A model for coal dust duct explosions

A model for coal dust duct explosions

COMBUSTION AND FLAME 44: 153-168 (1982) 153 A Model for Coal Dust Duct Explosions J. H. PICKLES Central Electricity Research Laboratories, Kelvin Av...

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COMBUSTION AND FLAME 44: 153-168 (1982)


A Model for Coal Dust Duct Explosions J. H. PICKLES Central Electricity Research Laboratories, Kelvin Avenue, Leatherhead, Surrey, K122 7SE

A theoretical discussion is given of the propagation of a dust explosion in a linear duct or pipeline. The particular aim is to investigate the experimental observation that propagating explosions are much harder to initiate in small laboratory scale ducts than in, say, coal mine galleries. A model is proposed in which a turbulent mixing phenomenon first identified by G. I. Taylor gives, for large ducts, very high flame velocities, which in turn lead to large fluid velocities and further increases in flame velocity. In small ducts, the time scale of the turbulent mixing is less than the time needed for the burning of individual coal particles. The particle burning time becomes an additional constraint on the rate of flame propagation and the development of explosions is inhibited.

INTRODUCTION An understanding of flame propagation in confined dust/air mixtures is needed for the safe design of industrial pneumatic conveying systems. The relevant experimental data are however concerned mainly with either coal mine galleries [1, 2] or with small scale laboratory ducts [3]. It is well known that in coal mine galleries, which are typically 2 m in diameter, flame propagation and acceleration can lead to serious explosions. Cybulski [4, 5], for example, reports explosions in an experimental gallery in which pressures of up to six or seven atmospheres were observed. In smaller ducts it appears to be much more difficult to initiate such explosions [6]. Nettleton and Stirling [3] found that an atmosphere of oxygen, rather than air, was necessary to sustain a propagating explosion in a 0.07-m-diameter duct. The apparent dependence of explosive propensity on duct diameter has important implications for the design of conveying systems, in particular pulverized fuel (p.f) systems in power stations, with duct diameters in the intermediate range, Examination of the question of diameter dependence leads to another, more fundamental, Copyright © 1982 by The Combustion Institute Published by Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, NY 10017

problem. It is not at all clear by what means the flame propagates, even in a large duct. There is no indication that the very large pressures observed by Cybulski [4, 5] resulted from a detonation wave, and the natural assumption E7] is that the explosion propagates as a deflagration, with the flame traveling into a flow field set up by its own motion. The observed pressures and flame speeds imply, at least on Artingstall's E7] simplified theory, that the flame's burning velocity--the rate at which it overtakes and consumes its fuel--attained values of 50 m/s or more. This is much larger, of course, than the laminar burning velocities of order 1 m/s predicted [8] or measured [9] for coal dust flames, or calculated on the basis of a radiative heat transfer mechanism [10]. The most likely mechanism for rapid flame propagation involves some form of turbulent mixing but no theoretical models have been proposed [6, 11]. The aim of this paper, then, is to develop a theoretical analysis of propagating coal dust flames within which these questions may be tackled. The emphasis is on rapidly burning flames leading to explosions with appreciable overpressures. A very wide range of phenomena is involved, in two-phase flow [12-1, in turbulent flame propagation [13-1 and in the physics and chemistry of the combustion




[14]. The model proposed is necessarily crude, especially as regards the chemistry, and retains only those factors which appear essential to an cxplanation of the experimental data. The results of Richmond and Liebman [2], using an cxpcrimental mine gallery, have guided the choice of theoretical model for the analysis of the dynamics of the flow set up by an accelerating flame of the type observed [2]. The methods used arc similar to those of Artingstall [7] and Jones [15], but are appliedtothefrecgeomctryofanopcn-endedduct, The propagation of the turbulent flame in the flow field is then examined; the main point here is that an attempt is made to allow for the dispersive 500

effect [16] of the mean velocity profile of the turbulent flow into which the flame travels. A heuristic picture follows of the acceleration which occurs as the flame feeds upon the turbulence in the flow set up by its own motion. Finally, the question of diameter dependence is considered. It is suggested that the dominant parameter here is the ratio of the particle burning time t Bto the time scale of the turbulent mixing t*. In large ducts, with tB < t*, flame propagation appears on a macroscopic scale much as it would in a gas, and is controlled by the rate of turbulent mixing. In small ducts, with tB>t*, particle burning begins to operate as the rate controlling factor in the propagation of the flame.




iG 300 O ..j uJ > 31: o J u. e-i z 200 <


z < ._j u,.




-,// 0










Fig. 1. Flame and flow speeds in a mine gallery explosion. (Data taken from Richmond and Liebman [21 .)





Figure 1 shows the dependence of flame speed w(t) on time t for the flame in an experimental mine gallery explosion regarded by Richmond and Liebman as a typical "strong coal dust explosion" (their test No. 3487). The time t is measured from the time of the initial "source" explosion which (at least in a mine gallery) is needed to initiate a developing coal dust flame. Also shown in Fig. 1 is the velocity ul(t ) of the flow immediately ahead of the flame, and the "burning velocity" w - u l, which is the rate at which the flame overtakes and consumes fresh fuel. The velocity u~(t) has here been calculated from Richmond and Liebman's measurements of the pressure p~(t) immediately ahead of the flame, on the assumption that the flow ahead of the flame is a "simple wave" [17]. Then

From the dynamical point of view, the most significant feature of the acceleration shown in Fig. 1 is that it is faster than it would be if it were conditioned by the presence of the closed end of the mine gallery. Substantial changes in flame speed and burning velocity take place in the interval of 0.2 s between t = 1.1 s and t = 1.3 s, when the flame has travelled about 100 m from the closed end. The flow changes produced by this acceleration would not be modified by the closed end until after a delay of 0.3-0.4 s, the time needed for a sound wave to travel from the flame (through combustion products at a temperature given by Richmond and Liebman as 900°K) to the closed end, and return after reflection. It appears that the flame does not accelerate by thrusting backward against the dosed end, which is the assumption implicit in the models of Jones [15] and Artingstall [7-J, but by thrusting against the inertia of its own combustion products. The model taken, therefore, will be that of a duct (mine gallery or p.f. pipeline) extending in both directions and with no closed end. Distance along the duct will be denoted by x and the motion treated as one-dimensional. The initial conditions chosen are an idealization of the conditions produced by a source explosion in a mine gallery, or by the sudden introduction of burning fuel into a p.f. pipeline. Thus, at time to, there is a uniform flow of speed uo along the duct, with (instantaneously) the region x > x o occupied by a preformed suspension of unburnt fuel at temperature To, and pressure P0, and the region x < x0 occupied by combustion products at the same pressure but a higher temperature Tf (see Fig. 2). As the interface between the unburnt fuel and the combustion products develops into a flame, and advances into the fuel ahead of it, it modifies the pattern of the flow. A more detailed description of the flame will be given later; here it is regarded simply as a surface of discontinuity in fluid properties, across which the energy of the fuel is released. The instantaneous pressures pl, P2, velocities u~, u 2, and densities p~, P2 immediately in front of and behind the flame are related by equations expressing the conservation of mass, momentum, and energy across the flame. When the flame has speed

2c~ t

ul(t ) = 7~

~(P1(t)/P~-)~- 1:2~._

l 1) ,


where Po~ and c~ are the pressure and sound speed in the undisturbed air far ahead of the flame and ), (= 1.4) is the adiabatic constant, The data in Fig. 1 are unlikely to be very accurate [Richmond and Liebman obtained the flame speed w by numerical differentiation of its position-time curve, and the pressures p~(t) needed for Eq. (1) were estimated from graphs in their published paper], but they indicate the general course of the development of the explosion. At first the burning velocity w - ul of the flame is very small, and it rides "on the back" of the flow generated by the source explosion. Little fresh fuel is consumed at the flame front, although it accelerates slowly, presumably (as suggested by Rae [1]) because of the continuing combustion of coal dust swept up from the gallery floor into the region behind the front. At about t = 1 s, the flame begins to advance into the fuel ahead of it and accelerates more rapidly. A flame velocity of 280 m/s and a burning velocity of 40 m/s are attained by t = 1.3 s. Finally, there is a very rapid increase in flame velocity (although the data here must be treated with caution because there is no corresponding increase in the flow velocity u~) before the flame runs out of fuel.




traveling wave ahead of the flame

Pl(W--Ul)= P2(W--U2), Pl + Pl(W-Ul) 2=p2 + p2(w-u2) 2,



l U--Uo~ 2~'/~'- 1 1+ } ( 7 - 1 ) ~ c ~ - ] ,


1 U--UO~ 2/3~- 1 1+ ~ ( 7 - 1 ) , Co /


7Pl + 1 (W__Ul)2 +q ( 7 - 1)pl .






1 U-Uo~ c=c o 1+ ~ ( 7 - 1 )

-'1- 1 ( w _ u 2 ) 2 .

(7 - 1)p2 In Eq. (4), 7, the ratio of specific heats, is taken for simplicity to be the same in both the unburnt fuel and the combustion products, q represents the heat energy released in the combustion of unit mass of fuel. The natural assumption is to take q as constant, but it is more convenient to adopt an artifice used by Jones [15] and to impose on q a slow variation which permits the construction of simple mathematical solutions for the motion. Here q will be supposed to vary in such a way that the charge in entropy of unit mass of fuel in passing through the flame AS=Cv{ln (p2p2-~') - In (PlP~-~)}


remains constant as the flame develops [Cv in Eq. (5) is the specific heat at constant volume]. In this special case the flows both ahead of and behind the flame can be represented as simple waves. Simple waves have the property that all the fluid variables are uniquely determined by the local flow velocity, so that their space- and time-dependence is determined by that of u(x, t). In the forward


Co ,/


where c represents the sound speed, and Po and co the density and sound speed of the unburnt fuel in the initial state. Similarly, in the backward traveling wave behind the flame


~2 7/~,- 1

1- 1 u-u o ~ ( 7 - 1) - - -cf- - j

p--pf ( l - 1 ,]2,.;,- l \ ~ (7 - l) u -cru° / ' / 1 U-Uo\ c = c f ~I - } (7 - 1) - - - - - ) , cf /


where pf=(To/Tf)p o and cf=Cox/(Tr/T o) are the initial density and sound speed of the combustion products. Typical velocity and pressure profiles for the motion are sketched in Fig. 3. The existence of a backward traveling as well as a forward traveling compression wave is confirmed in the wave aliagram for Richmond and Liebman's experiment (Fig. 5 of their paper [2] ), although further calculations would be needed to compare thetheoretical pressure profiles with the experimental data.







Fig. 2. Initial state for gas dynamicalcalculation.


15 7












Fig. 3. Pressure and velocity distributions for flame-driven motion. To get the relation between the burning velocity v = w - u I and the velocity u l of the flow, Eqs. (6) and (7) may be specialized to the fluid variables immediately ahead of and behind the flame, Elimination of the pressures and densities in Eqs. (2) and (3) then leads to simultaneous equations for the velocities ut and u 2


Tf (W-Ul) Too



1+ ~1 ( 7 - 1 ) - -u-1--- -u o Co /

\2/~- 1

=(w-u 0 x

1 - ~1 ( 7 - 1) -u-2---u- -0) cf /

, (8)



1+ ~ ( 7 _ 1 )

which may be solved numerically for any given w. Figure 4 shows the resulting relation between the burning velocity w - u ~ and the flow velocity u l for various initial flame temperatures Tf. Before proceeding it is necessary to determine the range of applicability of the results of Fig. 4. The simple wave solutions calculated are valid mathematically so long as no shock forms in the flows ahead of or behind the flame, and meaningful physically so long as Eq. (4) does not require an implausible variation in the heat energy q released at the flame. Figure 1 shows a concave graph of flow speed u~ against time, and so calculations of shock formation have been done for an assumed time dependence.

Co / Ul(t) - u o = A ( t - to) 2.



{ =(w-u 0

( cf z

1 u2 _Uo~2 1- } (7-1)-----cr /

+ 7(w- u2)21 ,

The method of Landau and Lifshitz [18] or Jones [15] then shows that a shock first forms in the forward traveling wave, when (for Tf=6To) (9)

u I - U o = 1.53%.






--:8 To

1.0 _


] i

u.I o




Tf ~=4

i '°





0.1 (W - U I )/Co




Fig. 4. Relation between burning velocity w - u 1 and flow speed u 1 for various initial temperatures.

Landau and Lifshitz's calculations for the very similar motion produced by an accelerating piston show that the maximum flow velocities attained before the shock forms are not very sensitive to the time dependence (10) assumed. The variation of q over this range of flow velocities is shown, together with the temperature T2 of the combustion products immediately behind the flame, in Fig. 5. The gradual increase in both q and T 2 as the flow (and flame) accelerates is not unreasonable and would be consistent with an increase in the efficiencyof the combustion process as the flame develops,

MIXING AND COMBUSTION IN TURBULENT DUST FLAMES Figure 1 shows that burning velocities of up to at least 40 m/s were attained for the flame. As already pointed out, these are substantially higher than normally observed, and cannot be accounted for by either a radiative heat transfer mechanism (for which Essenhigh and Csaba's treatment [10] predicts a burning velocity of about 1 m/s) or by adiabatic compression and heating in the fuel ahead of the flame (where even the highest over-





o" ~_


Z l-


10 i_. °




"' Q. Ig u,J I--


Ig .....1








F L A M E S P E E D (W - Uo ) / C o

Fig. 5. Heat input and flame temperature for flame with initial temperature ratio r ~ / r o = 6.

pressures observed, about 2 atm, would produce temperatures of about 400~K, still well below the particle ignition temperature). The rapid combustion is presumably initiated by turbulent mixing in the flow of about 100 m/s generated by the source explosion. A similar initial flow might be generated in a p.f. pipeline, where a pressure transient of, say, 0.2 bar would according to Eq. (1) temporarily increase the pipeline flow speed by 70 m/s. The following discussion of the turbulent combustion process applies strictly only in the limit when the burning velocity is small, and the flame does not appreciably perturb the initial flow, but it is to be expected that the processes leading to larger burning velocities will be qualitatively similar, In rapid, explosive combustion, the release of energy derives most probably from the burning of the volatile components of the coal particles, rather

than the carbon residue. Field et al. [14] discuss the processes which control the rate of release and combustion of the volatiles--heating of the particle by conduction from the hot combustion products of neighboring particles, chemical decomposition and devolatilization of the particle as its temperature rises, and diffusion-controlled burning of the volatiles around the particle. Their formulas (see Appendix) suggest that for a typical particle of diameter 50 # in a flame at a temperature 1800°K a total time TB of 10-20 ms is needed before volatile burning is complete. The inertial response time for the particle z n, as calculated from Stoke's resistance law, is about 12 ms in air at 300°K, or about 4 ms in combustion products at 1800°K. These time scales have to be compared with the time scales of the turbulent fluctuations of the bulk flow. For a flow uo in a duct of radius a, Owen [12] suggests a time



scale for the energy containing eddies t* ~ 10 la/u*,

mixing in a mean flow u can be represented by a longitudinal diffusion coefficient (11) D ~ 10au*.

where u* is the friction velocity at the duct wall. For the experimental conditions of Fig. 1, with a = 1.4 m and uo _~100 m/s, t* _~40 ms. These conditions approximate the limiting case where the particles follow the turbulent fluctuations in the flow, and burn rapidly on the scale of the eddy lifetimes. The controlling process in the propagation of the flame is likely to be the rate at which turbulent mixing brings unburnt fuel into contact with the hot combustion products. Many authors (see, for example, the review by Andrews et al. [13]) have proposed models which describe the role of turbulence in generating large flame surfaces at which unburnt fuel may be ignited. The simplest is that proposed originally by Damkohler [19], where the turbulent burning velocity is related directly to the velocity fluctuations in the approach flow. The present application is concerned with a flame which is carried along with the flow, rather than a fixed burner flame, but arguments of the same general kind suggest that in a flow of mean velocity u, vocu*,


where v = w - u is the effective turbulent burning velocity of the flame as discussed earlier. (The subscript 1, which labels the"approach" flow ahead of the flame, is now omitted.) An argument due to Taylor [16] suggests that the constant of proportionality in Eq. (12) will be large. Taylor shows that in a duct flow, mixing is dominated by the influence of the mean velocity profile. It may be imagined that parcels of flame are carried forward in the middle of the duct, where the mean velocity is higher; turbulent fluctuations then induce radial mixing with hot burning fuel mixing outward, igniting the unburnt fuel near the duct wall. Similarly unburnt fuel mixes into the middle of the duct where it is ignited, and, in turn, carried forward (see Fig. 6). According to Taylor the effect of the


Experimental work [ 16] with nonreacting flows (saline and dye solutions) confirms that very high rates of mixing can occur, greater by orders of magnitude than those induced by longitudinal turbulent fluctuations alone. Here the entities mixed are the parcels or turbulent eddies of burnt or unburnt fuel, with eddy lifetimes t* given by Eq. (11). Dimensional arguments suggest therefore that the effective turbulent burning velocity for the propagating flame is of order /D v~ ~/~ ,

or, from Eqs. (11)and (13), v~10u*,


in accordance with Eq. (12). With

Ul*~-u/30 for the Reynolds numbers of order interest here [16], Eq. (14) gives v~ ½ u.

10 7 which are



Turbulent mixing might thus sustain very high burning velocities, at least in the absence of any other constraints. FLAME ACCELERATION The preceding analysis leads to an idealized model of flame acceleration. For the assumed "simple wave" behavior of the flows generated by the flame, the flame-generated perturbation u - u0 of an initial flow u0 is determined, via Eqs. (8) and (9), from the effective burning velocity of the flame w - u = v . At the same time Eq. (15) gives the dependence, in a fully developed turbulent flow, of the burning velocity v on the total flow u. A flame burning at a




FUEL Fig. 6. Flame profde for propagatingturbulent flame.

speed v, and driving a total flow u, may be expected to accelerate so long as Eq. (]5) win permit, at that flow speed u, a further increase in burning velocity when the turbulence in the flow is fully developed, The dynamical relation of Eqs. (8) and (9) is plotted again in Fig. 7, for a range ofd~crcnt values

of the initial flow Uo. These curves have Tf/To = 6, and represent the development, say, of a flame moving into a coal dust suspension at a temperature T o = 300°K and ~ving combustion products at a tempcrture ]800~K. The kinetic relation of Eq. (15) may be plotcd on the same axes, ~ving the







r~ klJ

~. 0.5 0 ._J ii

U0 Co







0. I





Fig. 7. Flame speed-flow speed relations for a large duct.










0.8 o

o" Ill klJ rt

0.6 o / u.


l/ 0








Fig. 8. Flame acceleration controlled by flame speed-flow speed relations.

straight line OK as shown. For a large initial flow uo, the kinetic relation OK remains always to the right of the curve AA' representing the dynamical constraint. Turbulent mixing is thus always able to sustain a flame with a burning velocity v greater than that required to maintain the flow. Continued acceleration, leading to an explosion, therefore occurs. Figure 8 gives a schematic representation of the feedback process which drives the acceleration. For a small initial flow, the dynamical curve B B ' cuts OK at D and a stable state ensues with a constant flame speed v', at least for so long as the simple wave model remains valid and the initial flow uo persists. When the initial flow is withdrawn,

the flame will collapse. It will be seen that there is a critical initial flow Uo~, corresponding to the curve CC' in Fig. 7, which is the minimum flow speed required to initiate a self-sustaining explosion. Quantitative estimates of this critical flow have been derived from the argument outlined above. To simplify calculation the dynamical curves of Figs. 7 and 8 were approximated as U=Uo + F v +

G H-v/c o

Gc 0 H



where c o is the speed of sound, and the numerical coefficients F, G, and H chosen to match the exact



TABLE 1 Coefficientsfor NumericalApproximationto v--uCurve of Figure 7 Temperature ratio Tt,/To 4 5 6 7 8

F 0.57 0.64 0.70 0.76 0.81

G 0.075 0.081 0.085 0.086 0.087

H 0.38 0.34 0.32 0.30 0.28

curves are given in Table 1 for a range of values of the temperature ratio Tf/T o. A dimensionless factor 21 was introduced into Eq. (15) to allow for the numerical uncertainty in this order-of-magnitude relation; thus v=1021u*___ ~21u.


The critical initial flow is then found to be

Uoc= Co

(~/-~ (~_ ) ~HG)2 3 _F .


THE EFFECTS OF FINITE PARTICLE SIZE The dust particles have so far been treated as though they were molecules of a combustible gas, without consideration of effects arising from their finite size. In duct flows at high speed, however, the time scale of the turbulent velocity fluctuations 0. la 27" ~

modeled, within the analysis of the dynamics of flame acceleration given earlier, by a reduction in the turbulent velocity scale u*. On the other hand, the relative motion will tend to increase the rates of transfer of heat and oxygen to the surface of a burning particle, giving a small decrease in the burning time scale zB. More fundamentally, the finite particle burning time forces a qualitative change in the pattern of combustion. The picture of distinct parcels or eddies of burnt and unburnt fuel, as presented in Fig. 6, is valid only if the typical eddy lifetime is rather longer than the typical particle burning time. If this is not the case, the parcels lose definition and the particles are mixed together more rapidly than they can burn. Their finite burning time begins to operate as the ratecontrolling constraint on the propagation of the flame. The burning velocity of the flame in these conditions may again be estimated by using Taylor's theory of longitudinal mixing. The longitudinal diffusion coefficient D is, as before, given by Eq. (13), but it must now be supposed that the entities mixed are not parcels of fuel, with lifetime t*, but single particles with lifetime z B. The burning velocity therefore becomes. v ,-,




when the eddy lifetime is smaller than a time of the order of the particle burning time 273- Within this approximation the complete flame speed-flow speed relation, over the whole range of flow speeds u, becomes




becomes comparable with the inertial response time 27r of the particles and the time ts needed for their ignition and burning. This happens first in the smaller ducts. A variety of complicated effects will then occur. For example, the particles will not respond wholly to the turbulent velocity fluctuations, but will move relative to the surrounding air [12]. For a stoichiometric coal dust/air mixture the relative motion may damp the turbulence in the flow by as much as 10%. This effect might be

v~lOalu* l~au~ v~ 22 / - - ~/ 27B


u*<0.1 2227B, 2 2 2a for

u* > 0.1

~12~- B .


Dimensionless factors 21 and 22 have again been introduced into Eqs. (19) and (20) to allow for the numerical uncertainties in these order of magnitude relations. The transition between the two regimes

164 at u*'-~ 0.1



(22,]2 a --or u--- 3 -\ ' ~ 1 / / "CB ~11/ % is chosen to ensure continuity of v as function of u. The relations of Eqs. (19) and (20) can be plotted on the same axes as the dynamical relation (16) to investigatethestabihtyofaflamewithagivenvand u. The critical initial flow for the development of a self-sustaining explosion is again fixed by the

requirement that the two curves touch in the v--u plane. For large ducts, the curves touch in the low flow speed regime of Eq. (19), as before. In small ducts, the curves touch in the high-flow speed regime [Eq. (20)] as shown in Fig. 9. The critical initial flow is then increased. The variation of the critical initial flow with duct radius a, as obtained for a flame with temperature rate by numerical calculation from Eqs. (16), (19), and (20), is shown in Fig. 10.


IEq (20) for small duct I.O


/ / / Eq ( 19),large



duct limit



Q ,,,




0 _.1 u" ' ~





Uc/C O


~" 0








Fig. 9. Flame speed-flow speed relations for a small duct.








0 -1 u. ..I


~I -








DUCT RADIUS ~, a/C -,2

o B

Fig. 10. Critical initial flow as a function of duct radius.

RESULTS AND DISCUSSION The preceding analysis considers the stability of a rapidly propagating dust flame in an open-ended duct, given a particular set of assumptions regarding its interaction with the turbulent flow. The analysis, though idealized, suggests two points of practical importance for coal dust explosions in ducts or pipelines. First, even in large ducts an initial flow is needed to generate a self-sustaining explosion. The order of magnitude of the flow speed required can be estimated by setting 21 = 1 in Eq. (17); this gives u c ° - 4 0 m/s.


For comparison, Rae [1] reports that in his 2.5-mdiameter gallery, ignition sources giving an overpressure of less than 0.12 atm or an initial flow of less than 30 m/s failed to produce self-sustaining explosions. Sources with initial overpressures and flow speeds 50~ greater gave explosions whose

strength was limited only by the supply of fuel and the length of the gallery. Bartknecht [20] obtained accelerating flames in ducts of 1.4 m and 2.5 m diameter with an initial flow of 40 m/s, while the experimental data of Richmond and Liebman [2], as shown in Fig. 1, correspond to an initial flow of 50-100 m/s. These observations do not necessarily give a very precise test of the theoretical prediction (21), Rae's results in particular being obtained for explosions weaker and accelerating less rapidly than those envisaged here. The prediction is in any case itself very sensitive to changes in the factor 2,, introduced in Eq. (17) to characterize the quantitative uncertainties in the order-of-magnitude arguments used. [The predicted minimum flow (21) varies from 0 to 200 m/s, as 21 varies from 0.5 to 2.] Nevertheless, there is some experimental support for the idea that an initial "kick" is needed to trigger an explosion. The second point concerns the effect of the finite particle burning time on the propagation of explosions in small ducts. The variation of the critical



initial flow Uocwith duct radius, plotted in Fig. 10, is steep and in practice might appear as a cutoff at a radius a min 0.1 c°r B (22) ~-2 "~

below which the propagation of explosions becomes very difficult. Setting Co= 350 m/s, zB= 15 ms, ).2 = 1 gives as a rough estimate of the cutoff radius: amin~ 0.5 m. Variations in the factor 22, which characterizes the quantitative uncertainties in Eq. (13), will not affect the conclusion that the theoretical cutoff (17) lies within, or close to, the practically important range of duct diameters up to 1 m. Unfortunately no experimental data seem to be available for ducts of this size. From an ideal standpoint, this theoretical analysis rests rather too heavily on the representation of the turbulent flame in terms of Taylor's mixing coefficient D. Further theoretical elaboration would, however, carry little conviction in this complex problem without some experimental confirmation. The simple theory given here offers guidance in the design of experiments. The central role of the initial flow u o emphasizes that the properties of the explosion cannot be analyzed independently of the event which triggers it, or of the circumstances in which it occurs. Measurements of the strength of the ignition source are important and could perhaps be used, in the analysis of experimental results, as a substitute for the flow variable ordinate in Fig. 10. The simple theory also brings out the role of the particle burning time. Experimental results with fuels of different burning times should, when appropriately scaled, collapse onto the single curve of Fig. 10. In particular, following Nettleton and Stirling [3], small-scale experiments could be made, using very small particles or oxygen enriched mixtures to give appropriately small burning times, CONCLUSIONS A theoretical model has been developed for dust explosions, and in particular coal dust explosions,

propagating in a one-dimensional duct. The model describes the flow pattern along the duct generated by the explosion, the turbulent mixing which is presumed to control the rate of combustion in the explosion, and the constraints introduced on its propagation by the finite burning time of the particles. There are two theoretical predictions: 1. A dust flame in an open ended duct will not develop into a self-sustaining explosion with an appreciable overpressure unless driven by some initiating explosion, or some preexisting flow, in the direction of flame propagation. 2. The finite particle burning time restricts the development of explosions in small ducts. There is a cutoff diameter, estimated to be of order 0.5 m, below which it becomes markedly more difficult for explosions to propagate. Experimental work is needed to test these predictions. The analysis indicates that the cutoff diameter scales with the typical particle burning time, so that it would be possible to test the principles of the theory in small-scale experiments such as those of Nettleton and Stirling [3].

APPENDIX COMBUSTION TIME FOR A COAL PARTICLE The burning of the volatile components of an initially cold coal particle can be treated in three stages: i.


heating of the particle by conduction from the hot combustion products of neighboring burning particles;

chemical decomposition and devolatilization of the particle as its temperature rises; iii. burning of the volatiles released. The times associated with these processes, assuming that the initial temperature of the particle is To = 300°K, the temperature of the combustion products is Tr= 1800°K, and that stages (ii) and (iii) take place at an average intermediate temperature T~= 1200°K, can be estimated as follows: i. The temperature T of a particle at an initial temperature To, immersed in an atmosphere at temperature T~ increases with time according to



Eq. K. 4 of Field et al. [14] : T=To+(Tf-To)(1-

On the assumption that the speed of the chemical reaction is the rate controlling factor,

exp (-fit)); ~3 ~ 3ms.

with Diffusion control is, on these figures, probably the more important, so that the volatile burning time is better estimated by Eq. (A3). The total burning time z B for the particle is thus

12K fl~- pCpd 2 " Here K is the thermal conductivity of the surrounding atmosphere, evaluated at a "film" temperature ~(To + Tf), and p, Cp, and d are the density, specific heat, and diameter of the particle. In mks units, setting p = 103, C p = 1.5, K = 6 x 10- 5 gives fl = 200, and the time taken to reach T~= 1200°K is

z 1 _ 5 ms.

(A 1)

ii. The proportion f of volatiles released from a particle held at a temperature Tiincreases with time according to Eq. (4.3) of Field et al. [14]: f-

1 - exp [ - C 2 ( e x p

(--C3/Ti))t ].



al. [14] discuss the rate of reaction of the volatiles released from a coal particle at a temperature of 1273°K which approximates to the temperature T i = 1200~K chosen here. T w o assumptions are explored. On the assumption that the reaction rate is controlled by the laminar diffusion of oxygen inward to the volatiles, which form a layer round the surface of the particle, the volatile burning time is estimated as


This work was done at the Central Electricity Research Laboratories, Leatherhead, and is published by permission o f the Central Electricity Generating Board.


iii. Sections 3.1 and 3.2 of Chapter 5 in Field et

273 7 ms.

separately. For the purpose of the two sections preceding the conclusions, the total burning time for a particle with a typical diameter 50 #m will be taken as


and for T~= 1200°K, r 2 ~ 8 ms

Direct addition of Eqs. (A1)-{A3)gives z B~ 20 ms. In practice the motion of the coal particle relative to the surrounding combustion products will accelerate heat and mass transfer rates and reduce the times z I and z 3. There will also be some overlap between the processes (i)-(iii) which are here treated

ZB= 15 ms.

Their constants C 2 and C a are estimated as 1.5 x 105 s - ~ and 8900°K, respectively. The time constant for the release of volatiles is thus

~2 ~ C-2 exp

z ~ ~ z 1 + rz + z 3.


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