Model of accelerating coal dust flames

Model of accelerating coal dust flames

COMBUSTION AND FLAME 62:255-269 255 Model of Accelerating Coal Dust Flames D W I G H T P. C L A R K and L. D O U G L A S SMOOT Chemical Engineering...

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AND FLAME 62:255-269


Model of Accelerating Coal Dust Flames D W I G H T P. C L A R K and L. D O U G L A S SMOOT Chemical Engineering Department, Brigham Young University, Provo, Utah 84602

A model describing turbulent coal dust flame propagation and acceleration is based on the transient, macroscopic equations of change. The turbulent flame velocity was obtained from a simple correlative technique combining turbulence and chemical effects. Predictions indicate that a deflegrating coal dust flame can accelerate to high velocity and pressure, with increasing turbulence a major cause of the acceleration. A parametric study was conducted to identify key parameters in the model. The need for turbulent flame velocity data for particle-laden systems was identified and the effects of duct diameter, coal particle diameter, and various model parameters were described. The model is useful for describing relative effects of various parameters.

INTRODUCTION One o f the greatest hazards associated with the utilization o f coal lies in its ability to create selfsustaining coal dust explosions. Explosions in mines typically require an ignition source, usually p r o v i d e d by pockets o f natural gas trapped in the rock. Pressure waves generated by these initiating gaseous explosions entrain coal dust from the walls and floor of the mine. The coal d u s t - a i r mixture then ignites and the flame front accelerates down the mine p a s s a g e w a y , often causing extensive d a m a g e and injury. Coal dust explosions are not confined to the mining industry, however, and dust explosions are not limited to coal. C o a l - f i r e d p o w e r generating plants and other coal processing equipment also experience coal dust explosions and fires. Though often not as energetic as explosions e x p e r i e n c e d in mines, flames generated in coal pulverizing systems can be quite destructive. Safety measures in recent years have helped prevent p r o b l e m s both in coal mines and coal pulverizing systems which lead to explosions. H o w e v e r , explosions still occur and it is essential to understand, characterize, and eventually Copyright <~;' 1985 by The Combustion Institute Published by Elsevier Science Publishing Co.+ Inc. 52 Vanderbilt Avenue, New York. NY 10017

predict the nature and causes of all types of dust explosions. An attempt was made in this study to characterize some of the important p a r a m e t e r s in coal dust explosions through modeling o f the flame processes. The a p p r o a c h adopted c o m b i n e d aspects o f turbulence and chemistry through correlation o f the turbulent flame velocity with m a c r o s c o p i c mass, m o m e n t u m , and energy balances to d e t e r m i n e the p a r a m e t e r s important to an accelerating coal dust flame. A v a i l a b l e data were also sought in an attempt to incorporate realistic information for coal reactions. TURBULENT MODEL



The general m a c r o s c o p i c equations o f change for particulate systems ( e . g . , C r o w e and Smoot [i]) were used as the foundation for the accelerating flame model, along with a p p r o p r i a t e auxiliary relationships. P i c k l e s ' s [2] equations describing the flame front, which are essentially the Hugoniot relationships for deflagrations in a transformed coordinate system, are based on the general m a c r o s c o p i c equations for a single








Combustion Products

Compressed Reactants

P2 V2 T2


Undisturbed Reactants

p0 T1

Zone 2

3n{ F

Zone 1

v0 To

Zone 0

Fig. 1. Diagram of an accelerating flame.

phase where all source and sink terms other than heat release have been neglected. One of the important aspects of the present approach was the inclusion of source terms due to the particles. The flow field was divided into four regions as shown in Fig. 1, with arrows indicating the direction of flow. This transient flow is treated with two sets of equations; those relating zones 0 and 1 and describing properties of the pressure wave, and those relating zones 1 and 2 and describing properties of the flame.

Pressure Wave Equations (Zones 0, 1) Laboratory dust and mine gallery tests show that a pressure wave can develop in front of an accelerating dust flame [3, 4]. This pressure wave serves to entrain coal dust from the floor and walls of a mine passage. Although no attempt was made to include the process of coal entrainment, the pressure wave description was essential to the model. The pressure wave imparts a velocity to the reactant mixture which in turn alters the turbulence level and the flame velocity and hence flame propagation and acceleration. For a description across the pressure wave, between zones 0 and 1, equations without reaction source terms were used. Cybulski [5] describes this pressure wave propagation for unsteady, one-dimensional, shockless flow. These equations are derived from the transient continuity and momentum equations for a single phase. Time- and position-dependence are removed through a mathematical transformation. The flow is considered to be isentropic. The resulting relationships describing properties be-

tween zones 0 and 1 are [5] Pt =P0[1 + ( 3 , - 1)/2 (vi-vo)/ao] 2v/~-I,


pl = p0[1 + ( ' r - 1)/2 ( v l - Vo)/ao] 2/~- 1


The subscripts in these two equations represent the regions depicted in Fig. 1. Individual variables are defined in the nomenclature. These equations were intended for simple waves. Where solutions produce sharp pressure rises, results must be questioned. Further, these equations only describe the gas phase so the particles were assumed to remain unentrained until after the pressure wave. This assumption is not valid but the error is small for dilute particle systems. For an improved estimate, the mass of the particles in dynamic equilibrium with the gases can be included in the gas density.

Flame Equations (Zones 1, 2) The equations describing the flame front are derived from the general equations of change for flow in a duct. They were then reduced to a single dimension and integrated to form the macroscopic equations of change [1]. The flame front is consequently treated as a significant change in the flow field much like a compression wave [2, 6]. One of the major assumptions was that particles and gas were in dynamic equilibrium. Consequently, the momentum equation for the particle phase was not required. The analysis was transient but it was found that the gas-phase momentum equation contributed little to the transient wave and the time-dependent term was dropped. The gas-phase energy equation was slightly influential for transient calcula-



tions while the gas-phase continuity equation contributed the most to the transient wave. The particle phase equations were used to calculate source terms which modified the behavior of the gas phase. Both of these particle equations were presumed to be in quasi-steady state, thus adjusting instantaneously to the time-dependence of the gas phase. Additional assumptions were also made to arrive at the flame equations: (1) particles have negligible volume compared with gas; (2) axial body forces are negligible; (3) particle dilation is negligible; (4) particle pressure-volume work is negligible; and (5) particle transpiration effects are neglected. In the early stages of particle entrainment, the particles will have different velocities from the carrier gas and so dynamic equilibrium is not valid in this region. However, in the latter stages of entrainment, as the flame overtakes the particles, the velocity of the particles and the gas will be similar. Given enough distance between the pressure wave and the flame, the particles velocities will approach that of the gas at the onset of the flame. Dynamic equilibrium is therefore thought to be a reasonable assumption in the flame zone, especially for the smaller particle sizes which are likely to be entrained in the gas flow. Assumption 2 is valid for a dilute particle system. The remainder of the assumptions were adopted to simplify the equation set. After applying the symplifying assumptions, the equations were transformed to a fixed reference frame from a moving reference frame tied to the flame front [2]. The velocities shown in the following equations therefore describe flow with respect to the wall and not the flame. The continuity equation for the gas phase is dmg/dt=Ac[ol(w-

v l ) - o 2 ( w - U2)] "+"Rp.


The first term represents the accumulation of mass in the control volume represented by the flame. Since position-dependence is integrated out of the equations of change, the control volume was found by multiplying the crosssectional area of the duct by the flame thickness obtained from another relationship (discussed

later). Average properties (e.g., density) of the gas across the flame (regions 1 and 2) were used for calculating the accumulation of mass. The gas phase in all regions represented by Fig. 1 was assumed to be ideal. The next two terms represent the net convective mass flow and the final term is the generation of gas from particle reactions. In all equations, subscripted numbers indicate the region in Fig. 1 described by the particular variable. The corresponding gas-phase momentum equation is PJ + Pl (w - vl) 2 + wRp/Ac + r w A J A c =/92 + P2( w -- u2) 2.


The pressures and the momentum convective terms are self-evident. The third term from the left represents change in momentum due to mass addition to the gas phase from particle reaction. The fourth term represents change in momentum due to wall friction. Both of these terms describing processes in the flame were calculated using variables such as the velocity of the flame front, w, which applied directly to the flame. Again, individual variables are described in the nomenclature. The gas-phase energy equation is d / d t {mg[ig + (IV-- 0)2/21} = Ac{oI ( w - vx)[hl + ( w - ol)2/2] - 02(w - v2)[h2 + ( w - v2)2/2] + q'} + (Qw + Qrc)/A~ + RphjA~.


In Eq. (5), the term enclosed by the first [ ] is the accumulation of energy in the flame zone. As was the case for the gas-phase continuity relationship, the control volume was calculated from the duct diameter and the flame thickness (to be discussed). Average properties (e.g., density) across regions 1 and 2 (Fig. 1) were used to estimate the accumulation term. The next two groups of terms describe the flow of thermal and kinetic energy in and out of the flame zone. The term qt is the turbulent heat conduction across the flame. The next set of parentheses groups heat transfer to the wall (Qw)



with heat transfer between particulate and gaseous phases (Q~). The last term represents change in energy due to mass addition to the gas phase from particle reactions. For the particulate phase, the steady continuity equation shown below represents a simple mass balance for the particles across the flame: Rp/A~





This equation was used to calculate the macroscopic rate of reaction for the particles, Rp. The two variables at the far right, the coal loading (CL) and fraction of coal reacted (fr), were to be specified and remained constant. The coal loading term is the coal concentration on the reactant side of the flame. Coal devolatilization and subsequent combustion of the volatiles were presumed to dominate the reaction processes. The fraction of coal reacted, fr, is taken to be the fraction of volatiles released. The coal volatiles were assumed to react to their equilibrium constituents in the gas phase. A comprehensive equilibrium program was used for the calculation of product species [7]. It was found that the most important species were CO2, CO, H20, H2, 02, N2, Ar, and SO2. These species were assumed to comprise the complete set of gaseous product species in all calculations. The final equation comprising the flame zone equation set is the steady-state particle energy equation:

Auxiliary Equations Several auxiliary relationships were required to complete the characterization of the accelerating dust flame. Most of these relationships are summarized in Table 1. Ideal equations of state [Eqs. (8) and (9)] were used to describe the gas phase in both regions 1 and 2. The gas phase pressures on either side of the flame were set equal [Eq. (10)] for the deflagrating flame [8]. Enthalpy relationships were required for both gaseous and particulate phases as well as for the enthalpy addition to the gas phase from the reacting particles [Eqs. (11)-(13)]. In addition, equations were included to describe convective heat transfer to the wall as well as turbulent conduction across the flame. Convective heat transfer to the wall was assumed to occur through the gas phase alone. This is a valid TABLE 1 Auxiliary Relationships for Accelerating Flame Model Equations of State PIMI = piRTi


P2Mz = p2RT2


Static Pressure -°2 = P~


Enthalpy Gas Phase r

Ppl(W-- Vl)[hpl + (W-- Vl)2/2]

h = ~,, c~i ,iro Cri d T + ~, c~,hf,°

Particle Phase

= Pp2(W- b'2)[hp2+ ( W - 192)2/2]

1- Cpr d T + hip ° hp = I r0 + RphJAc

+ @c/A,..



The various groups of variables are readily identified by comparison with the gas-phase energy equation. Equation (7) was used to calculate Q~, the heat transfer between gaseous and particulate phases. This term was used to investigate the two alternatives of either proportioning the heat of reaction between particles and gas (thermal equilibrium) or assigning all of the heat of reaction to the gas phase (Q~c = zero).


Enthalpy Addition to Gas Phase h, = .iro Cvv d T + hu °


Nusselt Number for Turbulent Flow Nu = 0.023 Re°~° Pr 113~


Convective Heat Transfer to Gas from Wall Q~ = Nu kvr6'(T. - 7"2)


Turbulent Conduction across Flame q' = kt(T~ - Tz)/b'




assumption for a dilute particle-gas mixture. A standard heat transfer correlation was used for turbulent heat transfer in a duct [9] [Eqs. (14) and (15)]. The turbulent flame thickness was used to calculate the surface area of the duct exposed to the flame. The turbulent heat conduction across the flame was calculated from standard conduction equations except that turbulent coefficients replaced their laminar counterparts [Eq. (16)].

Flame Velocity Correlation

Laminar Flame Velocity

Correlative Equations The turbulent flame velocity is defined as the velocity normal to the flame front at which the flame propagates into the uncombusted fuel, relative to the velocity of the fuel itself. Reasonable success in estimating the turbulent flame velocity has been achieved in gaseous flames by correlating the ratio of the turbulent to laminar flame velocity with various turbulence parameters. Andrews et al. [10] proposed a correlation of the turbulent gaseous flame velocity as a function of the laminar flame velocity and the turbulent Reynolds number based on a turbulent microscale. Other investigators, such as Ballal and Lefebvre [11], specified different forms of the correlations for different regimes of turbulent combustion. Smith and Gouldin [12] conducted experiments in the wrinkled laminar flame regime and found that correlations of the same nature as Andrews et al. [10] were satisfactory. The effect of pressure on the propagation velocity of turbulent flames was measured by Golubev et al. [13]. Ballal [14] studied the influence of the laminar flame velocity on turbulent flame propagation. A review of various correlation attempts for gaseous data is reported by Andrews et al. [15]. The form in which they represented the data of several investigators was u t / u I oc


where Rex = u ' X/v.

This approach, applied to dust flames, is attractive for three reasons. First, the turbulent Reynolds number considers the turbulence effects. Second, the inclusion of the laminar flame velocity accounts, at least in part, for the chemical reaction effects on the turbulent flame velocity. Third, the form is simple. However, estimates of the laminar flame velocity and the turbulent Reynolds number are required. Further, use of the relationship for dust-containing flames has not been conclusively demonstrated.


A value for the laminar flame velocity is required in the correlation. The laminar flame velocity depends upon the volatiles content and moisture content of the coal, the particle size, the coal/air mass ratio (i.e., coal loading) [16], the initial temperature, and the pressure. Direct prediction of this quantity is possible [ 17] with a comprehensive numerical model, but the method is too cumbersome for inclusion in this treatment. Therefore, a method of data correlation has been adopted. An empirical correlation of available laminar burning velocity data [ 16] is used for the coal dust-air laminar flame velocity, as a function of mass mean particle diameter, dust concentration, and coal type. This correlation was modified to include initial temperature and pressure effects [18]. Since temperature and pressure effects have not been explicitly measured in coal dust systems, gaseous laminar flame velocity data were used to estimate trends associated with these variables [15, 19]. The comprehensive coal dust model [16] has been used to a limited extent to evaluate this proposition, with encouraging but incomplete results.

Turbulent Reynolds Number In order to apply the correlation of Eq. (17) to systems where data for turbulence scale and intensity were not available, a classical method for estimating the turbulent Reynolds number was used from universal velocity profile data for fully developed pipe flow. With e = u '/, and with the result from Dryden [20] for isotropic



turbulence, the turbulent Reynolds number is Rex = (49E/v) 0.5.

flow of coal dust flames exist. Thus, the data for gaseous turbulent flames of Ref. [15] have been used in the correlation, again with some uncertainty, together with limited values deduced from accelerating dust-laden experimental mine gallery data from Richmond and Liebman [4]. Although Richmond and Liebman did not investigate the turbulent flame velocity relative to the flow, they did calculate flame velocities with respect to the wall from their time-dependent measurements of the flame. Pickles's [2] method was used to estimate the velocity of reactants in advance of the flame, from which the turbulent flame velocities were computed for Richmond and Liebman data. Figure 2 compares the gaseous data with the coal dust flame velocities from experimental gallery tests. The coal dust data are for much larger Rex values than gas data and t l t / u I increases quite sharply beyond an Rex of 1400. The sharp change may be due to the breakdown of the reactant flow velocity estimation technique used to calculate the turbulent flame


An estimate of the maximum value of the eddy viscosity, c, is obtained from turbulent pressure drop data for fully developed duct flow and the maximum velocity gradient from the universal velocity profile [21]. The resultant value was corrected for effects of particle loading [22] and combined with Eq. (18) to give Rex = {24.3[(1 + 0.0096f[Re] 2)1/2_ 1] ×/[1


pb/Pg] 1/2} 1/2.


This result is subject to several assumptions too numerous to list [18]. At best, it is restricted to steady, isotropic, fully developed, nonreacting, gaseous turbulent pipe flow, and subsequently applied with some uncertainty to transient, reacting dust flames. With u I data in correlative form, the turbulent flame velocity is estimated from Eq. (17). However, little or no data for turbulent, steady










From coal explosion data (Richmond and Liebman [4])

Gaseous data [15]



120 Coal Dust Data ~



1 O0





Correlation based on all



~" 7



40 Gas Data

Correlation based on gas

20 -~/pel I 0

•~ V " 0

/ 200

, 400

to high Re~ values , 600

, 800

, 1000

, 1200

, 1400

, 1600


Turbulent Reynolds Number Re,~

Fig. 2. Correlation of the gaseous turbulent flame velocity data (Andrews et al. [15]) and turbulent coal dust flame velocity data estimated from Richmond and Liebman's [4] experimental gallery tests. (Correlation form: u t / u I = 1 + Rexa; for gas data only, o~ = 0.022, 3 = 1.12; for all data, tx = 0.016, /3 = 1.18.)



velocities at these higher turbulent Reynolds numbers. This estimation technique, although useful in demonstrating the relative range of the turbulent flame velocity data with respect to the gaseous data, is not very accurate for obtaining quantitative results. For this reason, the coal dust data points of Fig. 2 where Rex > 1400 were not used in the first correlation (line 1, Fig. 2). Rather, the gaseous data, were used to establish the correlation to calculate the turbulent flame velocity in the accelerating flame model for all Rex values, as shown by the first correlative line. Even so, the extrapolation of the correlation based on gas data agrees reasonably well with coal dust data below Rex = 1400. Figure 2 also shows a second correlation (line 2) which was based on all gas and coal dust data. Effects of differences in lines 1 and 2 were subsequently evaluated.

Flame Thickness One interesting point concerning the equation development was the description of the flame thickness, ~t. In the transient form of the differential equation set, the flame thickness was required for description of the control volume. Thus, a separate relationship was selected to describe the flame thickness for the transient equations of change. To obtain an expression, an analogy to laminar flames was selected where a simple energy balance relates the flame thickness to the thermal conductivity, heat capacity, and volumetric reaction rate [23]. The laminar coefficients were then replaced by their turbulent counterparts and the reaction rate was taken to be the macroscopic reaction rate assuming the rate to be constant across the control volume. The resulting expression is dtt=



It was also possible to solve the steady-state form of the model equations [Eqs. (3)-(7)] independently of the flame thickness [i.e., without Eq. (20)]. Calculated steady-state results were then compared with transient solutions which were allowed to achieve steady-state. Predicted flame velocities from the two methods

differed by less than 1%, which indicated that Eq. (20) represented a suitable approximation to the turbulent flame thickness. The question as to whether the transient equations could have been solved without Eq. (20) was also considered but not resolved.

Summary of Model Relationships Eqs. (1)-(7) comprise the basic equation set used in the model. All the auxiliary equations were required [Eqs. (8)-(16)] with Eqs. (11) and (12) applied to both the reactant and product sides of the flame. In addition, the equations describing the flame thickness and the turbulent flame velocity correlation were required for solution [Eqs. (17) and (20)]. The turbulent Reynolds number was obtained from Eq. (19). A simple correlation was used for the friction factor ( f = 0.0791/Re TM) and the Reynolds and Prandtl numbers were calculated using the velocity of the flame front and average gas-phase properties across the flame. The correlation for the laminar flame velocity [18] was also required. The wall shear stress, rw, was calculated from a simple force balance including pressure and frictional forces. The velocity of the flame front and the average gas density across the flame were used. All variables with a subscript zero (except a0) were specified. In addition, the heat capacity ratio, coal loading fraction, CL, coal reaction fraction, fr, duct diameter, D, initial particle density (assumed to remain constant across compression wave), coal heating value, coal composition, and particle diameter were specified initially. Gas-phase heat capacity values were calculated from standard temperature dependent correlations. The particle phase heat capacity was assumed to be constant. Product species mass fractions were computed from the equilibrium program previously mentioned [7]. Gas-phase viscosities were calculated from Chapman-Enscog theory using air as the working fluid, and the turbulent thermal conductivity was calculated from an analogy between mass and heat transfer (i.e., k t / p C p ~ f:). Finally, the area terms (A¢, As) were calculated using the

262 appropriate length (i.e., either duct diameter or flame thickness). These equations and variables constituted the complete model description. All remaining variables (i.e., temperatures, pressures, densities, velocities) described in the model equation set were calculated from solution of the model. Solution of Equations The transient system of equations was cast as an initial value problem and a fourth order RungeKutta algorithm, adapted for systems of equations, was used for solution. The equations were nonlinear and an algebraic numerical scheme was used to solve the functional relationships for the Runge-Kutta driver. The technique was used to solve the nonlinear, steady-state system of equations for each time-step. The steady-state system was solved by modifying the method-offalse-position to accommodate systems of algebraic equations. The solution technique was very stable, even for relatively large time steps. Initial conditions were required to begin the solution but a technique was developed whereby the starting values necessary for the modifiedfalse-position methods were calculated internally. Details of the solution technique as well as the model development are given in Clark [18]. P R E D I C T I O N S AND D I S C U S S I O N A parametric study was made to determine effects of key variables on the accelerating coal dust flame. Key variables investigated included the duct diameter, coal particle size, and the airto-coal ratio. The sensitivity of the model to the correlation for the turbulent flame velocity and to the various source and sink terms was also investigated. Table 2 provides a summary of the values of model parameters which were required for the various computations shown in the figures. Duct Diameter Figures 3a-3d show comparison plots of the flame velocity, the flame front velocity, the

DWIGHT P. CLARK and L. DOUGLAS SMOOT TABLE 2 Summary of Parameters for Model Computations A. Model Parameters Common to All Figures Parameter


To P0 Tw

298K 0.1 MPa 298K



CL 3/ Time step

0.30 kg/m 3 1.4 0.001 s Coal Data

Type Density Heating value Composition (daf, mass fraction)

Pittsburgh bituminous 1340 kg/m 3 33 MJ/kg C 0.7757 H 0.0575 0 0.0978 N 0.0137 S 0.0553

B. Unique Parameters for Each Figure Figs. 3a-3d, 5a Figs. 4a-4b, 5b Fig. 6

33 #m particle diameter 1 m duct diameter 33/~m particle diameter 2.67 m duct diameter

static pressure, and the product temperature as functions of the duct diameter and time for a Pittsburgh bituminous coal with a mean particle diameter of 33 /~m [18]. The flame velocity represents the velocity of the flame relative to the flow of reactants in front of it. The flame front velocity was defined as the velocity of the moving flame relative to the wall. The flame accelerates more rapidly and propagates at higher velocities in larger ducts. The most significant diameter-dependence in the model is contained in the turbulent Reynolds number. Increasing the diameter increases the turbulent Reynolds number, which increases the turbulent flame velocity. Since the turbulent Reynolds number is the only way in which turbulence is incorporated into the model, the results indicate that increased turbulence in the large diameter ducts is chiefly responsible for the predicted acceleration differences.







1 2 3 4





1 -





1 = 2.5 m Duct 2 = 2.0inDuct 3 = 1.5 m Duct

-- 2.5 m Duct -2.0mDuct - 1.5 m Duct - 1.0 m Duct

2~ I i

300 ¢

8~ i

200 4


1 O0

I, oL









1 2 3 4

= = = =

2.5 2.0 1.5 1.0

m m m m


olo8 Time (s)

Flame Front Velocity

b. Flame Velocity




0 12


1700 /


1 2 3 4

Duct Duct Duct Duct.

= = = =

2.5 2.0 1.5 1.0

m m m m

Duct Duct Duct Duct

g.® 16°°I B 1500~





Time (s)


1400 o



1200 0.04






c. Static P r e s s u r e R i s e

0.08 Time (s)

Time (s) d. P r o d u c t



Fig. 3. Predicted effects of duct diameter on accelerating flame properties for a Pittsburgh seam bituminous coal with monodispersed diameter of 33 #m (see Table 2 for other parameter values).



The transient change in the product temperature, as a function of diameter, is shown in Fig. 3d. The calculations were made for constant coal and air percentages but product temperature varies significantly with duct diameter. Under steady conditions, the product temperature should remain constant. However, in the early periods of acceleration, the temperatures are similar but then deviate noticeably as the flame accelerates. This temperature change is caused by increased compression of the reactant mixture due to increasing flame velocity which is greater for larger duct diameters.

velocity observed for the smaller particles is physically due to the increase in the surface-tovolume ratio as the particle size decreases. The smaller particles heat up faster and thus devolatilize at an enhanced rate. The effect of particle size on pressure is very similar to that on velocity. Particle size is thus a significant factor influencing the explosive nature of coal dust. The very fine coal dust is easily entrained in the air and can burn at a faster rate than larger particles, resulting in larger overpressures. Other Parameters

Particle Size Figure 4 shows the calculated dependence of the flame velocity and pressure on coal dust particle size for a Pittsburgh bituminous coal in a 1 m diameter duct. Particle size information is included in the model solely through the laminar flame velocity. Decreasing the particle size increases the laminar flame velocity. Then for a given turbulent Reynolds number, the turbulent flame velocity also increases. The increased

The dependence of the accelerating flame properties on air-to-coal ratio exhibited the same trends associated with laminar flames [24]. Peak flame velocities were slightly to the fuel-rich side of a stoichiometric ratio of unity and decreased as the lean-flame limit was approached. The flame velocity decrease was more gradual, however, on the fuel-rich side of the flame velocity curve (not shown). Figure 5 shows the dependence of the flame

10 0.24,

10 ~m f

10 um


7 6

?' 5 >





0.16 33/am


012 41 ,um 1

41 ,um

0 0







0 to







Time (s)

Time (s)

a. Flame Velocity

b Static Pressure Rise

Fig. 4. Effects of monodispersed particle diameter on accelerating flame properties for

Pittsburgh seam bituminous coal in a one meter duct (with parameters from Table 2).


0 10


265 0.50

0.40 o



I 10

I 20

I 30

I 40


Particle Diameter (pm) (b)











t.I 0


Duct Diameter (m) (a)

Fig. 5. Predicted effects of (a) duct diameter and (b) coal particle diameter on flame thickness.

thickness on duct diameter (5a) and particle size (5b). Shown are the peak (steady-state) flame thicknesses obtained for runs made at specific duct diameters and coal particle sizes. Figure 5a indicates that the flame thickness increases with increasing duct diameter. This is due to the faster acceleration and higher peak flame velocities in the larger ducts. In the model, the flame thickness is directly dependent on the turbulent thermal conductivity which was calculated from the eddy viscosity through an analogy between momentum and heat transfer discussed previously. Figure 5b indicates that the flame thickness increases with increasing particle size. This is due to the slower heat up and reaction rates in the larger particles causing the flame to thicken. Particle size dependence, as mentioned, comes

into the model through the laminar flame velocity correlation. Several additional model parameters were varied in order to determine the sensitivity of the calculations to each. Effects of selected source terms were tested by omitting a particular source term and then comparing the calculation with that made including it. As mentioned previously, the wall heat loss term (Qw) had negligible influence on flame acceleration, even when the maximum temperature difference (Tw T2) was used. In a small duct exhibiting slow velocities, the heat loss to the wall may become a factor. In duct diameters typical of mine galleries or pneumatic conveying systems, effects of the wall heat loss are predicted to be small. The turbulent heat conduction across the flame, qt, was also of minor influence. -



The flame front velocity increased about 33 % without the wall frictional effects (rw) in a I m duct. The effect exhibited by the momentum source term (oRp/Ac) was only slightly larger than this. The influence of these two terms may be overstated since the pressures on either side of the flame were set equal. The pressure difference will not be great across a deflagrating flame [8] but may be enough to offset some of the influence of gaseous mass addition and wall friction. The term describing heat transfer between particles and gas (Orc) affected the steady flame front velocities by as much as 44 % for a 1 m duct. Qrc is calculated directly from the particle energy equation. The coefficients in the turbulent flame velocity correlation (solid line in Fig. 2) were also varied. Maximum influence of variation in the coefficients for the correlation would occur at high turbulent Reynolds numbers. To illustrate, the correlation was recalculated including all of the coal dust data from Fig. 2, and the 2.5 m duct case was rerun. With the new correlation (dashed line in Fig. 2), the velocity of the flame front increased by about 33%. This emphasizes the need for good turbulent flame velocity data

I 600 --

- 0


in coal dust-air systems. This is especially true in the higher turbulent Reynolds number regions. C O M P A R I S O N WITH OBSERVATIONS Calculated flame pressure and the flame front velocity were compared with four sets of experimental data. Not all of the required parameters for the four sets of experiments were available for a rigorous comparison. Thus, the computations only approximately represent the four sets of tests. Figure 6 shows the predicted pressure versus flame front velocity for a 2.67 m duct, which corresponds approximately to the size of an experimental mine gallery for which Richmond and Liebman [4] reported both the static pressure rise and the flame front velocity. Pressure and velocity data from Richmond and Liebman [4] are in reasonable agreement with predictions for flame front velocities less than about 300 m/s. However, one point reported by Richmond and Liebman indicated overpressure of about 2.8 was experienced at a flame front velocity of roughly 575 m/s. This clearly deviates from the results of Fig. 5.


Predicted Reference 4


Reference 2 8


References 26, 27

500 --


[] V






1 O0 0

1 O0




[ 500


Flame Front Velocity (m/s)

Fig. 6. Comparison of predicted flame front velocity dependence on pressure with experimental data from the literature.

COAL DUST FLAMES Artingstall [25] reported pressure data from Cybulski [26, 27] and Fischer [28], showing pressure as a function of the flame front velocity also in a duct of about 2.7 m in diameter. Maximum overpressures recorded in the experimental studies for these mine gallery experiments were on the order of six times the initial pressure. This maximum overpressure occurs at a reported flame velocity of about 525 m/s. As can be seen from the figure, these points are also in reasonable agreement with the predicted curve. However, both Cybulski and Fischer report one and two data points, respectively, for flame front velocities approaching 1000 m/s. The recorded pressure data did not go beyond the 6 atm maximum overpressure achieved by the slower flames. This result could not be compared with predictions since the model predicted that a steady-state deflagration would occur at a flame front velocity of approximately 550 m/s. The agreement of the predicted transient pressure rise with experimental mine explosion data is interesting, although discrepancies exist. Several reasons could account for the observed discrepancies, in addition to the simplified reaction processes contained in the model. First, the model assumes that the duct or gallery is infinite in length and wave reflections from the end of the gallery do not affect the flow. Second, data collected were for tests where only a portion of the gallery was loaded with pulverized coal for combustion. In the tests conducted by Richmond and Liebman [4], the flame continued to accelerate until all of the available coal was consumed or until sufficient dilution was achieved at the leading edge of the flame. Nagy and Mitchell [29] indicated in their study of gaseous explosions in mine galleries that the maximum pressure developed by the gas explosion was proportional to the length of the zone containing the fuel and the gas concentration. In other words, the amount of fuel present dictated the maximum pressure achieved. Although Richmond and Liebman lacked sufficient data to make the same conclusion about coal dust explosions, they did say that the maximum pressure increased as the length of the dust zone

267 was increased. Finally, several approximations and assumptions were required in modeling the turbulent flame velocity and in providing model parameters from the four data cases.

CONCLUSIONS 1. The method for describing flame acceleration presented herein, together with the laboratory data for laminar flames, predicts some of the observed characteristics of turbulent accelerating dust flames. The most significant of these is the role turbulence plays in augmenting flame propagation and acceleration. Although not intended for precise predictions, the calculated relationship between the flame velocity and the pressure corresponds to the general observations from selected experimental coal mine explosion data. Results suggest that a major driving force behind flame acceleration in coal dust flames is the increasing level of turbulence in the accelerating fluid. 2. The diameter of the duct greatly affects the turbulent flame velocity through increasing turbulent Reynolds number. 3. Coal particle size affects both the rate of acceleration and the peak values for flame velocity and pressure, by changing coal reaction rate. 4. Flame acceleration rate is essentially independent of the product temperature. Increases in product temperature with increasing flame velocity can be attributed to differences in the extent of compression of gaseous reactants prior to combustion, and to a lesser extent, to heat transfer between particulate and gaseous phases. 5. The calculations are sensitive to the flame velocity ratio, (ut/ul), which suggests the need for good experimental data for turbulent coal dust flames. Important also were the heat transfer between gas and particles, the wall shear stress, and the momentum source term. Heat loss to the wall and turbulent heat conduction across the flame were negligible.




a Ac As

m s- 1 m2 m2


kg m-3


J kg-l K-l

Cpi J k g - l K - l Cpp J k g - l K - l f fr



J kg-l


J kg-I


J kg-l


J kg- l

hfp° J k g - i


J kg-l J kg-l


W m- 1 K- l


W m-l K-l

1 M

m kg kmole-1



Nu P Pr

-Pa --

hfr °

speed of sound duct cross-sectional area surface area of duct exposed to flame volumetric coal loading in reactants gas-phase heat capacity gas species heat capacity particle phase heat capacity fanning friction factor coal volatiles mass fraction (reacting portion of coal) gas-phase specific enthalpy particle phase specific enthalpy enthalpy addition to gas phase from particle reaction specific heat formation of gas-phase reactants specific heat formation of unreacted coal specific heat of reaction specific internal energy of gas phase gas phase thermal conductivity turbulent gas-phase thermal conductivity turbulent macroscale gas-phase molecular weight mass of gas phase in control volume Nusselt number pressure Prandtl number

Left column is the symbol; middle column is the unit; right column is the definition.


W m -2

turbulent heat conduction across flame Ow W m -2 wall heat loss Qrc W m - 2 heat transfer between gas and particles R J kgmol- l K - I universal gas constant Reynolds number Re - turbulent Reynolds Rex - number Rp kg s - l coal reaction rate time t s T K temperature enthalpy reference To K temperature wall temperature Tw K UI ms-I laminar flame velocity Ut ms-I turbulent flame velocity ms-I fluctuating component of U" velocity 13 ms-! flow velocity ms-I flame front velocity (with w respect to wall) O/i m species mass fraction heat capacity ratio 3' -5t m flame thickness m 2 s-1 eddy viscosity m turbulent microscale v m 2 s -1 kinematic viscosity p k g m -3 gas-phase density Pb kg m - 3 bulk density pg kg m - 3 gas-phase density (in correlation) pp kg m -3 particle phase density Zw N m -2 wall shear stress S u b s c r i p t s ( F i g . 1)

0 1 2

undisturbed region of flow reactant region of flow (compressed) product region of flow

The authors express appreciation to Utah Power and Light Co., Dr. Dee Rees, Project Officer, f o r partial funding f o r the project, Prof. Philip Smith, Chemical Engr. Dept., f o r invaluable assistance in numerical solu-


tion methods, and Lynda Richmond f o r assistance with preparation o f the manuscript.

269 14. 15.


2. 3.



6. 7.

8. 9. 10.

11. 12. 13.

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25. 26. 27. 28. 29.

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Received 29 October 1984; revised 29 May 1985