The role of the diffusion of oxygen in the ignition of a coal stockpile in confined storage

ELSEVIER Plh S0016-2361(97)00077-X

Fuel Vol. 76, No. 10, pp. 975-983, 1997 © 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0016-2361/97 $17.00+0.00

The role of the diffusion of oxygen in the ignition of a coal stockpile in confined storage

Ashley Hull, Jennifer L. Lanthier and Pradeep K. Agarwal* Department of Chemical and Petroleum Engineering, University of Wyoming, Laramie, WY 82070, USA (Received 14 October 1995; revised 30 October 1996)

This paper deals with the spontaneous combustibility and ignition behaviour of coal stored in confined spaces. The geometry, in which the stockpile is sealed on all sides except the top, which is exposed to the ambient conditions, is of interest in the transport of coals in barges or in rail cars. The role of oxygen diffusion as the cause of self-heating of coal is re-examined. In previous work, it was concluded that the diffusion in a one-dimensional stockpile can lead to a maximum temperature rise of -80 K; this conclusion is shown to be a consequence of the unrealistic boundary conditions imposed on the oxygen and energy balance equations. Simple calculations, with more realistic boundary conditions which reflect that the resistance to heat transfer may lie at the interface between the pile and the ambient, lead to a maximum possible temperature rise of -2150 K. Mathematical models are developed, assuming that the pile is isothermal, to determine the influence of various parameters--coal reactivity, stockpile dimensions and ambient conditions--on ignition behaviour. In particular, it is shown that storage of coal in this configuration is, at best, conditionally safe. Based on the calculations, methods which may be implemented to suppress spontaneous combustibility in confined storage are discussed. Application of the model is illustrated by detailed calculations pertinent to the transport of coal in barges or in rail cars. © 1997 Elsevier Science Ltd.

(Keywords: coal storage; spontaneous combustion; oxygen diffusion)

The safety of coal during storage and transportation is of considerable importance to coal producers and users. It is well known that low-rank coals--such as the subbituminous coals from Wyoming--and the upgraded products prepared from them exhibit strong self-heating characteristics in storage. Various exothermic processes such as low- temperature oxidation, microbial metabolism, heat of wetting and oxidation of pyrite can contribute to the self- heating of coal ~. Current research efforts in this laboratory are directed towards obtaining a better understanding of, primarily, the role of oxidation and heat of wetting in promoting hazardous conditions during coal storage.

Self-heating of coal in storage due to low-temperature oxidation is a complex problem. The issues which require consideration are the mechanisms for access of oxygen within the stockpile, the reaction of oxygen with the coal, and the mechanisms for removal of heat generated by the uptake of oxygen. For open stockpiles, it has been suggested that diffusion of oxygen alone cannot lead to spontaneous combustion; reactant consumption leads to oxygen depriva- tion and steady-state temperatures which do not pose hazardous conditions 2"3. Consequently, convection mechanisms for delivering oxygen within the stored coal pile are considered essential. Brooks and Glasser 3 suggested, using a steady-state one-dimensional analysis, that natural convection must be the dominant mechanism for oxygen

* Author to whom correspondence should be addressed

access. On the other hand, Krishnaswamy et al. 4 recently showed that natural convection is unlikely to provide oxygen during the initial stages of heat-up, as the temperature differences required to bring about convection currents may not be high enough.

This paper focuses on the safety of coal stored in confined spaces with the levelled top surface of the pile exposed to the ambient environment. This geometry is of importance when coal is transported in barges or in open cars by rail. Though volumetric forced convection can certainly explain access of oxygen in open stockpiles, it cannot be used in the context of confined storage of coal. This is because forced convection, by motion of the car and/or wind, is in a direction normal to the body of the coal and could influence the behaviour of the stockpile only at the exposed boundary. If natural convection and diffusion mechanisms are ruled out as well, then, based on previous arguments, the inevitable conclusion is that low-temperature oxidation by itself cannot lead to the ignition of coal in confined storage. Heat of wetting or moisture condensation 5 could explain the field observations which clearly show that dangerous self- heating is possible. However, Chen6--through extension of the approach of Brooks and Glasser3--concluded that combined diffusion of moisture and oxygen also leads to a maximum temperature rise similar to that from diffusion of oxygen alone.

This paper re-examines the role of oxygen diffusion as the cause of the self-heating of coal in confined storage. In

Fuel 1997 Volume 76 Number 10 975

The role of the diffusion of oxygen in the ignition of a coal stockpile in confined storage: A. Hull et al.

particular, it is shown that the maximum temperature rise of - 8 0 K obtained by Brooks and Glasser 3 is a consequence of the restrictive boundary conditions imposed on the oxygen and energy balance equations; similar arguments apply to the conclusions of Chen 6. Simple calculations, with more realistic boundary conditions which reflect that the resistance to heat transfer may lie at the interface between the pile and the ambient, lead to a maximum possible temperature rise of -2150 K. Mathematical models are developed, assuming that the pile is isothermal, to determine the influence of various parameters--coal reactivity, stockpile dimensions and ambient conditions--on ignition behaviour. In particular, it is shown that storage and transport of coal in confined spaces is, at best, only conditionally safe. Based on the calculations, methods which may be implemented to suppress spontaneous combustibility in confined storage are discussed. Application of the model is illustrated through detailed calculations pertinent to the transport of coal in barges or in rail cars.

THEORY

The idealization used to develop the mathematical model for a coal pile in confined storage is illustrated in Figure 1. The pile is sealed on all sides except the top surface, which is exposed to the environment. As noted earlier, this situation is encountered, for example, when levelled stockpiles of coal are transported in barges or in rail cars. Given that coal is in a confined storage vessel, there is no volumetric convective component for either heat or mass transfer; consequently, the one-dimensional oxygen and energy balances can be written as:

0Co2 b ~ 02Co2'b -- (1 - Eb) ( -- Ro2) (1) ~b Ot ' ~ /302' b OX 2

orb 02rb (1 -- %)ppCp, p - - ~ = )k b ~ .4- ( _ Z~k/-/R)( 1 _ 6b)( _ No2)

(2)

In these equations, eb is the bed porosity, Co2,b is the oxygen concentration within the pile and T b is the temperature; Xb and OoE,b are respectively the effective thermal conductivity and mass diffusivity of oxygen within the pile, and ( - AHR) is the heat of reaction.

The reaction-diffusion approach developed earlier 7 is used to obtain the rate of oxygen consumption per unit particle volume, ( - Ro:), as:

( - Ro2 ) = T/GKRCo2,b (3a)

where ~TG is the global effectiveness factor, which can be calculated for the dry coal under consideration from:

1 1 KR I- - - (3b)

r/G 7/ 3kg, p/rp

r /= tanh(~bp)

q~p = r p ~ (3d)

K R is the first-order reaction rate constant, kg,p is the mass transfer coefficient at the external surface of individual coal particles of radius rp, and Do2,p is the effective diffusivity of oxygen within individual coal particles.

The boundary conditions are:

X OTb 0Co2 ,b__0 b O-'X- x = 0 = DO2' b OX

(4)

OZb = h i ( Z b , i - Za) - X b - ~ x x = L

(5a)

0 Co2, b -- Oo2, b O-----~lx= L = kg, i ( f o 2 , b , i - C02,a ) (Sb)

where hi and kg.i are the heat and mass transfer coefficients, respectively, at the interface, x -- L, between the stockpile and the ambient environment.

The initial conditions are:

T b = To; C02,b = C02,a a t t = 0 (6)

Most often, one would expect that To = Ta. However, when an upgraded coal product is loaded/stored, the temperature

l _ D ! I -- ," / interface: kgi, h i

- - x = L

Figure 1

r - - x = O

,, I

Idealized representation of coal in confined storage with exposed top

Coal pile - voidage: e b -- particle size: rp - Lb

- Do2,b - temperature: T b -oxygen conc.: Co2,b

976 Fuel 1997 Vo lume 76 Number 10


of the coal may be different from the ambient. Similarly, if coal brought from the mine in rail cars is loaded into a barge, the initial temperature may be different since some self- heating may have occurred already.

These equations form a set of coupled partial differential equations which cannot be solved analytically; some simplified solutions are examined which provide insight into the relevant parameters and variables of the system.

Steady-state analysis and ignition criterion Neglecting the time-dependent accumulation terms, the

mass and energy balance equations take the form:

d2 Co2, b DO2,B dx 2 -- r/G(l -- %)KRCo2, B (7)

d2Tb -- ~k b ~ ---- ( -- AHR)~G(1 -- %)KRCo2, b (8)

Substitution of Equation (7) in Equation (8) leads to:

d2 Tb dCo2, b -- ~ k b - ~ = ( - - A H R ) D o a , b dx 2 (9)

Integration of Equation (9) with substitution of the boundary conditions given by Equations (4), (5a) and (5b) leads 8'9 to:

( kg, i V b - T a = ( - AHR) / - '~- i ( C o 2 , a - Co2, b,i )

D o 2 ' b / r Co2, b)} (10a) -~- - " ~ b ~t~O2, b, i --

Brooks and Glasser 3 imposed the boundary condition that C02,b,i = Coza and Tb,i = Ta. These boundary conditions imply that the heat and mass transfer coefficients at the interface are very high. Consequently, the first term on the right-hand side of Equation (10a) drops out. If conduction through the gas film and forced convection at the interface are the dominant mechanisms for heat and mass transfer, then Sh/Nu ~ (Pr/Sc) j/3 ~ 1. This in turn implies that (kg,i/hi) -~ (Do2,m/Xg), where Doz,m is the molecular diffusivity of oxygen and Xg is the thermal conductivity of air. We can therefore rearrange Equation (10a) as:

Do2, m) {(Co2, a __ Co2, b, i) rb-Ta= (--AHR) --~

Xg (Do2'b') (~bb)(Co2,b,i--Cogb) } q- \ D o 2 , m ] "'

(10b)

As shown in Appendix 1, (Do2,b/Do2,m) ~ ffb/Tb ~ 0.1 for eb "~ 0.3 and (Xg/Xb) "~ 0.1. It would appear from these estimates that the major resistance to heat transfer is at the interface; thus it is unlikely that Tb, i = Za. It is not possible to make an a priori estimate for Co2,b,~ without considering the more detailed problem. For slow reaction, the compara- tively higher resistance to mass transfer within the pile would lead to Cozb,i ~ Co2,b,a. On the other hand, for com- paratively faster reaction, oxygen would be consumed with very little penetration into the pile, such that Co2,b,~ ~ Co2,b,a. For such reactions, with the second term on the right-hand side being negligible, and using the typical values given in Appendix l:

Tmax - -Ta ~ ( -AHR)D~ '2 'mc02 '~ a ~ 2150 K Ag

( l l )

This value, remarkably different from the maximum temperature difference of - 8 0 K estimated by Brooks and Glasser 3, shows that diffusion of oxygen can lead to conditions under which coal storage in confined spaces is hazardous. Though the consumption of oxygen is taken as first-order in the model development, note that the results of Equations (10a), (10b) and (11) are independent of the form of the reaction rate.

If the resistance to heat transfer is concentrated at the interface as a first-order estimate, the pile may be considered isothermal. Since the pile is at a uniform temperature, r/c and KR are constant. The steady-state oxygen balance can therefore be integrated with the boundary conditions of Equations (4), (5a) and (5b) to yieldg:

C°2 ' ac°sh(dPbX/L) (12a) C°2' b(X) = DO2, b~bb sin ~ b

cosh 4) h + Lkg, i where 0b, the Thiele modulus for the entire pile, is defined as:

q~b L /~G(1 -- ~b)KR

= V (12b)

Integration of the energy balance yields:

I i CO2' b x (x) (Ix

Substitution of the boundary conditions and Equations (12a) and (12b) for Co2.b(X) leads to:

r/G(1 -- eb)KR( -- AHR) 7L Co2 a ~b ' (13a)

hi(Tb - Ta) = Do2, bob coth q~b q- - -

Lkg, i

Substitution of ~bb according to Equation (12b) then yields:

hi(T b - Ta) = ( - AHR)Co2'akg'i (13b) Lkg, icoth 4~b + 1

0bDo2, b

It can be verified that Equation (13b) will also be obtained through the use of Equation (10b), assuming that Xb is high, as required by the isothermal pile assumption.

The temperature dependence of the right-hand side of Equation (13b) is complex. However, as established in Appendix 2, ~b >> ~bp. Consequently, it can be assumed that coth ~b ---* 1 for conditions which may lead to ignition; note that ~bb --> 3 for coth ~bt, --'* 1. Also, Do2,b~DO2,m(eb/7"b). As a result:

hi(T b - T a) ~ ( - AHR)C°e'akg'i (13c) L kg, i rb

q-1 ~bb Do2, m %

~bb >> qSp has interesting practical and theoretical implications. The result suggests that the stockpile experi- ences severe mass transfer limitations in its interior: the

Fuel 1997 V o l u m e 76 N u m b e r 10 977


stockpile is starved of oxygen. Upon reflection, this is not surprising. It has often been observed in practical situa- tions that temperature excursions in coal piles stored in holds occur when the hatch is opened m. q~b >> q~p also indi- cates that the transport limitations within individual coal particles are of secondary importance; this in turn suggests that particle size effects will not be very significant in determining the overall oxygen consumption rate and ignition behaviour in confined storage. This is dramatically different from the strong dependence obtained for open stockpiles 4.

Now define:

6 = ( - - A H R ) C o 2 ' akg' i E (14a) hiRr 2

E 0 = --~-¢~ (Tb - Ta) (14b)

/~la

1/2 - - E / R T a

(KR0 e . . . . aDo2, meb. ~

/3=t, ) (14c)

Since transport limitations with respect to individual particles are small, we can assume that rig ~- 1. Then, with KR = KRoe-emrL Equation (13) can be rearranged to:

Oe - 0/2 6/3 = - - (15)

1 - 0/&

Equation (15) can be differentiated with respect to 0 to determine its critical value; thus:

6_+ -86 0c - 2 (16)

Transient analysis A major advantage of the isothermal pile assumption is

that the temperature history can by determined readily. This temperature history in turn can be used to determine the period in which the cargo shipment must be completed before hazardous conditions arise. Integration of Equation (2) with respect to x, with coth ~bb "--* 1, gives:

(1 - ~.b)ppCp, pL@t ~-- - hi(T b - Ta) + ( - - A H R ) C o 2 , akg, i

L k g , iT" b +1

~b Do2, m~b

(17a)

Defining the dimensionless time ¢ = (hit)~(1 -- ~:b)ppCp,pL, it can be shown that Equation (17a) reduces, in dimensionless form, to:

dO-o+6( /3 } (17b) d--~ = e - 012 . . ~ / 3

with 0 = 00 at ~b = 0.

RESULTS AND DISCUSSION

This section first considers the implications of the ignition

1000

100 5=100

5=1000

8=10

1 8=1

0.1 i , , ~ t , , , , I , , , , r , ~ , ~ i i i , , , , , , ,

0.0 0.5 1.0 1.5

Figure 2 Steady-state dimensionless temperature in the stockpile according to Equation (15)

criterion developed through steady-state analysis. Results for the time dependence of the temperature of the stockpile and the safe storage times are then presented.

Equation (15) is a dimensionless steady-state energy balance. Its solution will provide the expected steady-state temperatures of the coal pile. A plot of 0 as a function of (/36) is shown in Figure 2. For 6 = 8 there is only one critical value: 0c = 4. For 6 > 8, 0c has two real roots and these manifest themselves in the form of inflection points on the 0 versus (/36) plot of Figure 2. Using the parameters listed in Appendix 1, it can be shown that realistic values of 6 are large: 6 ~ 170. For these values, the smaller root of 0c, which is conventionally taken to represent ignition, is 0c ~ 2. In fact, 0c ~ 2 is the asymptotic solution (for the smaller root) of Equation (16) for large 6, as shown in Figure 3, and this asymptotic solution appears to be a reasonable approximation for 6 > 100. Note that the classical ignition theory 11, in which reactant consumption is not taken into account, predicts that 0c = 1. The strong mass transfer limitations within the stockpile are responsible for the critical value obtained here.

Consider the calculations shown for 8 = 10 in Figure 2. It is clear that for 38 > 0.96 or /3 > 0.096, the stockpile will ignite at a steady state since there is only one solution for 0 ~ 10. For 0.7 </36 < 0.96 or 0.07 </3 < 0.096, there are three steady-state solutions; that is, the pile will be only conditionally safe and whether it will ignite or not depends on the initial conditions. For/36 < 0.7 or/3 < 0.07, there exists only one low-temperature solution at steady-state and the pile will not ignite. These critical values of/3 can be determined readily from the model. For a specified value of 6, the two critical values of 0c can be calculated from Equation (16) or Figure 3. The lower value of 0c, when substituted in Equation (15), permits calculation of /3 beyond which only the ignited solution exists. The higher value of 0o when substituted in Equation (15), permits calculation of/3 below which only the non-ignited solution exists. For intermediate values of 3, the pile will be only conditionally safe. A plot of these calculations is shown in Figure 4. For 6 < 8, of course, /3 has no critical values. Regression of calculated values leads to:

/3 -> 1.24156-1.09 for unsafe conditions (18a)

12.225e -°4941. </3 < 1.24156-1.09

for conditionally safe storage (18b)

978 Fuel 1997 Volume 76 Number 10


100

10

_ 0 c = 4

_ 0:=2

0 c = 8 + ' ~ " ~ 2

2

I I

10 100 1000

Figure 3 Critical Equation (16)

8

dimensionless temperature according to

for turbulent flow. D is the length of the exposed surface in the direction of convection and U~ is the freestream convective velocity relative to the exposed surface. Clearly, D must be as small as possible. Since Us is a relative velocity between the barge/rail car and the atmospheric wind, its magnitude can vary quite substantially; its effect is discussed in greater detail later. Higher values of U~ will assist in removing heat from the system. This holds, of course, for a coal pile with a levelled surface. When the surface is not levelled, higher U~ values will lead to penetration of oxygen within the pile, as discussed by Krishnas- wamy et al. 4, leading to increased potential for spontaneous combustion, especially with steep side slopes. This explains the field observation that the levelling of surfaces dramatically reduced the temperature rise during shipment of coal in barges J°.

If the values of (/36) cannot be controlled and ignition is predicted at a steady state, then the only recourse is to deliver the cargo to its destination before criticality is

0.1

0.01

0.001

0.0001

0.00001

Non-critical

Figure 4

I I Unsafe

^ . I c=12'225e-°'49418

iSafe / Conditionally safe

10 100 1000

8

Critical values of/3 as a function of 6

As noted earlier, realistic values for storage of coal lead to 5 ~ 170. From Figure 4, it is evident that the coal will at best be only conditionally safe.

Now examine the practical implications of Equation (18) in terms of procedures that may enhance safe storage. For 5 ~ 170, the stockpile will ignite when:

-- z~r-/RCo2 a E /KRoe-eliOt"(1 - % ) D o 2 , m % > 0 .75

V - (19)

For a particular type of coal being stored, the coal reactivity, KR0, is fixed; the ambient conditions fix Ta, Co2,a and Do2,m. The only term which may be controlled, perhaps, is (1 - 6 b ) E b / 7 " b, If rb "-~ %- 1, then it is clear that lower values--through compaction--will lower the risk of ignition. The heat transfer coefficient at the interface, h~, is another variable which may be controlled to some extent. Using correlations for convection past a flat plate ~2, it can be shown that:

n l - -n hi oc U g / D (20)

where n = 1/2 for laminar flow (Re <- 5 × 105) and n ~ 0.8

reached. Clearly, it is important to estimate the safe storage times through calculation of the induction period. Attention is therefore turned to the transient analysis. The dimensionless energy balance given by Equations (17a) and (17b) was numerically solved using the Runge-Kutta algorithm. Temperature histories, for constant values of and with 00 = 0, are plotted as a function of ~k for several values of 6 in Figure 5(a)-(c). It is clear that for low values of 6, a steady-state, non-ignited solution is obtained. For higher values of 6, ignition is predicted: as 6 increases, the magnitude of the temperature jump increases and the time at which this jump occurs decreases. For decreasing values of/3, as expected from Figure 4 and Equation (19), the ignited state is attained for increasingly higher values of & With 00 = 0, the steady-state temperature in the conditionally safe regime will invariably be the unignited solution. However, if the initial temperature is higher (than the middle branch of the steady solutions plotted in Figure 2), then ignition will occur. The effect of the initial temperature, 00, is illustrated in Figure 6. For these calculations, /3 = 0.0028 and 5 = 173; consequently, the operation is under the conditionally safe regime. It is evident that for 00 < 4.518, ignition will not occur; higher values lead to ignition.

An illustration is now given of the use of the



1000

f 8=5oo 8=Ioo I00 .

10

8=15

l ~ 8= 8

0.1 " 13=0"1

( a ) 0.01 + "1 +

~=500

1 0 0 , t ) o / f 8=I00

]o 111 1 [3=0.01

8=15

5=8 ( b )

t ÷

0.1

0.01

100

10

0.I

8=1000

~=O.OOl

8=500

8=100

0.01 4- + + +

0 10 20 30

( e )

40

Figure 5 Variation of dimensionless temperature, 0, as a function of dimensionless time, if, with (5 as a parameter

1000

100.

10.

CD

1

0.1

0.01

0 0 = 10

I ~ o° = 4519 /

....... 00 = 4.518 0 - 1 ~ ' " ~ , _ ..0..2,._ " • . . . . . . . . . . .

Oo=0

I I I 5 10 15 20

Figure 6 Effect of initial temperature, 0, on variation of temperature history with dimensionless time;/3 = 173, (5 = 0.0028

mathematical model for calculation of the time below which coal storage may be considered safe for conditions pertinent to the transport of coal in barges down the Mississippi. A typical barge loads --1350 tonnes of coal.

100 (a)

I0 ~. / K R 0 =8"83x106, eb=0.3 /

• . ,~ ~ ~ "-. g ~...~ KR0=8.83X10, eb=0.2

KRo=8.83x105 eb=0.3 , I , " '2" ~ - " k,'-,

~ . 0.01 , , - - - t - " " ~,'¢ " Eqnl8a

Conditionally s a f e "':,~,,:~,,~. :,.., 0.001

KR0=8.83x105, eb=0,2 / ' / "

,-:.. 0.0001

0.0000[ I I I I I

' - . . ~ . / T = 429K, T O = 300K ( b )

~ " " ..,, T = 364K, T o = 300K

<

lOO ;

"~ 1 4 ~ T = 36 , -

0.1 I I I I I 0.0001 0.001 0.01 0.1 1 10 100

-i U®, ms

Figure 7 Effect of convective velocity, U=: (a) on /3, with reactivity, KRO, and voidage, ~b, as parameters; (b) on safe storage time, t, with safe temperature, T, and initial temperature, To, as parameters, at KR0 = 8.83 × 10 6 s - I and ~b = 0.35

The depth (L) of the pile, which is frequently not levelled (and, as discussed earlier, therefore poses greater potential for spontaneous combustion), is - 7 m, the length is - 3 0 m and the width is - 1 0 m. Calculations were performed for specific values of U~ (which permit estimation of the relevant heat/mass transfer coefficients) and with parameters listed in Appendix 1, to determine the stability of the coal at steady state as well as the times for safe storage. At the outset, it is worth noting that (kg,i/hi) in the definition of 8 is insensitive to the value of U~; thus 6 does not vary with U~. On the other hand, kg,i increases and ~ decreases significantly as U~ increases. In these calculations, the wind is assumed to flow along the length of the barge; thus the length dimension used in the calculation of Re, Sh and Nu is D --~ 30 m. The results are shown in Figure 7(a) and (b). In Figure 7(a), values of /3 have been plotted, for several values of eb, as a function of U=. To determine the stability at a steady state, values of J3c were also calculated using Equation (18); for/3 > He, the pile will ignite at a steady state. Note that since 8 is independent of U~, /3 c is approximately constant. The results clearly show that higher values of U~ ( > 1 m s -~) will lead to conditional stability; lower values of coal reactivity (through KR0) and eb enhance safe storage. When ~ > ~ , the only option available is to ensure that the coal is transported well before self-heating becomes serious. The level of self-heating that may be permitted can vary somewhat; for example, coal shipped in barges m may not be loaded into the holds of ocean liners if the temperature of the pile in the barge exceeds, typically, - 3 2 3 K.

From transient energy balance calculations, the time required to reach a specified temperature can be readily calculated. Examples of these calculations are shown in

980 Fuel 1997 Vo lume 76 Numbe r 10


Figure 7(b) for 6 = 170. In the first set of calculations, the initial temperature, To, was fixed at 300 K, and the times required to reach 364 and 429 K were plotted. In the second set of calculations, the initial temperature was fixed at 360 K. Several interesting features are revealed. First, there exists a threshold value of U~, above which the pile can be safe but below which the period for safe storage is reduced to a minimum. Further, this threshold Us depends on the initial temperature of the pile. For non-isothermal objects, the dimensionless time, often called the Fourier number, is defined as Xt/pC~ 2. For the isothermal pile under consideration here, hi replaces the ratio (~L) in the definition of the dimensionless time. Higher values of the Fourier number invariably indicate that the steady state will be reached rapidly; thus, higher hi values resulting from increasing U~ lower the times required t.o reach the unsafe conditions at steady state. Consequently, the time for safe coal storage is reduced. However, as Us increases, /3 decreases. Eventually when/3 is less than the value/3c given by Equation (18), a transition from the unsafe to the conditionally safe regime occurs, with the possibility of a lower steady-state temperature. At this point, the pile can become safe, depending on the initial temperature: if the initial temperature is higher, a correspondingly higher value of Us is required to render the pile safe, During the course of the barge's journey, the atmospheric winds can vary quite significantly in magnitude and direction; control may be difficult. It is clear that potential exists for dangerous levels of self-heating. Finally, the initial temperature has a profound effect, with a temperature difference of 60 K leading to a reduction in safe storage time by an order of magnitude. These results are of significance for industrial processes where low-rank coals are heat- treated to decrease the moisture content. Clearly, it would be wise to reduce the temperature of the coal product as much as possible, otherwise it can self-heat to dangerous levels in a short time.

In the calculations presented, it has been assumed that the wind blows along the length of the barge. On the one hand, cross-winds would reduce the effective length in the calculation of Re, Nu and Sh; as pointed out earlier, this would increase h i and enhance stable storage for a levelled pile. On the other hand, for non-levelled piles, cross-winds would force greater penetration of oxygen within the pile and enhance the possibility of spontaneous combustion. It has also been assumed that the coal pile is flush with the top edge of the barge. In practice, there is - 1 m gap between the top edge of the side of the barge and the surface of the levelled coal pile. Stagnant regions, close to the edge of the barge perpendicular to the direction of wind flow, will experience lower local heat transfer coefficients, leading to hot spots observed in practice ~°.

The simple model presented in this paper provides valuable insight into the issues relating to the stability of coal in confined storage. A framework for interpretation of results from more detailed calculations has also been established. In the analysis the sides of the stockpile are assumed to be insulated. In practice there is some heat loss from these sides. The model then provides the worst-case analysis, since additional heat loss will reduce the risk of dangerous self-heating.

Finally, two major limitations of the model must be mentioned. The first relates to the effect of moisture. In the calculations reported here, moisture migration has been neglected. In the transport of coal in barges, moisture migration must play a significant role and, as shown in a

recent paper 5, condensation in environments of high relative humidity can enhance the potential for self-heating. However, inclusion of this effect requires a non-isothermal unsteady-state formulation to handle the dynamics of moisture movement. The second major limitation is the neglect of radiation. It has been shown here that the convective boundary condition at the exposed surface, in conjunction with diffusion, can lead to ignition. Inclusion of radiation effects at the boundary again requires numerical solution of the non-isothermal pile problem, since the effective heat transfer coefficient would be augmented and (kg,i/hi) will no longer be approximated by (Do2,m/~kg). Such calculations are currently being performed and the results will be reported in due course.

CONCLUSIONS

This paper examines the role of the diffusion of oxygen in leading to spontaneous combustion of coal stored in confined spaces such that the top surface is exposed to the environment. This geometry applies to levelled piles of coal transported in barges or in rail cars. It is shown, in contrast to some previous work, that diffusion of oxygen in one-dimensional piles can lead to hazardous conditions with a maximum possible temperature rise of -2150 K.

Through a steady-state analysis which assumes that the stored coal behaves as an isothermal slab, it is shown that the coal pile is oxygen-starved. Ignition criterion are developed which show that compaction and higher convective wind velocities at the exposed surface will enhance the safety of a levelled coal pile. An explanation is also provided for the field practice of levelling piles in barges. For conditions which will invariably lead to ignition at a steady state, the only recourse is to ensure utilization of the stored coal well within the time required for the pile to become critical.

For illustration of the use of the model, detailed calculations pertinent to the transport of coal in barges are presented. These calculations show that the relative velocity between the barge and the atmospheric current is of paramount importance in determining criticality. Higher values of the relative velocity lower the steady-state temperature as well as the induction time.

ACKNOWLEDGEMENTS

The authors thank the National Science Foundation and the Department of Energy for financial support of the research programmes of which this investigation is a part. Special thanks are due to Mr Paul Woessner, who provided invaluable guidance on the practical aspects of coal handling and storage operations. The authors also thank Ms Kirse Kelly for preparation of the manuscript.

REFERENCES

1 Mackenzie, K. J. D., Harrison, W. J. and Walker, I. K., NZ Journal of Science, 1974, 17, 93.

2 Nordon, P., Fuel, 1979, 58, 456, 3 Brooks, K. and Glasser, D., Fuel, 1986, 65, 1035. 4 Krishnaswamy, S., Gunn, R. D. and Agarwal, P. K., Fuel,

1996, 75, 344.



5 Bhat, S. and Agarwal, P. K., Fuel, 1996, 75, 1523. 6 Chen, X. D., Combustion and Flame, 1992, 90, 114. 7 Krishnaswamy, S., Bhat, S., Gunn, R. D. and Agarwal, P. K.,

Fuel, 1996, 75, 333. 8 Lee, J. C. M. and Luss, D., Industrial & Engineering

Chemistry Fundamentals, 1969, 8, 597. 9 Froment, G. F. and Bischoff, K. B., Chemical Reactor

Analysis and Design. Wiley, New York, 1990. 10 Woessner, P., Paper 84-JPGC/FU-F to Joint Power Genera-

tion Conference, ASME, 1984 (and private communication, 1995).

11 Bowes, P. C., Self-heating: Evaluating and Controlling the Hazards. Elsevier, New York, 1994.

12 Holman, J. P., Heat Transfer McGraw-Hill, New York, 1989.

NOMENCLATURE

Co2 Cp D

002 E h ( - AHR) kg KR KR0 L n Nu Pr r

R

( -- RO2 ) Re Sc Sh t T U

U= x

oxygen concentration (mol m-3) specific heat capacity (J kg-lK -1) dimension of the pile along the direction of wind flow (m) diffusivity of oxygen (m2s -~) activation energy (kJ mo1-1) heat transfer coefficient (W m-2K -1) heat of reaction [J (mol O2) -l] mass transfer coefficient (m s -l) first-order reaction rate (s -l) pre-exponential factor (s -l) depth of pile (m) exponent, eqn (19) Nusselt number, hiD/•g

Prandtl number radius (m) molar gas constant rate of oxygen consumption per unit particle volume Reynolds number, DUJIpg

Schmidt number Sherwood number, kg,iD/Do2,m

time (s) temperature (K) variable in Appendix 1 convective wind velocity at exposed surface (m s-l) axial coordinate (m)

Greek symbols

t~ E

~/o #

X

P

P 7"

dimensionless parameter, eqn (14) dimensionless parameter, eqn (14) porosity or voidage effectiveness factor, Equation (3) generalized effectiveness factor, Equation (3) dimensionless temperature thermal conductivity (W m- lK -~) kinematic viscosity (m2s -~) density (kg m-3) tortuosity Thiele modulus dimensionless time

Subscripts

a ambient b pile or bed

c critical g gas i interface m molecular max maximum p particle 0 initial, at t = 0

APPENDIX 1

E/R KRO D o2,rn Do2,p Do2,b Nu

Sh

Co2,a ( - aHR) Xg

Xb

= 7000, from Bowes I l and Krishnaswamy et al. 4

= 8.83 × 10 6, from Krishnaswamy et al. 4 = 2 × 10 -5 m2s -t

-1 Do2,mep/'r p with ep ~ 0.3 and rp ~ ep ~-" Do2,meblrb with rb ~ e~- I = 0.664Re°SPr °33 for Re < 5 × 105 = (0.037Re °'s - 850)Pr °'33 for Re > 5 × 105, from Holman 12 = 0.664Re°5Sc °33 for Re < 5 × 105 = (0.037Re °s - 850)Sc °'33 for Re > 5 × 105, from Holman 12 = 8.53 mol m -3 = 330 kJ (tool O2)-1, from Brooks and Glasser 3 = 0.026 W m- lK -l = 0.2 W m- lK -l, from Brooks and Glasser 3

APPENDIX 2

According to Equation (12):

- ~ = V/r/G0 - % ) K R (A1)

Also:

1 1 KRrp - - - F " - -

~'/G r/ 3kg, p

With no convection within the bed, Shp leading to:

(A2)

2kg, prp = 2,

DO2, m(Eb/'/'b)

2 2 KRrp KRrprb (gp('rb. ep~

3kg, p 3Do2,m % 3 ~kEb "rp.] (A3)

Also:

1 ~b~ 1 (A4) ~7 2 [~pc o th (~p ) - 1]

Substituting Equations (A3) and (A4) into Equation (A2):

~{ 1 rb%'[ -~ r/G ----- ~bpcoth(~bp) - 1 ~- - - - - e b rp j

(A5)

Denoting u = (rbep/ebrp), substitution of To in Equation

982 Fuel 1997 Volume 76 Number 10


(A1) leads to:

~p~ r u4~pc°th(%)-u ] ~b b = 3(1 - eb) [ Uq~pcoth(~bp----~ ~ (1 -- ui (A6)

When ~)p is large, coth(~bp) --* 1, so:

~bb = L / - ~ _ eb) (A7) rp

In this limit, (~b is independent of temperature. When 4~p is small, ~/c ~ 1; thus:

= rLp--~b V/(i -- %)u (AS) ¢p

Since in practice L >> rp, it is reasonable to say that q~b >> ~p regardless of q~p, and consequently coth(qbb) ---* 1.


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