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On the mathematical modeling of the transient process of spontaneous heating in a moist coal stockpile
114
COMBUSTION A ND F LA ME 90: 114-120 (1992)
On the Mathematical Modeling of the Transient Process of Spontaneous Heating in a Moist Coal Stockpile X. DONG CHEN* Department of Chemical and ProcessEngineering, UniL,ersityof Canterbury, Christchurch, New Zealand The influence of moisture transfer on the maximum temperature rise in a coal stockpile has been analyzed using the simplified one-dimensional differential equations that govern the spontaneous heating process. Analytical solutions of the maximum temperature rise have been obtained using well-established coefficients. The solutions suggest that, if the saturation of the gas stream in a coal stockpile with moisture is assumed, the numerically predicted temperature of the stockpile will be well below 100°C. However, as the relative humidity of the gas stream reduces, the predicted maximum temperature increases to over 100°C. These analytical solutions strongly support the idea of introducing the equilibrium relationship between the relative humidity of the gas and the moisture content of coal into the transient mathematical model of the heating process that has been developed over 20 years at the University of Canterbury.
INTRODUCTION
Experimental Background Comprehensive experimental work has been carried out on the spontaneous heating of coal. Similar temperature-time curves have been obtained by different researchers in different types of laboratory experiments. Stott's classical experiments demonstrate the effect of the moisture content on the temperature rises attained during spontaneous ignition processes of subbituminous coal samples [1, 2]. In these experiments, a certain mass of coal, which had previously been dried under vacuum at a temperature of 105°C, was placed in a cylinder through which moist oxygen or moist air was passed from the bottom of the cylinder to the top. A moderate-scale apparatus, described by Smith and Lazzara [3], allowed a coal sample of 3 kg to be evaluated in an adiabatic device. The spontaneous heating temperature (which is the minimum inlet air temperature required to lead to a continuous temperature rise) of different U.S. coal samples was determined while passing moist air through each sample.
* Present address: New Zealand Dairy Research Institute, Palmerston North, New Zealand. 0010-2180/92/$5.00
Jones et al. [4] determined the critical conditions of a sample lignite in stainless steel cubic baskets. Stott et al. have completed spontaneous heating experiments on eight different New Zealand coals in a 2-m column system [5-8] designed so that the heat flow in the radial direction was minimized by computer controlled heaters. Compressed air was passed through the 2-m length of the coal bed at 1 cm/min. The spontaneous heating proceeded from room temperature (about 20°C) to above 110°C and was then terminated. This created a one-dimensional spontaneous heating process along the length and showed that moisture transfer from one region to another is extremely important to the development of the heating in stockpiled coal. Three distinct regimes exist during the spontaneous heating of coal to over 100°C (Fig. 1). In period I, the heat provided by coal oxidation brings the temperature to about 50°C before moisture evaporation starts to affect the heating significantly. (This has been confirmed by Stott and Chen [8, 9]; moisture appeared in a condenser located at the outlet of the 2-m column when the maximum temperature in the coal bed exceeded 50°C in a variety of different coals.) In period II, when the temperature rises from 70° to 100°C, a relatively large quantity of water was removed by the air stream. Copyright © 1992 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc.
T R A N S I E N T H E A T I N G IN M O I S T C O A L PILE
I
115
~ . . ~ 1 °C0 I0
/
~a
[..,
Curve 1
l -
....
~-
20 °C
-
III
L
I Time
Fig. 1. Three regimes of spontaneous heating in a moist coal stockpile. During this period, the drying process significantly delays the temperature rise. After period II, if the size of the stockpile is large enough and the ambient temperature is about 20°C, and if coal oxidation (after such extended aging) is still capable of evaporating all the water at one location in the stockpile, the temperature increases, as shown by curve 1 in period III. Otherwise, the temperature gradually levels off, as shown by curve 2, with the removal of the heat generated by oxidation.
Theoretical Background In addition to the experimental work on the spontaneous heating of moist coals, a number of mathematical models have been proposed for predicting temperature rise during heating. To date, however, only two transient mathematical models have been proposed in which the moisture transfer along a certain length is taken into account, as follows. ModelA. This assumes that the gas is saturated with moisture at every point in a moisture coal stockpile. The vapor concentration in the gas stream is a function of temperature only [10, 11]. Model B. This assumes that the moisture in the gas stream is in equilibrium with the moisture in the coal, but that saturation is not necessary. The vapor concentration in the gas stream is therefore a function of both
temperature and the equilibrium moisture content of the coal [9, 12]. The models share the following characteristics. 1. Heat transfer is assumed to occur by both conduction in the coal bed and convection in the gas stream. 2. Mass transfer of oxygen in the gas stream is assumed to occur by diffusion and convection. Mass transfer of moisture in the gas stream is assumed to occur by convection in the gas stream. Diffusion is also considered in Chen's work [9]. 3. The dry gas flowrate at the standard temperature and pressure is assumed to be constant and independent of space and time. 4. The local temperatures of the gaseous and solid phases are assumed to be equal. 5. The heat of oxidation is assumed to be constant and the heat of drying is assumed to have the same value as the heat of wetting. The heat of drying is defined as the amount of energy needed to drive off 1 kg of water from the coal. The heat of wetting is defined as the amount of energy released when the coal absorbs 1 kg of water. Model A, which was proposed in 1971 [10], was not acknowledged until Schmal et al. published a similar model [11] in 1985. Although Stott and Quan published an advanced model
116
X.D. CHEN
(Model B) in 1974, no attention has been drawn to this model [12]. The following basic differential equations, similar to those proposed and later modified by Stott and colleagues [10, 12], Chen [9], and also by Schmal et al. [11], governing the one-dimensional spontaneous heating of a moist coal stockpile, are used in present study. Energy Equation dT 32T Cps Ps d t = k-~Tx2
I
OT - CpgpgVg d x
II
III
+ Hor o - Hwr w,
IV
(1)
V
where I is the local rate of enthalpy change in the coal, II is the conductive heat transfer via coal particles, III is the convective heat transfer in the gas stream, IV is the rate of heat generation due to oxidation of the coal, and V is the rate of heat release due to drying or the rate of heat absorption due to wetting. Here, Cps is the specific heat capacity of bulk coal per unit mass, Ps is the bulk density of the coal stockpile, T is the local temperature, t is time, k is the thermal conductivity of bulk coal, x is distance, CPu is the specific heat capacity of the gas stream per unit mass, pg is the density of the gas, Vg is the velocity of the gas stream, which includes the dry and vapor flows, r o is the rate of oxidation of the coal, r w is the rate of drying/wetting (r w is positive when drying occurs). The heat of oxidation per mole of 0 2 consumed is 11o and this is assumed to be constant at 300 k J / m o l of 0 2 absorbed by the coal, and H w is the heat of drying/wetting, which varies, depending on the moisture content of the coal. The heat of drying is very close to the latent heat taken up during free moisture vaporization [9, 13-15]. The mass balances of oxygen transfer and moisture transfer, in which both diffusion term and convection term are included, are given below. Mass Balances Oxygen 02Co r o = D Oe
OX 2
Of o Vg
OX
(2)
Moisture 02C~ rw = -Dwe--~Z--x2
+ Vg'd
OCw,d 8X
(3)
In Eqs. (2) and (3), D o is the diffusion coefficient of oxygen in the gas stream, e is the void fraction of the stockpile, D w is the diffusion coefficient of water vapor in the gas stream, Vg,d is the dry gas velocity at STP, and Cw.d is the vapor density per unit volume of dry gas (weight of water contained at T in moist gas and resulting from 1 m 3 of dry gas at 273 K). In Eq. 2, r o is the rate of oxidation (mol m -3 s 1), which depends nonlinearly on the temperature, T, in the Arrhenius relationship and on the oxygen concentration, Co, in the gas; r o may be expressed as follows: Oxidation R a t e E q u a t i o n r o = A exp( - E / R T ) C o
n.
(4)
The "wet reaction" suggested by Gray [16] and by Gray and Wake [17] or the physical process of wetting [18] could occur when some parts of previously dried coal are moistened and this contributes to the temperatiare rise. The wetting adds moisture to the coal that has to be redried later. The interaction between the oxidation and the drying of coal continues until a certain temperature is reached. Beyond this temperature, moisture will not restrain the temperature rise at the hottest spot at nearatmospheric pressures. Numerical simulations using both Model A and Model B have been carried out previously. With Model A [10, 11], the predicted maximum temperature did not exceed 80°-90°C. Schmal et al. [11] attributed this to the long calculation time and the instability of the computing procedures. With Model B [12], a temperature exceeding 100°C was predicted without difficulty. Stott and Quan [12] concluded that the saturation assumption in Model A by Stott and Murtagh [10] was not appropriate. However, Model B has not been widely recognized. As suggested by Brooks and Glasser [19], the steady-state solutions can be used as an indication of the safety of coal stockpiles. A large discontinuous change in the steady-state regime occurring for small changes in the parameters,
T R A N S I E N T H E A T I N G IN M O I S T C O A L PILE for example, ambient temperature, could indicate a "critical behavior" that may lead to ignition, as described by Burnell et al. [20]. In this study, the maximum steady-state temperatures under different humidity have been obtained. The steady-state energy equation is 32T
3T
k--~x 2 - Cpg pgVg--~x + H o r o - H w r w = O.
(5) SOLUTIONS AND DISCUSSION (1) It is assumed that the molecular diffusion is the only mass transport mechanism, then the rate of oxidation, ro, and the rate of drying/wetting, r~, are equal to the diffusion terms in Eqs. 2 and 3. If r o and r w are substituted in Eq. 5, the energy equation can be rewritten as O2T
32Co
k-~x2
+ HoDoe~
+ H~D w
117 and Cw are equal to zero: Tma x -< T a + m o O o e / k
+ HwDwe/k
OZT
aT
- Cpg pyg-
2
Z
=0,
(6)
0C o
-
O2Cw •
(8)
Cw, a.
With the parameters provided in the Appendix, the maximum temperature rise is 77.9 o C. This calculation is very similar to those of Nordon et al. [21] and Brooks and Glasser [19] on dry char and dry coal, respectivelyl from which a maximum temperature of about 70°C was obtained. Brooks and Glasser [19] noted that diffusion alone is insufficient to explain spontaneous ignition. It is suggested that, in the absence of wind, the diffusion mechanism may dominate the heat and mass transfer during the initial stages of self-heating. Natural convection will take over subsequently. (2) If it is assumed that convection is the only mass transport mechanism, r o and r w are equal to the convection terms in Eqs. 2 and 3. If r o and r w are substituted in Eq. 5, the energy balance can be rewritten as
k--UZ - 3-X
Co.a
HoVg
OC~,d -- HwVg,d
gx
- O.
(9)
with the following boundary conditions: atx=0, atx=
T = T., 1,
Tx .
CO =Co,a,
ac o . . 3x
C~ = Cw,a;
3T
ac~ .
3x
a t x = 0,
O.
It is emphasized here that all the parameters in Eq. 6, except the derivatives, are constant. These boundary conditions correspond to ambient conditions at the top (x = 0) surface and bottom (x = 1) surface of a stockpile that is perfectly insulated [19]. When Eq. 6 is integrated twice with respect to x, and using the above boundary conditions: T=
The boundary condition at the air inlet are as follows:
T~ + H o D o e / k
(Co, a - c o )
+ HwDwe/k
(Cw,a - C w ) .
(7)
Thus, it can be seen that the maximum temperature will be less than, or equal to, the temperature that occurs at a point where C o
C O = C ....
k - - = C p g p g V g ( T i - Ta), 3x
Cw,d=Cw,d,a
The boundary condition at the outlet (x = 1) may vary. The gas velocity along the length of the pile, Vg, is assumed to be equal to the inlet air velocity. The heat of drying/wetting is also assumed to be the same as the normal latent heat [9-12] (2334 k J / k g of water evaporated; see Appendix). Integration of Eq. 9 once, with respect to x, gives aT k 3x
- C p g p g V g T - H o ~ C o - H~Vg,dCw, d
= Constant.
(10)
118
X.D. CHEN
The constant on the right of the equation can be determined using the boundary conditions at the air inlet. Therefore, OT k Ox - Cpg pgVg(T - Ta) - HoVg(C o - Co,,, ) -HwVg,d(Cw, d
-
(11)
Cw,d,a)=0.
The maximum temperature occurs at a point where the first derivative of temperature with respect to x is zero. Rearrangement of Eq. 11 gives (Tma x -
Ta)fpgpgVg
= HoVACo.o - Co)
(12)
+ HwV~.,( Cw.~.`, - Cw.~)
and the maximum temperature is given by Tmax
=
T`, + H o / ( C p g pg) +Hwf/(CpgPg)
(Co,`, - Q ) (Cw,d,`, -- Cw.d)
(13) where f is the ratio of the dry gas volume at the standard temperature and pressure to the
wet gas volume at T`,. That is, f is equal to Vg, d/Vg. T h e ambient temperature, T`,, is assumed to be 293 K, which is similar to the ambient temperatures used previously by Stott et al. [10, 12], Chen [9], and Schmal et al. [11]. If Cw, d is assumed to be the saturated vapor concentration at atmospheric pressure (as in Model A), then, as C O decreases gradually towards zero, the maximum temperature approaches 90°C using Eq. 13 and the constants listed in Appendix; see Fig. 2. This temperature corresponds well to the "equilibrium" temperatures found in practice [22, 23] and to experimental observations of stored coals [24, 25]. The numerical simulations using Model A gave similar temperatures [10, 11]. It would appear that the reason that these calculations could not give temperatures higher than 100°C, or could not "dry" the coal mass completely, was the saturation assumption in Model A itself, rather than the computing procedures, as claimed by the previous authors [11]. Model A is unsuitable for predicting ignition in a moist coal stockpile, in the sense that the time to dry the coal completely, when its tempera-
100
j____~Sy// 80
6O
[.,.,
/
4O
20
0
k
0
2
_
I
4
__
r
P
6
8
Co,, - C. (mol ma) Fig. 2. Temperature rise due to the change in oxygenconcentration of gas (expressed as the difference between inlet oxygenconcentration, Co,,,, and local oxygenconcentration, Co).
T R A N S I E N T H E A T I N G IN M O I S T C O A L PILE
119
110
100
c
80
v0 10
20
I
L
30
40
I
50
60
70
80
9{
oo
(100 - RH) (%) Fig. 3. Temperature rise due to change in relative humidity of gas (expressed as the difference between saturated humidity (RH = 100%) and local relative humidity, RH).
ture rises above 100°C, cannot be predicted this way. As the relative humidity, R H (Cw,d//Cw,d,s, where Cw,d,s is the saturated vapor density), decreases, the temperature predicted by Eq. 12 increases to above 100°C and then starts to rise significantly (Fig. 3), similar to curve 1 in Fig. 1. Model B, in which the gas stream during spontaneous heating is not necessarily saturated with water vapor and the relative humidity of the air is a function of both the moisture content of the coal and the temperature, predicts a similar t e m p e r a t u r e - t i m e curve to that shown in Fig. 1. The analytical solutions in this work confirm that this model is valid. Stott and Quan [12] could not obtain accurate data on the relationship between the relative humidity of the air and the moisture content of the coal at various temperatures for their prediction of self-heating in coal. The predicted temperat u r e - t i m e and temperature-distance profiles [12] were not in good agreement with the experimental results obtained 13 years later by Stott et al. [6]. However, similar numerical calculations by Chen [9] using more reliable
data for the drying characteristics and the rate of oxidation of the coal were in excellent agreement with experimental t e m p e r a t u r e time and t e m p e r a t u r e - d i s t a n c e profiles. Therefore, Model B may be taken to reflect the practical situation of spontaneous heating in a moist coal stockpile more accurately than does Model A.
This work is a by-product of a project "Spontaneous Heating of Coal--Large Scale Laboratory Assessment and Supporting Theory," funded by Coal Research Association of New Zealand (Inc.) and superuised by Dr. J. B. Stott, Reader in Chemical and Process Engineering Department, University of Canterbury, who greatly helped all the experimental work. The author is particularly grateful for helpful suggestions made by Professor G. C. Wake, Department of Mathematics and Statistics, Massey University; and by the referee of this article. REFERENCES
1. Stott,J. B., Nature 188:54(1960). 2. Stott, J. B., Contract JO395-146, United States Bureau of Mines, Pittsburgh (1980).
120
X.D. CHEN
3. Smith, A. C., and Lazzara, C. P., U.S. Bureau of Mines, Report of Investigation 9079 (1987). 4. Jones, J. C., Djakaria, R., and Ong, H., N Z J. Tech. 3:195-197 (1987). 5. Stott, J. B., Proceedings of the 21st International Conference on Safety in Mines Research, Sydney, 1985, pp. 521-527. 6. Stott, J. B., Harris, B. J., and Hansen, P. J., Fuel 66:1012-1013 (1987). 7. Stott, J. B. and Chen, X. D., Proc. Symp. 3rd Coal Research Conference, Wellington, New Zealand 2:209-219 (1989). 8. Stott, J. B., and Chen, X. D., A Laboratory Scale Apparatus to Measure the Tendency of Coal to Fire Spontaneously, Colliery Guardian, in press.
9. Chen, X. D., Ph.D. thesis, University of Canterbury, New Zealand, 1991. 10. Stott, J. B., and Murtagh, B. A., Australian and NZ Association for the Advancement of Science, 39th Congress, 1971. 11. Schmal, D., Duyzer, J. H., and van Heuven, J. W., Fuel 64:963-972 (1985). 12. Stott, J. B., and Quan, N. T., NZ Inst. of Mining Seminar, Paper 1, 1974. 13. Stott, J. B., Proc. Syrup. on Nature of Coal, Central Fuel Research Institute, Lealgor, India, Paper 22, 1959, pp. 173-184.
14. Bhattacharyya, K. K., Fuel 51:214-220 (1972). 15. Glanville, J. O., Hall, S. T., Messick, D. L., Newcomb, K. L., Phillips, K. M., Webster, F., and Wightman, J. P., Fuel 65:647-649 (1986). 16. Gray, B. F., Proc. R. Soc. Lond. A 429:449-458 (1990). 17. Gray, B. F., and Wake, G. C., Combust. Flame 79:2-6 (1990). 18. Berkowitz, N., and Schein, H. G., Fuel 30:94-96 (1951). 19. Brooks, K., and Glasser, D., Fuel 65:1035-1041 (1986). 20. Burnell, J. G., Lacey, A. A., and Wake, G. C., J. Austral. Math Soc. Ser. B 24:374-391 (1983). 21. Nordon, P., Young, B. C., and Bainbridge, N. W., Fuel 58:456-464 (1979). 22. Emera, T., and Nabeya, H., J. Mining Metallurg. Inst. Jpn. 74(842):83-87 (1960). 23. Jegar, C., and Froger, C., XV/International Conference on Coal Mine Safety Research, Washington, 1975, pp. 22-26. 24. Kok, A. KEMA Report WSK/20649-11 (also Ref. to Schmal et al. [11]) (1983). 25. Ogimoto,T., Gijutsu Daijesuto 18(12):32 (1982). Receit,ed l July 1991; rer,ised 17August 1992
APPENDIX
T h e following physical a n d chemical data were used in this article. Specific heat capacity of gas D e n s i t y of gas D e n s i t y of oxygen in gas s t r e a m atx=0m D e n s i t y of w a t e r v a p o r in dry gas s t r e a m at x = 0 m H e a t of drying H e a t of oxidation Void fraction of the coal b e d Ambient temperature T h e r m a l conductivity of coal Diffusion coefficient of oxygen in gas s t r e a m Diffusion coefficient of w a t e r v a p o r in gas stream R a t i o of the dry gas v o l u m e at 0°C a n d wet gas v o l u m e at 20°C
1 x 103 1.1 8.48
Jkg 1K-1 kg m -3 moi m - 3
0.0191
kgm
kJ k g - 1 kJ mol 1 0 2
DO
2334 300 30% 293 0.2 0.2
DW
0.25
cm 2 s - i
Cpg
p~ Co, ~ Cw, j, a
Hw Ho
E T, k
f
0.909
3
K W m -1 K cm 2 s - ~
l
COMBUSTION A ND F LA ME 90: 114-120 (1992)
On the Mathematical Modeling of the Transient Process of Spontaneous Heating in a Moist Coal Stockpile X. DONG CHEN* Department of Chemical and ProcessEngineering, UniL,ersityof Canterbury, Christchurch, New Zealand The influence of moisture transfer on the maximum temperature rise in a coal stockpile has been analyzed using the simplified one-dimensional differential equations that govern the spontaneous heating process. Analytical solutions of the maximum temperature rise have been obtained using well-established coefficients. The solutions suggest that, if the saturation of the gas stream in a coal stockpile with moisture is assumed, the numerically predicted temperature of the stockpile will be well below 100°C. However, as the relative humidity of the gas stream reduces, the predicted maximum temperature increases to over 100°C. These analytical solutions strongly support the idea of introducing the equilibrium relationship between the relative humidity of the gas and the moisture content of coal into the transient mathematical model of the heating process that has been developed over 20 years at the University of Canterbury.
INTRODUCTION
Experimental Background Comprehensive experimental work has been carried out on the spontaneous heating of coal. Similar temperature-time curves have been obtained by different researchers in different types of laboratory experiments. Stott's classical experiments demonstrate the effect of the moisture content on the temperature rises attained during spontaneous ignition processes of subbituminous coal samples [1, 2]. In these experiments, a certain mass of coal, which had previously been dried under vacuum at a temperature of 105°C, was placed in a cylinder through which moist oxygen or moist air was passed from the bottom of the cylinder to the top. A moderate-scale apparatus, described by Smith and Lazzara [3], allowed a coal sample of 3 kg to be evaluated in an adiabatic device. The spontaneous heating temperature (which is the minimum inlet air temperature required to lead to a continuous temperature rise) of different U.S. coal samples was determined while passing moist air through each sample.
* Present address: New Zealand Dairy Research Institute, Palmerston North, New Zealand. 0010-2180/92/$5.00
Jones et al. [4] determined the critical conditions of a sample lignite in stainless steel cubic baskets. Stott et al. have completed spontaneous heating experiments on eight different New Zealand coals in a 2-m column system [5-8] designed so that the heat flow in the radial direction was minimized by computer controlled heaters. Compressed air was passed through the 2-m length of the coal bed at 1 cm/min. The spontaneous heating proceeded from room temperature (about 20°C) to above 110°C and was then terminated. This created a one-dimensional spontaneous heating process along the length and showed that moisture transfer from one region to another is extremely important to the development of the heating in stockpiled coal. Three distinct regimes exist during the spontaneous heating of coal to over 100°C (Fig. 1). In period I, the heat provided by coal oxidation brings the temperature to about 50°C before moisture evaporation starts to affect the heating significantly. (This has been confirmed by Stott and Chen [8, 9]; moisture appeared in a condenser located at the outlet of the 2-m column when the maximum temperature in the coal bed exceeded 50°C in a variety of different coals.) In period II, when the temperature rises from 70° to 100°C, a relatively large quantity of water was removed by the air stream. Copyright © 1992 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc.
T R A N S I E N T H E A T I N G IN M O I S T C O A L PILE
I
115
~ . . ~ 1 °C0 I0
/
~a
[..,
Curve 1
l -
....
~-
20 °C
-
III
L
I Time
Fig. 1. Three regimes of spontaneous heating in a moist coal stockpile. During this period, the drying process significantly delays the temperature rise. After period II, if the size of the stockpile is large enough and the ambient temperature is about 20°C, and if coal oxidation (after such extended aging) is still capable of evaporating all the water at one location in the stockpile, the temperature increases, as shown by curve 1 in period III. Otherwise, the temperature gradually levels off, as shown by curve 2, with the removal of the heat generated by oxidation.
Theoretical Background In addition to the experimental work on the spontaneous heating of moist coals, a number of mathematical models have been proposed for predicting temperature rise during heating. To date, however, only two transient mathematical models have been proposed in which the moisture transfer along a certain length is taken into account, as follows. ModelA. This assumes that the gas is saturated with moisture at every point in a moisture coal stockpile. The vapor concentration in the gas stream is a function of temperature only [10, 11]. Model B. This assumes that the moisture in the gas stream is in equilibrium with the moisture in the coal, but that saturation is not necessary. The vapor concentration in the gas stream is therefore a function of both
temperature and the equilibrium moisture content of the coal [9, 12]. The models share the following characteristics. 1. Heat transfer is assumed to occur by both conduction in the coal bed and convection in the gas stream. 2. Mass transfer of oxygen in the gas stream is assumed to occur by diffusion and convection. Mass transfer of moisture in the gas stream is assumed to occur by convection in the gas stream. Diffusion is also considered in Chen's work [9]. 3. The dry gas flowrate at the standard temperature and pressure is assumed to be constant and independent of space and time. 4. The local temperatures of the gaseous and solid phases are assumed to be equal. 5. The heat of oxidation is assumed to be constant and the heat of drying is assumed to have the same value as the heat of wetting. The heat of drying is defined as the amount of energy needed to drive off 1 kg of water from the coal. The heat of wetting is defined as the amount of energy released when the coal absorbs 1 kg of water. Model A, which was proposed in 1971 [10], was not acknowledged until Schmal et al. published a similar model [11] in 1985. Although Stott and Quan published an advanced model
116
X.D. CHEN
(Model B) in 1974, no attention has been drawn to this model [12]. The following basic differential equations, similar to those proposed and later modified by Stott and colleagues [10, 12], Chen [9], and also by Schmal et al. [11], governing the one-dimensional spontaneous heating of a moist coal stockpile, are used in present study. Energy Equation dT 32T Cps Ps d t = k-~Tx2
I
OT - CpgpgVg d x
II
III
+ Hor o - Hwr w,
IV
(1)
V
where I is the local rate of enthalpy change in the coal, II is the conductive heat transfer via coal particles, III is the convective heat transfer in the gas stream, IV is the rate of heat generation due to oxidation of the coal, and V is the rate of heat release due to drying or the rate of heat absorption due to wetting. Here, Cps is the specific heat capacity of bulk coal per unit mass, Ps is the bulk density of the coal stockpile, T is the local temperature, t is time, k is the thermal conductivity of bulk coal, x is distance, CPu is the specific heat capacity of the gas stream per unit mass, pg is the density of the gas, Vg is the velocity of the gas stream, which includes the dry and vapor flows, r o is the rate of oxidation of the coal, r w is the rate of drying/wetting (r w is positive when drying occurs). The heat of oxidation per mole of 0 2 consumed is 11o and this is assumed to be constant at 300 k J / m o l of 0 2 absorbed by the coal, and H w is the heat of drying/wetting, which varies, depending on the moisture content of the coal. The heat of drying is very close to the latent heat taken up during free moisture vaporization [9, 13-15]. The mass balances of oxygen transfer and moisture transfer, in which both diffusion term and convection term are included, are given below. Mass Balances Oxygen 02Co r o = D Oe
OX 2
Of o Vg
OX
(2)
Moisture 02C~ rw = -Dwe--~Z--x2
+ Vg'd
OCw,d 8X
(3)
In Eqs. (2) and (3), D o is the diffusion coefficient of oxygen in the gas stream, e is the void fraction of the stockpile, D w is the diffusion coefficient of water vapor in the gas stream, Vg,d is the dry gas velocity at STP, and Cw.d is the vapor density per unit volume of dry gas (weight of water contained at T in moist gas and resulting from 1 m 3 of dry gas at 273 K). In Eq. 2, r o is the rate of oxidation (mol m -3 s 1), which depends nonlinearly on the temperature, T, in the Arrhenius relationship and on the oxygen concentration, Co, in the gas; r o may be expressed as follows: Oxidation R a t e E q u a t i o n r o = A exp( - E / R T ) C o
n.
(4)
The "wet reaction" suggested by Gray [16] and by Gray and Wake [17] or the physical process of wetting [18] could occur when some parts of previously dried coal are moistened and this contributes to the temperatiare rise. The wetting adds moisture to the coal that has to be redried later. The interaction between the oxidation and the drying of coal continues until a certain temperature is reached. Beyond this temperature, moisture will not restrain the temperature rise at the hottest spot at nearatmospheric pressures. Numerical simulations using both Model A and Model B have been carried out previously. With Model A [10, 11], the predicted maximum temperature did not exceed 80°-90°C. Schmal et al. [11] attributed this to the long calculation time and the instability of the computing procedures. With Model B [12], a temperature exceeding 100°C was predicted without difficulty. Stott and Quan [12] concluded that the saturation assumption in Model A by Stott and Murtagh [10] was not appropriate. However, Model B has not been widely recognized. As suggested by Brooks and Glasser [19], the steady-state solutions can be used as an indication of the safety of coal stockpiles. A large discontinuous change in the steady-state regime occurring for small changes in the parameters,
T R A N S I E N T H E A T I N G IN M O I S T C O A L PILE for example, ambient temperature, could indicate a "critical behavior" that may lead to ignition, as described by Burnell et al. [20]. In this study, the maximum steady-state temperatures under different humidity have been obtained. The steady-state energy equation is 32T
3T
k--~x 2 - Cpg pgVg--~x + H o r o - H w r w = O.
(5) SOLUTIONS AND DISCUSSION (1) It is assumed that the molecular diffusion is the only mass transport mechanism, then the rate of oxidation, ro, and the rate of drying/wetting, r~, are equal to the diffusion terms in Eqs. 2 and 3. If r o and r w are substituted in Eq. 5, the energy equation can be rewritten as O2T
32Co
k-~x2
+ HoDoe~
+ H~D w
117 and Cw are equal to zero: Tma x -< T a + m o O o e / k
+ HwDwe/k
OZT
aT
- Cpg pyg-
2
Z
=0,
(6)
0C o
-
O2Cw •
(8)
Cw, a.
With the parameters provided in the Appendix, the maximum temperature rise is 77.9 o C. This calculation is very similar to those of Nordon et al. [21] and Brooks and Glasser [19] on dry char and dry coal, respectivelyl from which a maximum temperature of about 70°C was obtained. Brooks and Glasser [19] noted that diffusion alone is insufficient to explain spontaneous ignition. It is suggested that, in the absence of wind, the diffusion mechanism may dominate the heat and mass transfer during the initial stages of self-heating. Natural convection will take over subsequently. (2) If it is assumed that convection is the only mass transport mechanism, r o and r w are equal to the convection terms in Eqs. 2 and 3. If r o and r w are substituted in Eq. 5, the energy balance can be rewritten as
k--UZ - 3-X
Co.a
HoVg
OC~,d -- HwVg,d
gx
- O.
(9)
with the following boundary conditions: atx=0, atx=
T = T., 1,
Tx .
CO =Co,a,
ac o . . 3x
C~ = Cw,a;
3T
ac~ .
3x
a t x = 0,
O.
It is emphasized here that all the parameters in Eq. 6, except the derivatives, are constant. These boundary conditions correspond to ambient conditions at the top (x = 0) surface and bottom (x = 1) surface of a stockpile that is perfectly insulated [19]. When Eq. 6 is integrated twice with respect to x, and using the above boundary conditions: T=
The boundary condition at the air inlet are as follows:
T~ + H o D o e / k
(Co, a - c o )
+ HwDwe/k
(Cw,a - C w ) .
(7)
Thus, it can be seen that the maximum temperature will be less than, or equal to, the temperature that occurs at a point where C o
C O = C ....
k - - = C p g p g V g ( T i - Ta), 3x
Cw,d=Cw,d,a
The boundary condition at the outlet (x = 1) may vary. The gas velocity along the length of the pile, Vg, is assumed to be equal to the inlet air velocity. The heat of drying/wetting is also assumed to be the same as the normal latent heat [9-12] (2334 k J / k g of water evaporated; see Appendix). Integration of Eq. 9 once, with respect to x, gives aT k 3x
- C p g p g V g T - H o ~ C o - H~Vg,dCw, d
= Constant.
(10)
118
X.D. CHEN
The constant on the right of the equation can be determined using the boundary conditions at the air inlet. Therefore, OT k Ox - Cpg pgVg(T - Ta) - HoVg(C o - Co,,, ) -HwVg,d(Cw, d
-
(11)
Cw,d,a)=0.
The maximum temperature occurs at a point where the first derivative of temperature with respect to x is zero. Rearrangement of Eq. 11 gives (Tma x -
Ta)fpgpgVg
= HoVACo.o - Co)
(12)
+ HwV~.,( Cw.~.`, - Cw.~)
and the maximum temperature is given by Tmax
=
T`, + H o / ( C p g pg) +Hwf/(CpgPg)
(Co,`, - Q ) (Cw,d,`, -- Cw.d)
(13) where f is the ratio of the dry gas volume at the standard temperature and pressure to the
wet gas volume at T`,. That is, f is equal to Vg, d/Vg. T h e ambient temperature, T`,, is assumed to be 293 K, which is similar to the ambient temperatures used previously by Stott et al. [10, 12], Chen [9], and Schmal et al. [11]. If Cw, d is assumed to be the saturated vapor concentration at atmospheric pressure (as in Model A), then, as C O decreases gradually towards zero, the maximum temperature approaches 90°C using Eq. 13 and the constants listed in Appendix; see Fig. 2. This temperature corresponds well to the "equilibrium" temperatures found in practice [22, 23] and to experimental observations of stored coals [24, 25]. The numerical simulations using Model A gave similar temperatures [10, 11]. It would appear that the reason that these calculations could not give temperatures higher than 100°C, or could not "dry" the coal mass completely, was the saturation assumption in Model A itself, rather than the computing procedures, as claimed by the previous authors [11]. Model A is unsuitable for predicting ignition in a moist coal stockpile, in the sense that the time to dry the coal completely, when its tempera-
100
j____~Sy// 80
6O
[.,.,
/
4O
20
0
k
0
2
_
I
4
__
r
P
6
8
Co,, - C. (mol ma) Fig. 2. Temperature rise due to the change in oxygenconcentration of gas (expressed as the difference between inlet oxygenconcentration, Co,,,, and local oxygenconcentration, Co).
T R A N S I E N T H E A T I N G IN M O I S T C O A L PILE
119
110
100
c
80
v0 10
20
I
L
30
40
I
50
60
70
80
9{
oo
(100 - RH) (%) Fig. 3. Temperature rise due to change in relative humidity of gas (expressed as the difference between saturated humidity (RH = 100%) and local relative humidity, RH).
ture rises above 100°C, cannot be predicted this way. As the relative humidity, R H (Cw,d//Cw,d,s, where Cw,d,s is the saturated vapor density), decreases, the temperature predicted by Eq. 12 increases to above 100°C and then starts to rise significantly (Fig. 3), similar to curve 1 in Fig. 1. Model B, in which the gas stream during spontaneous heating is not necessarily saturated with water vapor and the relative humidity of the air is a function of both the moisture content of the coal and the temperature, predicts a similar t e m p e r a t u r e - t i m e curve to that shown in Fig. 1. The analytical solutions in this work confirm that this model is valid. Stott and Quan [12] could not obtain accurate data on the relationship between the relative humidity of the air and the moisture content of the coal at various temperatures for their prediction of self-heating in coal. The predicted temperat u r e - t i m e and temperature-distance profiles [12] were not in good agreement with the experimental results obtained 13 years later by Stott et al. [6]. However, similar numerical calculations by Chen [9] using more reliable
data for the drying characteristics and the rate of oxidation of the coal were in excellent agreement with experimental t e m p e r a t u r e time and t e m p e r a t u r e - d i s t a n c e profiles. Therefore, Model B may be taken to reflect the practical situation of spontaneous heating in a moist coal stockpile more accurately than does Model A.
This work is a by-product of a project "Spontaneous Heating of Coal--Large Scale Laboratory Assessment and Supporting Theory," funded by Coal Research Association of New Zealand (Inc.) and superuised by Dr. J. B. Stott, Reader in Chemical and Process Engineering Department, University of Canterbury, who greatly helped all the experimental work. The author is particularly grateful for helpful suggestions made by Professor G. C. Wake, Department of Mathematics and Statistics, Massey University; and by the referee of this article. REFERENCES
1. Stott,J. B., Nature 188:54(1960). 2. Stott, J. B., Contract JO395-146, United States Bureau of Mines, Pittsburgh (1980).
120
X.D. CHEN
3. Smith, A. C., and Lazzara, C. P., U.S. Bureau of Mines, Report of Investigation 9079 (1987). 4. Jones, J. C., Djakaria, R., and Ong, H., N Z J. Tech. 3:195-197 (1987). 5. Stott, J. B., Proceedings of the 21st International Conference on Safety in Mines Research, Sydney, 1985, pp. 521-527. 6. Stott, J. B., Harris, B. J., and Hansen, P. J., Fuel 66:1012-1013 (1987). 7. Stott, J. B. and Chen, X. D., Proc. Symp. 3rd Coal Research Conference, Wellington, New Zealand 2:209-219 (1989). 8. Stott, J. B., and Chen, X. D., A Laboratory Scale Apparatus to Measure the Tendency of Coal to Fire Spontaneously, Colliery Guardian, in press.
9. Chen, X. D., Ph.D. thesis, University of Canterbury, New Zealand, 1991. 10. Stott, J. B., and Murtagh, B. A., Australian and NZ Association for the Advancement of Science, 39th Congress, 1971. 11. Schmal, D., Duyzer, J. H., and van Heuven, J. W., Fuel 64:963-972 (1985). 12. Stott, J. B., and Quan, N. T., NZ Inst. of Mining Seminar, Paper 1, 1974. 13. Stott, J. B., Proc. Syrup. on Nature of Coal, Central Fuel Research Institute, Lealgor, India, Paper 22, 1959, pp. 173-184.
14. Bhattacharyya, K. K., Fuel 51:214-220 (1972). 15. Glanville, J. O., Hall, S. T., Messick, D. L., Newcomb, K. L., Phillips, K. M., Webster, F., and Wightman, J. P., Fuel 65:647-649 (1986). 16. Gray, B. F., Proc. R. Soc. Lond. A 429:449-458 (1990). 17. Gray, B. F., and Wake, G. C., Combust. Flame 79:2-6 (1990). 18. Berkowitz, N., and Schein, H. G., Fuel 30:94-96 (1951). 19. Brooks, K., and Glasser, D., Fuel 65:1035-1041 (1986). 20. Burnell, J. G., Lacey, A. A., and Wake, G. C., J. Austral. Math Soc. Ser. B 24:374-391 (1983). 21. Nordon, P., Young, B. C., and Bainbridge, N. W., Fuel 58:456-464 (1979). 22. Emera, T., and Nabeya, H., J. Mining Metallurg. Inst. Jpn. 74(842):83-87 (1960). 23. Jegar, C., and Froger, C., XV/International Conference on Coal Mine Safety Research, Washington, 1975, pp. 22-26. 24. Kok, A. KEMA Report WSK/20649-11 (also Ref. to Schmal et al. [11]) (1983). 25. Ogimoto,T., Gijutsu Daijesuto 18(12):32 (1982). Receit,ed l July 1991; rer,ised 17August 1992
APPENDIX
T h e following physical a n d chemical data were used in this article. Specific heat capacity of gas D e n s i t y of gas D e n s i t y of oxygen in gas s t r e a m atx=0m D e n s i t y of w a t e r v a p o r in dry gas s t r e a m at x = 0 m H e a t of drying H e a t of oxidation Void fraction of the coal b e d Ambient temperature T h e r m a l conductivity of coal Diffusion coefficient of oxygen in gas s t r e a m Diffusion coefficient of w a t e r v a p o r in gas stream R a t i o of the dry gas v o l u m e at 0°C a n d wet gas v o l u m e at 20°C
1 x 103 1.1 8.48
Jkg 1K-1 kg m -3 moi m - 3
0.0191
kgm
kJ k g - 1 kJ mol 1 0 2
DO
2334 300 30% 293 0.2 0.2
DW
0.25
cm 2 s - i
Cpg
p~ Co, ~ Cw, j, a
Hw Ho
E T, k
f
0.909
3
K W m -1 K cm 2 s - ~
l