple (see below). Ignitions were readily distinguished as a sharp temperature pulse (approximately 150 K), which occurred after an induction period, when compared with the gradual temperature excess (approximately 15 K) that developed over several minutes under marginally subcritical conditions. The minimum ignition temperature was identified as the lowest oil bath temperature at which ignition could be brought about in a given composition (see Results).

Reactant Temperature Temperature excess-time records within reacting mixtures were monitored using chromel-alumel thermocouples (type K, 0.15-ram-diameter wire) located at the center of each vessel. The 5-din 3 reaction vessel used for the oxidation rate measurements (see below) was provided with an additional thermocouple positioned on the vertical diameter midway between center and top of the vessel (referred to, hereafter, as "the top junction"). The thermocouples were connected to electronic thermometers with internal cold-junc-

tion compensation. Signals from the thermocouple systems and the oxygen analyzer (see below) were monitored continuously and, following a-d conversion, were stored using a microprocessor systom.

Oxidation Rate The temperature and concentration dependence of reaction rate was investigated using an apparatus that is shown schematically in Fig. 2. Experiments were carried out in the 5-rim 3 spherical vessel that was first partially evacuated and then fuel was injected followed by sufficient air to restore the flask to atmospheric pressure. A peristaltic pump was used to draw a sample of gas from the center of the reactor through a condenser to a polarographic oxygen analyzer (Draeger El2). The condensate, and exhaust from the analyzer, were recombined and returned to the reaction vessel via a heating coil that was immersed in the oil bath. At a gas sampling rate of 22 cm 3 rain- ', changes in oxygen concentration in the flask could be detected by the analyzer after a delay of 25 s. When subject to a stepwise change in composition in the flask, for example, from nitrogen to air, the transient response of the gas sampling system

~ONDENSER

IMMERSION

HEATER

S'L'CONEO' t L

C>(

STIRRER

Fig. 2. Diagrammatic representation of the apparatus, including the oxygen sampling system.

386 between 10% and 90% of the final reading was approximately 12 s, corresponding to a frequency response (at - 3 dB) of approximately 0.03 Hz. This implies that rates of change of oxygen concentration at temperatures below the minimum ignition temperature that were typically less than 7% min-~ could be measured accurately and, at temperatures above the minimum ignition temperature, changes during the induction period could be followed with reasonable precision. The gas sampling system, which has a total volume of approximately 17 cm 3, did not appear to cause perturbation of the reaction insofar as the minimum ignition temperature was unaffected by the operation of the peristaltic pump. In each experiment the oxygen concentration and central flask temperature were monitored from the moment of cyclohexane injection until the oxygen concentration had fallen to its final value. Measurements were made at a series of oil bath temperatures up to, and including, the spontaneous ignition temperature. The vessel was cleaned with a nonflammable solvent between each experiment. Cyclohexane and air were mixed in the molar proportions 1:2 in these experiments.

Product Compositions The concentration of carbon monoxide and carbon dioxide in the residual mixtures from ignition and slow oxidation experiments were measured at the conclusion of an experiment by drawing the mixtures into a partially evacuated 1-dm 3 flask that was connected to the 5-dm 3 vessel. The sample was allowed to cool and air was admitted to restore the 1-dm 3 flask to atmospheric pressure. The concentration of oxygen, carbon monoxide, and carbon dioxide in the sample was determined by circulating gas from the sampling flask through infrared gas analyzers (ADC Ltd.), connected in series, and back to this flask until steady readings were obtained. The readings were corrected for the dilution with air by comparing the final oxygen in the 1-dm 3 flask with that measured in the 5-dm 3 reaction vessel before the sample was taken.

Characterization of Heat Loss Rates The cooling characteristics of the 5-, 10-, and 20dm 3 reaction vessels were investigated by moni-

T . J . SNEE and J. F. GRIFFITHS toring changes in central temperature following ignition. The cooling curves were fitted to a Newtonian cooling equation of the form

dT~ 1 ---(T~- T~), art tN

(9)

where Tc = central temperature, Ta = ambient (oil bath) temperature, and tN = Newtonian cooling time. Assuming a relationship between Tc and the average internal temperature (7,) of the form

Tc= 7,+M(T¢- T~),

(10)

where M is a constant, Eq. 10 represents the difference that exists between the average temperature and central temperature while still permitting the evaluation of a global Newtonian cooling time. The assumption is that the shape of the temperature distribution does not vary substantially once the large transient spatial variations following ignition have subsided. If the decay in 7" is Newtonian, then

d7,

HS 1 ( 7 , - T o ) = = ( 7 , - Ta) d-~-= VaCp tN

(11)

and tlv = VoCp/HS, o = molar density, and Cp = heat capacity at constant pressure. The characteristic cooling time for each vessel could be determined from a plot of log(To - Ta) versus time. RESULTS

Spontaneous Ignition Temperatures and Delay Times The lowest spontaneous ignition temperatures were observed in a composition comprising cyclohexane and air in the molar proportions 1:2. This corresponds to the fuel-rich composition 1 C-CtHI2 + 0.4 02 + 1.6 N2. However, within the range of compositions cyclohexane:air = 1:4 ~ 1:1 the spontaneous ignition temperature varies very litfie; tests in the 5-dm 3 vessel reveal an increase in Ta(cr) of not more than 1 K over this range of mixtures. The minimum spontaneous ignition temperatures for the composition c-CtI-Ii2:air = 1:2 in

CRITERIA FOR SPONTANEOUS IGNITION TABLE 1 Minimum Ignition Temperature and the Associated Ignition Delay Time for Cyclohexane-Air in the Molar Proportions 1:2

Vessel size (din3) and shape 0.2 conical (0.14 equivalent sphere) 1 spherical 5 spherical 10 spherical 20 spherical

To (K)

ti~ (s)

520 512 509 503 500

417 411 875 1214 1326

each of the vessels used are given in Table 1. They show a significant diminution as the vessel size is increased: minimum ignition temperatures below 500 K may be expected in vessels of an industrial capacity (V ~, 20 dm3). We may note that the autoignition temperature for cyclohexane determined in a method derived from the ASTM crucible-type test procedure is 529 K: an ignition delay time was 102 s under ASTM conditions [13]. The ignition delays measured at the minimum ignition temperature in the present series of experiments are also given in Table 1. Perhaps the most striking feature is the stretching of the induction time at criticality to beyond 22 min in the 20-drn 3 vessel. Very much longer times may accompany spontaneous ignition in larger industrial vessels. The ignition delay time in each of the vessels is much diminished as the temperature is raised beyond the minimum spontaneous ignition temperature. Apparent activation energies obtained from data of these kinds [16] (In ti~ vs 1/ Ta) vary from 216 to 160 kJ tool -1 over the range of vessel sizes 0.2-5.0 dm 3.

Temperature Changes Accompanying Slow Oxidation and Ignition Temperature-time records during reaction in the 5-din 3 vessel at different bath temperatures are displayed in Fig. 3B. In conditions remote from criticality (Ta ~ 509 K) reaction is virtually isothermal. But as the minimum spontaneous ignition temperature is approached there is increasing evidence of self-heating during the slow

387 reaction; the central temperature exhibits an acceleratory rise to its maximum and then falls as the reaction proceeds. The highest temperature excess reached is ca. 15 K. Similiar temperature profiles accompany the induction period during supercritical reaction. The measured temperature change when ignition itself takes place is not great, the central temperature rising by about 130 K (Fig. 4). Also displayed in Fig. 4 is the profile for the temperature excess during an ignition measured at the top thermocouple. Not only does the temperature during the induction time rise higher than at the center, indicating buoyancy effects, but also the rapid rise during ignition begins about 1 s prior to that at the center. The temperature change recorded at the top junction exhibits a double-peaked character, symptomatic of the propagation of a combustion wave perhaps moving around the walls from the top of the vessel before entering the central region and returning towards the top junction from below.

Cooling Characteristics for Sperical Vessels The characteristic cooling time determined by linear regression analysis of a plot of log(To - Ta) against time for the 5-dm 3 vessel is (1.8 +_ 0.1) s. Corresponding values for the 10- and 20-din 3 flasks are (6.5 ± 0.2) and (8.2 _+ 0.3) s, respectively. Cooling times were determined for excess temperatures up to approximately 50 K, and there was good correlation between the experimental data and an exponential decay law over this range. At higher excess temperatures the decay rate was more rapid. This is attributed to large transient spatial variation in temperature caused by propagation of a combustion wave. Heat transfer characteristics determined from the postignition cooling curves refer to vessels containing the products of decomposition. Criticality is related more closely to the heat transfer coefficients for the reactive initial compositions, but these are more difficult to determine because of the effect of self-heating. The present study is concerned with very fuel-rich conditions in which the fuel concentration varies only slightly over the course of reaction and the heat transfer characteristics of the final compositions would not be

388

T.J.

S N E E a n d J. F. G R I F F I T H S

A

1.0

0.5

D8K

.

.

i

i

525. B

"-'500. r

475 o

'

5o

'

~oo

'

~(o

t/rain Fig. 3. Oxidation of cyclohexane in the 5-din 3 vessel at different vessel temperatures. A. Proportion of oxygen remaining, expressed as a fraction of the initial concentration (C/Co) versus time. B. Reactant temperature [T (K)] versus time as measured by the center thermocouple.

650.

.

,

il • 0

J,ill,, i

501

t/s Fig. 4. Reactant temperatures measured during supercritical reaction by the center thermocouple (broken line) and by the top thermocouple (solid line). The history for the proportion of oxygen remaining is shown ( .). The time interval between the attainment of maximum temperature and maximum reaction rate is due to the delay in the gas sampling system (25 s).


389

expected to differ very substantially from those of the starting materials.

Rates and Extents of Oxygen Consumption in Subcritical Conditions and Overall Activation Energy The change of oxygen concentration measured during a series of experiments in the 5-din 3 vessel over a range of bath temperatures is also shown in Fig. 3A with corrections for the delay in transfer of samples to the oxygen analyzer (25 s). The Sshaped concentration-time profiles, which are characteristic of an autocatalytic process [2], become progressively steeper as the bath temperature is increased. The times to attainment of the highest temperature excess in subcritical conditions coincides with the time at which the maximum gradient of each curve is achieved. Rates of change of oxygen concentration were determined by digitizing and differentiating the concentration time curves using a microcomputer. The dependence of reaction rate on time and on concentration is shown in Fig. 5 for oil bath temperatures of 488.5 and 508 K. The parabolic

dependence of rate on concentration observed at a bath temperature of 488.5 K with its maximum occuring when approximately half of the oxygen has been consumed, is direct evidence for the quadratic autocatalysis taking place under virtually isothermal oxidation: there is no significant extent of self-heating accompanying this reaction. The relationship of Figs. 5A and 5B to those derived from analytical expressions (Figs. IB and 1C, respectively) is clear. However, at a bath temperature of 508 K the maximum reaction rate is not achieved until more than half of the oxygen has been consumed. This distortion to the parabolic form is due to selfheating accompanying reaction: the reaction rate (and with it the extent of reaction) is beyond what would be achieved under isothermal conditions [17]. Owing to the enhanced reactant consumption, there is more rapid fall in rate after the maximum internal temperature has been reached. If an overall rate constant for oxidation is assumed to follow an Arrhenius temperature dependence, the overall activation energy (E) and, subject to a correction for extent of reaction, a preexponential factor (A) can be determined from

B

A

1.1°

I

!

II II II II II II II II II II II 0

0

'

s'o

' 100 t/min

'

150 1.0

0'.5

'

C/C o

Fig. 5. Relationshipsbetweenreactionrate, scaledto maximumrate = 1 and (A) time and (B) proportionof oxygenremaining(C/Co). The solid lines representisothermalreaction(To = 488.5 K). The broken lines representnonisothermalreaction(To = 508 K).

390


the gradient and intercept, respectively, of the graph ln(dl~/dt) versus I / T at a fixed extent of reaction. Strictly, T represents some averaged value in excess of the bath temperature (To), the difference from To representing the effect of selfheating and the spatial variation due to natural convection that foUows because of it. The distinction is extremely important because it is known that activation energies obtained in nonisothermal conditions can be greatly in error if self-heating is not taken into account [17]. Formal theories permitting fully quantitative corrections for simple exothermic reaction pertain only to spatially uniform temperature (Semenov) [18] or purely conductive heat transfer (Frank-Kamenetskii) conditions [19]. The rate of oxidation reached at 15.7% oxygen (that is, at 25 % reaction, prior to the attainment of the maximum rate of consumption) and measured at seven different bath temperatures is plotted in Fig. 6 not only as a function of ambient (or bath) temperature (To) but also as a function of central reactant temperature (Tc). The effect of selfheating is quite clear, undoubtedly yielding a substantial over-estimate when E is derived from dln(dr;/dt)/d(1/To): it corresponds to 272 __ 15 kJ mol-1. An activation energy of 196 _ 5 kJ mo1-1 and a preexponential factor of (2.4 __ 0.1) x 1019 s -1 were obtained from the relationship

based on reciprocal center temperature (dln(d~/ dt)/d(1/Tc)). These latter values are used in the Discussion that follows because samples withdrawn to the oxygen analyzer were also taken from the vicinity of the vessel center. Results of these kinds are also accessible to other extents of reaction. However, the plot against reciprocal central temperature at 15.7 % O: showed the greatest linear correlation with a coefficient of 0.999. By contrast there was a correlation coefficient of 0.975 for the plot at 10.5% 02, for example, at which stage the reaction rate was close to its maximum value. The conditions that yield greatest precision would seem to be consistent with those at which the rate of reaction is becoming appreciable, but before the measured rate of change of oxygen concentration is also affected by convection in the vessel due to marked self-heating.

Extents of Consumption of Oxygen Accompanying Ignition and Stoichiometry of Reaction Spontaneous ignition occurs in the 5-dm 3 vessel at bath temperatures above 508 K. At 509 K the proportion of oxygen remaining falls at an accelerating rate through the ignition delay time to approximately 0.6 at the moment ignition occurs

°.251

%

°°1

/ /

%

-~0.10'

0.05 0-1104

49o ' T/K

5oo

Fig. 6. Fractionalrate of oxidationat 25 % conversionof oxygenversusbath temperature(To) or reactant center temperature (To). The symbols represent the respective experimental measurements.

CRITERIA FOR SPONTANEOUS IGNITION (Fig. 4). The displacement of this curve from that representing the record of central temperature, shown in Fig. 4, is due to the time lag transfer to the oxygen analyzer. As far as measurement at the center of the vessel is concerned, the consumption of only ca. 80% of the remaining oxygen occurs: there appears to be a further period of "slow oxidation" subsequent to the ignition. The residual mixture after ignition at an oil bath temperature of 509 K was found to contain 10.8% carbon monoxide, 1.8 % carbon dioxide, and less than 0.1% oxygen. These concentrations are expressed as percentages by volume after the removal of components that condensed at 293 K. The final gas concentration in subcritical conditions, at an oil bath temperature of 504 K, were found to be 8.7 % carbon monoxide, 2.3 % carbon dioxide, and less than 0.1% oxygen. The similar yields of carbon oxides, measured during subcritical reaction and following ignition, suggests that mechanisms of reaction do not differ greatly between the two conditions.

DISCUSSION Reaction Stoichiometry, Enthalpy Change, and Adiabatic Temperature Excesses The stoichiometry for complete combustion of cyclohexane in air corresponds to c-C6H~2 + 9 02 + 36 N2 --*6 CO2+6 H 2 0 + 3 6 N2.

(12)

Cyclohexane and air in the molar proportions 1:2 correspond to the initial stoichiometry 2.5 c-C6H12+O2+4 N2.

(13)

Clearly only very limited progress towards the complete oxidation of cyclohexane is possible in this mixture in any circumstances, and chemical analysis of the concentration of carbon oxides formed in the present study confirms that there is not much distinction between the chemistry accompanying subcritical and supercritical reaction. From the data presented here and previous chemical studies by Zeelenherg and Bickel [8] and by Bonner and Tipper [9] we may infer an overall

391 stoichiometry approximating to 2.5 c-C6H12+O2+4 N2--'1.9 c-C6H12+0.5 CO +0.10 CO2+0.6 H 2 0 + 0 . 6 (CsHIoO) + 4 N2,

(14) where C5H100 represents partially oxygenated intermediates. The purpose of deducing an expression of this kind is solely to establish, within reasonable precision, the enthalpy change and, from it, an adiabatic temperature excess (A Tad). The nature of the "partially oxygenated intermediates" could he quite diverse, involving species containing between one and five carbon atoms; lower hydrocarbons may also he formed. The overall enthalpy change will not be especially sensitive to variations in these components, however; it will be most affected by changes in the extent of oxidation to carbon oxides and water. The stoichiometry given in Eq. 14 yields an enthalpy change (AH;00) of ca. - 150 kJ mol- 1 of oxygen reacted or - 250 kJ mol- 1 of cyclohexane consumed. This modest but rather typical exothermicity for the partial oxidation of a hydrocarbon [1] in excess cyclohexane gives rise to a calculated adiabatic temperature rise of only ca. 180 K because the product composition has a very substantial heat capacity (Cv(700 K) > 125 J K- l mol-1 of product). Thus the thermocouple record seems to he a very satisfactory result (AT - 130 K) because a fraction of the enthalpy change (ca. 40% according to the extent of oxygen consumption) is brought about nonadiabatically during the induction period, and the temperature rise at ignition is therefore proportionately lower. The principal significance of this result is that although criticality is achieved it is not followed by a hot ignition driven by autocatalysis via H and O atoms. For this to he possible reactant temperatures would have to rise beyond 850 K and there would have to be sufficient oxygen available for very substantial oxidation of the fuel [20]. In the present circumstances the nature of reactions that take place throughout the course of subcritical or supercritical reactions does not differ greatly. Indeed, the experimental results show that not all of the oxygen is consumed at supercritical condi-

392


tions (Fig. 4). Spatial inhomogeneity may have contributed to this measurement, in part, the residual oxygen being swept from extremities of the vessel to the vicinity of the oxygen probe in the gas motion induced by ignition. At marginally supercritical conditions there is no visible manifestation of a flame, t Only at bath temperatures appreciably higher than that required for criticality was a flame observed issuing from the neck of the reaction vessel. It seems likely that in such a case criticality occurs sufficiently vigorously to cause "pressure venting" and so bring about a mixing of excess hot cylcohexane vapor with additional air.

and decomposition of the alkylperoxy radical can be avoided. Criticality and spontaneous ignition is brought about due to quadratic autocatalysis augmented by self-heating. At the reactant densities of the present investigation, buoyancy and natural convection are accompaniments even during subcritical, nonisothermal reaction. This means that internal temperatures are not entirely uniform in space, nor do distributions evolve symmetrically due to the effect of conductive heat transport alone. Nonetheless global characteristic loss times can be assessed, and we use these as a basis for interpreting conditions of criticality.

Reaction Mechanism, Quadratic Autocatalysis, and Self-Heating

Scaling Relationships for Criticality Under Quadratic Autocatalysis and Self-Heating

The present results are among very few to provide direct evidence of quadratic autocatalysis, which, under isothermal conditions, gives rise to a parabolic dependence of rate on concentration, with the maximum reaction rate occurring at 50% reactant consumption. Self-heating perturbs this dependence but nevertheless still yields the highest temperature excess very close to the maximum rate of subcritical reaction under constant pressure: no time difference can be discerned between the two in the present results (Fig. 3). Despite the complex mechanism of hydrocarbon oxidation, the simple form of autocatalysis emerges from the present study for two reasons. The first is that for cyclohexane:air = 1:2 the reactant is in very great initial excess over oxygen, so that not only is the extent of oxidation restricted but also the overall rate may be regarded as pseudo-zero-order with respect to the concentration of eyclohexane. (The concentration of cyclohexane varies by less than 5% throughout reaction.) The second is that reaction is investigated at temperatures sufficiently low that kinetic complexities that may otherwise arise due to isomerization

Our purpose in this section is to establish to what" extent interpretations at the Semenov limit of spatially uniform internal temperatures may be used to assess the effect of scale on the conditions for criticality. The major constraint at the present time is that there is yet no way to interpret thermal or chain-thermal criticality using analytical methods when natural convection takes place. Even, numerical methods designed to cope with temporal and spatial variations, such as those used to assess heat and mass transport under turbulent gas motion in reciprocating engines, require deep physical insight and are extremely demanding on computer resources [21]. When spatially uniform temperatures prevail, criticality is predicted to occur in an exothermic autocatalytic reaction under stationary-state conditions given by Eq. 6. Its general form is expressed in terms corresponding to the Semenov parameter ~¢r for thermal criticality under stationary-state conditions (Eq. 7) and is thus governed by a scaling relationship directly proportional to the characteristic dimension of the reacting syster~ (given by V/S). This quantitative basis for scal;~ to make a comparative prediction of critical vess¢,, temperature relative to that for reaction in the 1din 3 vessel is shown in Table 2. The overall scaling ratio of radii is nearly threefold. An extension of the range also down to the 0.2-din 3 conical vessel is based on the radius of the

i The possibility that criticality leads to a cool flame rather than fully fledged ignition cannot be discounted because the associated temperature changes would be rather similar to those measured in this work. No optical measurements were made to clarify the distinction. Further studies are in progress.

CRITERIA FOR SPONTANEOUS IGNITION TABLE 2 Comparisons Between Predicted and Experimental Minimum Vessel Temperatures for Ignition in Each Vessel. The Predicted Values Are Scaled Relative to That for the l-rim 3 Vessel

To(cr)

T,(cr)

V (dm 3)

r (m)

Exp.

Sealing

0.2 I 5 I0 20

(0.032)a 0.062 0.106 0.134 0.169

520 512 509 503 500

520 512 506 503 500.5

393 2.5 s. The measured thermal relaxation time in the 5 din 3 (tN = 1.8 S) is broadly consistent with this, and more than an order-of-magnitude shorter than the Fourier time, confLrming that natural convection makes a major contribution to heat transfer, and sufficiently so to generate a characteristic loss rate similar to that for spatially uniform conditions. The Semenov parameter at criticality (~bcr) may be expressed in the alternative form tN" AT~d" E " A e x p ( - E / R T ~ (cr)) ~[/cr --

RTo (cr) 2

(15)

a ( ) signifies radius of equivalent sphere.

Semenov equivalent sphere, that is, the sphere of an identical V / S ratio. Subject to some inconsistency for the 5-din 3 vessel, there is excellent accord between the experimental and predicted critical vessel temperatures over the range 0.2-20 dm 3. These results suggest that a scaling rule based on a linear dependence for the radius of the Semenov-ezluivalent sphere may be applicable to criteria for the criticality of the nonisothermal, quadratic autocatalytic reaction in vessels of large dimension. On this basis the minimum ignition temperature in a volume approaching 5 m 3 (r = 1 m, say) is predicted to be 482 K. Further tests to establish validity of this scaling relationship at much greater volumes and for different shapes are desirable. The same scaling relationship should also be applicable to criteria for purely thermal runaway in gaseous systems at similar pressures, which itself has not been tested to such large volumes. This prediction may seem surprising, but some justification for it emerges in the next section.

Ab Initio Calculations of ~ • Purely conductive heat transfer for c-C6Ht2:aJr = 1:2 at atmospheric pressure in the 5-din 3 sperical vessel would lead to a characteristic cooling time tF = o'Cp/'02/'ff2K --~ 128 S. Taking a heat-transfer coefficient extrapolated from independent experiments in a 0.5-din 3 well-stirred vessel [22] ( H = 60 W m -2 K-l), the Newtonian cooling time under the same conditions is tN( = aCu V / H S ) =

Thus taking the parameters above, at 509 K criticality in the 5-dm 3 vessel yields ~b= = 5.4. The magnitude given by the stationary-state theory of Melentiev and Todes is ~cr =

4 exp- i = 1.47.

The fourfold discrepancy between theory and experiment underlines the difficulty of successful matching on an ab initio basis. Fourfold errors are unlikely to arise in thermochemical, physcial data or the vessel dimensions. Nor is the critical ambient temperature likely to be in error by anything like 10 K, as would be required for c o r r e ~ o n to be made. The major constraint lies in the precision of the kinetic parameters. For example, the activation energy need only be enhanced by 3 % in order to bring theory and experiment into accord. The magnitudes of the overall activation energies for complex kinetic systems are rarely known to a precision that is as good as this. The same difficulties have been recognized in the ab initio interpretation of criteria for thermal ignition under purely conductive heat transfer [23]. Nevertheless, that criticality can reasonably be predicted for ignition in large vessels assuming Newtonian heat transfer would seem to be well supported. This is also borne out by the magnitude of the Rayleigh number Ra (see Appendix), which approaches 105 during subcritical conditions in the 5-din 3 vessel, implying that vigorous gas motion occurs due to natural convection. Since this parameter has a cubic dependence on the characteristic vessel dimension, in the smallest vessel

394 (0.2 dm 3) Ro is roughly 100-fold lower even at its maximum value but remains in the range at which convection is beginning to be an effective mode of heat transport [24]. CONCLUSIONS 1. The gas-phase oxidation of cyclohexane in air at atmospheric pressure in very fuel-rich conditions at vessel temperatures up to 520 K follows a quadratic autocatalytic rate law governed by the oxygen concentration. 2. Self-heating accompanies reaction at subcritical conditions close to the autoignition temperature. Criticality and ignition results from the branched chain-thermal interaction. 3. The minimum temperature required to bring about spontaneous ignition diminishes by 20 K when the size of the vessel is increased from 0.2 to 20 dm 3. The lowest autoignition temperature measured was 500 K, and still lower temperatures may be expected in industrial systems ( - 5 m3). The autoignition temperature determined by the ASTM method is 532 K. Extremely long induction times were measured in large vessels. 4. The prediction of the spontaneous ignition temperature in vessels of different size from a scaling rule based on the Semenov criterion for criticality under spatially uniform conditions matches the experimental results very satisfactorily on a comparative basis, but Arrhenius parameters of extremely high precision are required for accurate ab initio calculations to be exploited. Corrections to the Semenov thermal explosion criteria for the contribution due to autocatalysis were based on the method by Melentiev and Todes. 5. A justification for assuming spatially uniform temperatures emerges from the scale of natural convection that takes place, going some way to smoothing out temperature gradients and conferring on the system characteristic cooling times that are close to the well-stirred values. 6. The potential combustion hazard in large-scale industrial Systems when fuel-rich, vapor + air conditions prevail is highlighted. Rather small temperature rises are associated with ignition

T . J . SNEE and J. F. GRIFFITHS itself, owing to the high thermal capacity of the product compositions. The authors wish to t h a n k P r o f e s s o r P. Gray and Dr. S. K. Scott f o r helpful discussions, a n d the N a t i o n a l Engineering L a b o r a t o r y f o r provision o f data f o r transport properties.

REFERENCES 1. Griffiths, J. F., Advances in Chemical Physics (I. Prigogine and S. A. Rice, Eds.). Wiley, New York, 1986, Vol. 64, p. 203. 2. Semenov, N. N., Chemical Kinetics and Chain Reactions. Clarendon, Oxford, 1935. 3. Melentiev, P. N., and Todes, O. M., Acta Physicochim. USSR 14:27 (1941). 4. Gray, P., and Lee, P. R., in Oxidation and Combustion Reviews (C. F. H. Tipper, ed.). Elsevier, Amsterdam 1967, Vol. 2, p. 1. 5. Bowes, P. C., Se(f-heating: Evaluation and Controlling the Hazards. Her Majesty's Stationery Office, London, 1984. 6. Clothier, P. G. E., Glionna, M. T. J., and Pritchard, H. O., J. Phys. Chem. 89:2992 (1985). 7. Cullis, C. F., and Foster, C. D., Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, 1972, p. 423. 8. Zeelenberg, A. P., and de Bruijn, H. W., Combust. Flame 3:281 (1965). 9. Bonner, B. H., and Tipper, C. F. H., Combust. Flame 3:317 (1965). 10. Griffiths, J. F., Skirrow, G., and Tipper, C. F. H., Combust. Flame 12:443 (1968). 11. Berezin, I. V., Denisov, E. T., and Emanuel, N. M., The Oxidation of Cyclohexane (K. A. Allen, trans.). Pergamon, Oxford, 1966. 12. British Standards Institute, BS 4056 (1966). 13. ASTM, Designation D2155 (1966). 14. Drysdale, D. D., Combust. Flame 17:261 (1971). 15. Gray,B. F., and Yang, C. H., J. Phys. Chem. 69:2747 (1965). 16. Snee, T. J., to be published. 17. Griffiths, J. F., and Singh, H. J., J. Chem. Soc. Farad. Trans. 1 78:747 (1982). 18. Gray, P., Griffiths, J. F., and Hasegawa, K., Int. J. Chem. Kin. 13:817 (1981). 19. (a) Boddington,T., and Gray, P. Proc. R. Soc. A320:71 (1970). Co) Boddingtan, T., Gray, P., and Tyler, B. L, Int. J. Chem. Kin. 6:531 (1974). 20. Griffiths, J. F., and Scott, S. K., Prog. Energy Comb. Sci. 13:161 (1987). 21. Gosman, A. D., in Thermodynamics and Gas Dynamics of Internal Combustion Engines, O. H.

Horlock and D. Winterborne, Eds). Oxford University Press, New York, 1984, Vol. 2.

CRITERIA FOR SPONTANEOUS IGNITION 22. 23.

24.

Gray, P., Griffiths, J. F., and Moule, R. J., Faraday Syrup. Chem. Soc. 9:103 (1974). Egeiban, O. M., Griffiths, J. F., Mullins, J. R., and Scott, S. K., Nineteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, 1982, p. 825. (a) Tyler, B. J., and Tuck, A. F., Int. H. Heat Mass Trans. 10:251. (b) Ashmore, P. G., Tyler, B. J., and Wesley, T. A. B., Eleventh Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, 1967, p. 1133.

395 A T = temperature excess at center of the vessel

(K) density (kg m -3) viscosity (kg m - l s - 1) thermal conductivity (W m - l K - l ) molar heat capacity at constant pressure (J mol-I K-l) M = molecular weight (kg mol-l) p /~ K Cu

= = = =

(i) evaluation of/3 From

Received 22 January 1988; revised 10 June 1988

p__V l+B(r---2+... RT

APPENDIX Calculation of the Rayleigh number

+/3_-! dV

V

V dt

1 B(T)/V 1 =- + = - to sufficient precision. T T T

~/3ro3 Cpp 2 Ra =

xxM

AT,

where g = acceleration due to gravity (m s-2) /3 = coefficient of cubical expansion (K- l) ro = radius of vessel (m)

(ii) evaluation of # and x For the composition c-C6Hl2 + 2 02, at 105 N m-2 and 500 K, # = 1.89x 10-5 kg m - l s - l x = 3 . 6 2 x 10 -2 W m -1 K -1.