Energy & Fuels 1994,8, 598-606
598
Articles The Characteristics af Coal Char Gasification at Around Ash Melting Temperature S. Y. Lin,* M. Hirato,? and M. Horio Department of Chemical Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Koganei, Tokyo 184, Japan, and Energy Research Office,5-5-15 Kugayama, Shuginamiku, Tokyo 168, Japan Received February 1 1 , 1993. Revised Manuscript Received January 1 1 , 1994'
The high-temperature C02 gasification (1173-1773 K) of coal char was investigated by using TGA and SEM and by measuring inner surface area concerning four different coals having much different ash melting temperatures. In regard to the unreacted char, the surface areas of micropores (diameter < 0.01 pm) and mesopore (diameter 0.01-0.1 pm) decreased with increasing temperature but the macropore (diameter > 0.1 pm) surface area did not. The true gasification rate constant was evaluated based on effectiveness factor prediction. The activation energy thus determined was 399 kJ-mol-' which is 1.88 times as large as the apparent activation energy value obtained without the correction of effectiveness factor. When carbon conversion was greater than 50%, an unusual reduction in the reaction rate from values expected from the low-temperature gasification rate took place over certain temperature range, in the region of f200 K of the maximium reduction temperature T x for each char. Since T Xwas about 150 K below the ash softening temperature, ST, of each coal, the rate reduction can be explained by ash melting. The macropore plugging by molten ashes of high viscosity was found at around the temperature Tx.Ash-carbon reactions of char which would help pores to grow were investigated by the product gas analysis and found to become significant above the temperature where complete ash melting takes place.
Introduction In recent years the reactivity of coal in gasification has been studied intensively with respect to chemical and physical properties of coal and coal char.1-'3 In most of the previous studies experiments were done below 1273 K, which correspond$ to the operating conditions of fixed bed and fluidized bed gasifiers. This implies that kinetic information has been lacking concerning the entrained bed gasifiers which are usually operated at much higher temperatures14Js providing a highly efficient gasification atmosphere. The effect of temperature should arise in the followingmajor aspects: (1)reduced micropore surface
* Author to whom correspondence should be addressed.
t Energy Reaearch Office. *Abstract published in Aduance ACS Abstracts, March 1, 1994. (1)Huttinger, K. J. Fuel 1983,62,166. (2)Kapteijn, F.;Moulijn, J. A. Fuel 1983, 62, 221. (3)Radovic, L. R.; Walker, P. L.; Jenkins, R. G. Fuel 1983,62,849. (4)Mulligan, M. J.; Thomas, K. M. Fuel 1987,66,1289. ( 5 ) Kasaoka, S.;Sakata, Y.; Shimada, M. Fuel 1987,66,697. (6)Ballal, G.; Zygourakie, K. Ind. Eng. Chem.Res. 1987,26, 1787. (7)Adechiri, T.; Furueawa, T. Chem. Eng. Sci. 1987,42,1313. (8)Kyotani, T.; Zhang, Z. G.; Hayashi, S.; Tomita, A. Energy Fuels 1988,2,136. (9)Chi, W. K. MChE J . 1989,35, 1791. (IO) Goyal, A.;Zabransky, R. F.; Rehmat, A. Ind. Eng. Chem. Res. i989,28,i767. (11)Tseng, H.P.;Edgar, T. F. Fuel 1989,68,114. (12)Hurt, R. H.; Sarofim, A. F.; Longwell, J. P. Fuel 1991,70, 1079. FueCJpn. Jpn. 1986, 1986,65, 65,789. (13)Hasimoto, K.; Miura, K. J. Fuel (14)Soelberg, N. R.;Smoot, L. D.; Hedman, P. 0. Fuel 1986,64,776. (15)Brown. B. W.:Smoot. L. D.; Smith, P. J.; Hedman, P. 0. AIChE J . 1988,34, 435.
0887-0624/94/2508-0598$04.50/0
area due to carbon rearrangement, (2) effect of incomplete diffusion, (3) ash melting, and (4) ash-carbon reactions. Among all, the ash melting would have one of most significant effects on gasification kinetics.16J7 Some indication of the significant influence of ash melting on the reaction kinetics of char gasification can be already found in the previous investigations although the complete information has not yet been presented.laZ0 Makino et a1.18haveobserved that the resistance of gaseous reactants to diffusion through the ash layer increased at around the ash melting temperature. Koyama et al,lgand Kasaoka et al.2O also reported that porosity and reactivity of chars gasified at around the ash melting temperature decreased from the values at lower temperatures. In this work, to clarify the characteristics of hightemperature gasificationversus those a t low temperatures, attention was focused on the C02 gasification at a temperature range of 1173-1773 K with respect to four kinds of coal chars which have much different ash melting temperatures. (16)Koyama, S.;Miyadera, H.; Nogita, S.; Hishinuma, Y.; Tamura, Z. 18th IECEC 1983,455. (17)Lin, S. Y.; Kameyama, S.; Ninomiya, Y.; Hirato, M. J . FuelJpn. 1990,69,191. (18)Makino, M.; Kaiho, M.; Kobayashi, M.; Yamada, S.; Yamashita, Y.Sunshine J . 1986,7 , 28. (19)Koyama, S.;Matuo, M.; Miyadera, H. J . Fuel Jpn. 1986,65,660. (20)Kasaoka, S.;Sakata, Y.; Tong, C. L. J.Fuel Jpn. 1983,62,335.
0 1994 American Chemical Society
Energy & Fuels, Vol. 8, No. 3, 1994 599
Characteristics of Coal Char Gasification Table 1. ProDerties of Coals and Coal Chars (wt %. dry)' BL NE PL TA coal proximate anal 40.8 44.2 28.8 37.4 volatile matter 45.3 57.5 50.0 63.4 FC 10.5 5.1 9.2 7.8 ash ultimate anal C 70.87 73.50 63.68 64.82 4.52 4.80 6.25 H 4.13 N 1.10 1.60 1.87 0.88 coal char ultimate anal C 84.73 75.52 85.20 79.43 H 1.57 2.15 3.80 2.23 1.52 1.07 N 1.78 1.79 ash 11.24 8.74 15.75 21.36 ash chemical comp (wt %) 57.32 42.75 60.47 47.73 Si02 25.15 35.44 29.64 15.03 Alz.03 .5.83 8.29 3.46 7.00 Fez03 3.97 8.68 1.17 8.12 CaO 0.44 1.75 1.63 2.03 MgO 0.49 1.54 0.82 2.40 NazO KzO 0.19 0.10 1.47 1.51 ash melting temp (K)b ST 1773 1763 1498 1463 HT >1873 >1873 1633 1559 >1873 1703 1578 FT >1873 a
BL = Blair Athol; NE = Newlands;PL = plateau; TA = Taiheiyo.
* Reduced atmosphere.
balance
Figure 1. Schematic diagram of experimental apparatus.
Experimental Section Coal Char Samples. Table 1 shows the properties of coals and coal chars used in this work. Coal particles of about 1 mm diameter were first carbonizated under Nz atmosphere at 1173 K (heating rate 5 K-min-l) about 30 min and then the char thus obtained was crushed and sieved into about 0.2 mm particles (80-48 mesh) for experimentation. Thermogravimetric Experiments. A high-temperature thermobalance (maximum temperature 1973K, 1 pg sensitivity, RIGAKU-TG-DTA-8823) was used to measure the gasification reaction rate. Figure 1 illustrates the apparatus. Gasification temperature was established from 1173 to 1773 K with 100 K intervals. Gasification was performed with a Ar/COZ mixture gas stream (COz60%). According to the following Dobner21 correlation for COz gasification rates rate =
kdfl = 7.45 X 104(T/2000)0.75/(P,d,)
Y = rJR; k,
[g-~m-~~atm-'~s-'] [g.~m-~.atm-~d]
(21) Dobner, S. EPRI,Palo Alto, CA 1976, (Jan. 15).
=kd2.'
where rc is the radius of the unreacted core and R is the radius of the whole particle including the ash layer; km is the gas film diffusion constant; k, is the reaction constant; k d is the ash f i i diffusion constant; e is voidage in the ash layer and T is the temperature of gasification; the smallest gas film diffusion constant can be predicted to be also 26 times larger than the overall COz gasification constant in the temperature range of the present experiments. We also confirmed that the negligible contribution of the gas film diffusion effect from the fact that the gas flow rate change from lo00 to 2000 cma(STP).min-l did not change gasification rate appreciably. Accordingly, we have chosen lo00 cm3(STP)-min-' for the gas flow rate in the present experiment. The experiment was conducted in the following procedure: a char sample of about 5 mg was put in a basket made of Pt/Rh (10%)gauze (150 mesh), and set in the thermobalance, and then heated with the heating rate of 10 K-min-' under Ar atmosphere. When the furnace was heated to a certain predetermined temperature, the gas stream was changed from Ar to Ar/COZ and gasification was started. At the same time the weight change measurement was started. Char Characterization. In order to inspect the effect of high-temperature heat retreatment of char on its pore structure, the higher temperature char samples were made by heating the 1173 K char further at temperatures same as gasification temperature. To quantify the pore structure of char, pores were divided into three ranges according to the ordinary terminology,22i.e., micropore (diameter < 0.01 pm), macropore (diameter > 0.1 pm) and mesopore. The micropore surface areas of the present char samples were determined from COn adsorption equilibrium measurement in a conventional volumetric adsorption apparatus. The surface area Sdm0of micropores was then calculated with the following Dubinin-Medek equationaasu
S,
Thermo
l/[l/kdm + l / ( k , p ) + (1 - Y')/(k,,Y')]
k, = 247 exp(-21060/2') [g-~m-~-atm-'~s-']
= 0.925583 X ~w,,(E/K)'/~
where wo is the total micropore volume, E is the characteristic energy constant, and K was estimated by Medeku as 3.145 kJ.nm3.mol-1 for COz absorption. Mercury porosimetry was used to obtain size distributions of meso- and macropores. The meso- and macropore volumes measured from mercury porosimetry was converted to surface area by the following equation derived based on the assumption of cylindrical pores of a uniform sizesz6
S = 4z(Vi/Di) I
where S,D,V, and i denote surface area (m2),pore diameter (m), pore volume (m3), and the interval number of pore diameter, respectively. The morphology of macropore surface of char taken after gasification was examined by a scanning electron microscope (SEM). Since the ashes produced by gasification accumulate on or/and adhered to the macropore surface area of char and are an obstacle observing the macropore surface of char, we steeped the samples of char taken after gasification in HF-HC1 acid (10% HC1) at room temperature about 6 h to remove the ashes which accumulated on or adhered to the macropore surface of char. (22) Chemical Engineers Handbook, 5th ed.; The Societyof Chemical Engineers: Japan, 1988; P1111. (23) Dubinin, M. M. Chem. Rev. 1960,60, 235. (24) Medek, J. Fuel 1977,56, 131. (25) Smith,J. M. Chemical Engineering Kinetics; McGraw Hill: New York, 1970.
Lin et al.
600 Energy & Fuels, Vol. 8, No. 3, 1994
8 00
-
n ln
>/I 1 c
U
5
* c 9
4QC
0
a
100 Surf
Temperature C K 3
Figure 2. Effect of carbonization temperature on micropore surface area of char.
,
200
300
0 400
area Cm2/g-initial-mbonI]
Figure 4. Gasificationrate corresponding to micropore surface area. 90 Newlands cool char
x =o.o
0 i
0
\
! !
\
\ \
',
0
i
0
Mesopore
0 Macropore
1100
1300
1500
1700
1900
0
0.1
0.2
Surf. area E m 2 / g initial carbon1
Temperature C K 3
Figure 3. Effect of carbonization temperature on meso- and macropore surface area of char.
Figure 5. Gasificationrate corresponding to macropore surface area.
Results and Discussion
area decrease took place not only in the micropore range but also in the mesopore range (0.014.1 pm). However, the graphitization may not be the only cause of the decrease of surface area of micro- and mesopores. Kasaoka et aL27 reported that even when graphite content was not increasing very much, the surface area decreased much with increasing temperature though the reason was not elucidated. In this work we would like to point out the contribution of ash melting to the pore structure change. Ash melting would induce pore blockage and ashcarbon reactions as discussed in later sections. Accordingly, the surface area of char should be influenced both by graphitization and ash melting particularly in high temperature gasification. For no decrease in the macropore surface area with increasing temperature, this is reasonable because the ash volume in original char 9.2 X lV m3*kg-l was much less than the macropore volume 8.38 X 1V m3.kg1. Effect of Incomplete Diffusion into Micropores. Figures 4 and 5 show the relationship of gasification rate and the surface areas of micro- and mesopores. Samples of different gasification rate and surface areas were obtained at various conversions. Figure 4 shows the relationship between gasification rate and micropore surface area. It can be seen that, in
Pore Structure of Char at HighTemperature. Prior to the gasification tests, the variation of char pore surface area with temperature was examined as shown in Figures 2 and 3. Figure 2 shows the micropore surface areas. It can be seen that the surface area decreased with increasing temperature. However, above 1573 K, the decrease in surface area levels off. All kinds of coal chars had high values of surface area of about 300-500 m2& at lower temperature such as 1273 K, and then greatly lost their surface area with increasing temperature up to 1573 K, reaching small values of 7-40 m2& at higher temperature such as 1773 K. Figure 3 shows the mesopore and macropore surface areas. At 1273-1473 K, mesopore surface area was about 2-2.2 mz*gl,but at 1573-1773 K the surface area fell about one-half and became about 1.2-1.4 mz.gl. There was no detectable decrease in the macropore surface with increasing temperature. Carbon in char is known to lose its micropore surface area through high-temperature heat treatment. It is generally agreed that this surface area reduction is caused by the structural rearrangements of carbon (graphitization).26 In the case of Newlands coal char, the surface
(27) Kaaaoka,S.;Sakata,Y.;Matsutomi,T.Proc.5fthMtg.Soc. Chem.
(26) Chiche, P.; Durif, S.;Pregermain, S. Fuel 1965, 44,5.
Eng., Jpn. 1986, 412.
Energy & Fuels, Vol. 8, No.3, 1994 601
Characteristics of Coal Char Gasification
1273 K
1773 K
1573 K 10 pm
U Figure 6. SEM photograph of Newlands coal char at X = 0.8 (reashed). Table 2. Thermogravimetric Data from Char Gasification and Pore Characteristics Determined by Mercury Porosimetry and BET Surface Area Measurement (Newlands Coal Char)
Table 3. Diffusivities of Cot, Thiele Modulus, Effectiveness Factor and True Rate Constants (Newlands Coal Char)
~
1273 1573 1773
2.78X l(r 0.0673 0.137 1.30X le2 0.115 0.0272 8.10X 10-2 0.112 0.0112
4.94 5.85 5.36
1273 1573 1773
7.40 7.76 7.96
0
the temperature range of 1373-1773 K, the gasification rate shows a linear dependence on the micropore surface area. As shown in Figure 5,no relationship between reaction rate and macropore surface area was found for the temperature range of 1273-1573 K, although at 1773 K some correlation seemed to exist between them. Taking 1273, 1573, and 1773 K as the reference temperatures corresponding to the three ranges, 1673 K,respectively, the morphology of the macropore surface of deashed gasified char was examined with a scanning electron microscope. Figure 6 shows the SEM photographs. In the 1273 K sample, a number of pores and cracks can be found on the macropore surface. In the 1573K sample, numerous circular pits can be seen on the macropore surface of the char, but none of them grew deeply. In the 1773 K sample, some deeply grown pores are seen on the macropore surface. The gasification at temperatures 1573 and 1773 K appeared to have occurred mainly in the region close to the macropore surface. This suggests that the effect of incomplete diffusion could be significant in high-temperature gasification. Accordingly, it is necessary to examine the "effectiveness factor" concerning the contribution of micropore to the overall gasification rate. The results from thermogravimetric experiments and the values for diffusivities, porosities, and pore diameters of Newlands coal char are shown in Table 2. The relationshipbetween the gasificationrate constant kv, (s-l) and the overall gasification rate dX/dt (s-l) can be expressed by dX/dt = kvpcCt.t.lXco*EfJPc
(3)
where MC is the mass of carbon (kg*kmol-l), is the totalgas density (kmol*m3), xc% is the COzmolar fraction, pc is carbon density in char (kg-m-9, and Ef, is the effectiveness factor for a sphere defined by
1.77 2.52 3.07
3.86 4.50 4.90
7.42 30.3 34.5
0.273 0.995 5.53 1.09 0.929 3.58 X 102 3.33 0.633 3.82 X 109
Cf. Appendix I. XCO,= 0.5.
Ef, = [real reaction rateI/ [reaction rate when the internal condition is equivalent to the external] (4) If we assume uniform temperature and diffusivity in a char particle, Ef, can be expressed Ef, = (3/m8) [l/tanh(m8)- l/m81 (5) where m, is the Thiele modulus for a sphere defined by = Tx ash melts completely to form spheres. In these two cases ash did not adhere to macropore surface. However, around the temperature Tx, the ash adhesion on the macropore surface can be observed. Hirato et al.38~~~ reported that with increasing temperature the ash melting proceeds from solid to slurry to liquid states. Coal ash is composed of oxides and other compounds. In general, the melting temperature of such a system is lower than those of pure oxides. Lin et ala40 observed the ash melting behavior with a high-temperature microscope using Newlands coal ashes and reported that the ash grain begins melting a t about 190 K below ST. Sagf et al.41and I ~ a n a g reported a~~ that the viscosity of molten ash was a function of ash composition and temperature. reported that the liquid-state slag of molten ash has a contact angle to carbon surface about 131-165 "C. Slag drops always contracted forming spheres and they did not soak into char matrix. Based on these (35) Miura, K.; Silveston, P. L. Energy Fuels 1989, 3, 243. (36) Calemma, V.; Radovic, L. R. Fuel 1991, 70, 1027. (37) Kashiwaya, Y.; Ishii, K. J. Iron Steel, Jpn. 1990, 76, 1254. (38) Hirato, M. Q. Rep. Inst. Appl. Energy 1986,8, 25. (39) Hirato, M.; Suzuki, A.; Ninomiya, Y. J . Fuel SOC.Jpn. 1983,62, 889. (40) Lin, S. Y.; Hirato, M.; Horio, M. J. Jpn. Inst. Energy 1992, 71, 272. (41) Sagf,W. L.; Mcllroy, J. B. J . Eng. Power 1960, April, 145. (42) Iwanaga, Y. Iron Steel 1982, 68, 2223. (43) Raask, E. Trans. ASME 1966, January, 40.
Characteristics of Coal Char Gasification
Energy & Fuels, Vol. 8, No. 3, 1994 603 10-1
4
10"
n VI \
5 lo-: .L
?
4 10-~ 10-
5
6
7
8
9
1 I T x i 0 4 CKI
1 / T x10 C K J
(a)
( c )
10-1
1
10-2
u'
10-2
n In
n In
z
\
5
;;
c
. I -
? 4 IO-^
?
4
10-5
5
x =o .l
10-4
10-~
6
7
8
9
1 I T xioh CKI
5
6
7
8
s
1 I T x104 C K 3 (d 1
(b)
Figure 8. Arrhenius plot for various carbon conversions. Table 5. Internal Surface Area Ratio of X = 0.8 to X = 0.0. Sx-o.slSx=o.o (m2/g)
Blair Athol Newlands Plateau Taiheiyo
1% (1573) 0.70 (1273) 1.55 (1273)
0.95 (1073)
5
1.85 (1673)
2.10 (1773)
0.55
0.90
(1573) 1.05 (1473) 0.75 (1173)
(1773) 1.75 (1673) 1.20 (1373)
Numbers in parentheses are temperatures in kelvin.
findings as well as ours (Figure 9), the ash melting in char can be described as a process of the following three steps as illustrated in Figure 1 0 (1) dry solid ash, (2) highviscosity melting ash adhered to carbon, and (3) lowviscosity spherical slag. When ash adhetad to carbon surface, it prohibita reactant gases to diffuse into macropore and lead to gasification rate reduction. The percentage of ash increases with increasing conversion. This is why in Figure 8 the effect of ash can been seen more evidently at higher conversion. Since the viscosity of molten ash depends largely on ash composition, different ash composition coal chars have, of course, different temperature
TX. Table 5 shows the values of the ratio of char internal surface areas a t X = 0.8 and X = 0.0. Since the specific surface area in Table 5 is based on the mass of char at each
conversion,it can be larger than the initial value depending on coal types. However, the important thing in Table 5 is that a t temperature Tx the value of this ratio becomes the smallest. Figure 11shows the relationship between the temperature TX and ash softening temperature ST measured according to JIS method (M8801) where ST, HT, and FT in JIS M8801 are equivalent to the IT, HT, and FT in ASTM D1857-68, respectively. It is found that the temperature TXlies about 150 K below ST. A s h 4 a r b o n Reaction in Char at High Temperatures. Nishi et al.44investigated ash-carbon reactions by measuring the amount of Sic, a-quartz, sillimanite, and mullite of coke ashes d t e r reaction and confirmed the impdrtance of ash-carbon reactions. The following ash-carbon reactions are possible in the high-temperature range as proposed by Nishi et ale4 for 1273-1773 K
SiO,
+ 3C = Sic + 2CO
for 1773-2023 K: SiO, 3A1,03.2Si0,
+ 3C = Sic + 2CO + 15C = 2SiC + 6A1+ 13CO
(44) Nishi, T.; Haraguchi, H.; Okuhara, T. J.Fuel Jpn. 1990,69,126.
604 Energy & Fuels, Vol. 8, No. 3,1994
Lin et al.
1673 K
1473 K
{OWJ
(a) Blair Athol coal char
1273 K
1573 K
, l o pI
1773 K
(b) Newlands coal char
1273 K
1473 K (c)
iOcu”;1
1673 K
Plateau coal char
1073 K
1173 K
pllm J
(d) Taiheiyo coal char Figure 9. SEM photograph of chars at X = 0.8.
1373 K
Characteristics of Coal Char Gasification Temperature up
-
T e Tx
T-Tx
e..,*:.*
?:.Y,..*;d:?
unmelting ash
T > Tx
molien ash
~
Figure 10. A model for behavior of ash melting in char.
-:
Conclusions
1673 -
tjn
y 1473
-
1273
-
t! $
0 0
Blair Athol
A
Plateau
0
Taihelyo
Newlands
2.01.5-
co -0coz -4-
Basegas: Ar 300mllmin
C
0 .-.-c
E" 0.5 8
By such an additional generation of macropores, the overall effectiveness factor should be improved and the diffusion of gases into mesopores, and consequently into micropores, should be enhanced. In fact, some ~ o r k s ~ ~ , ~ ~ have reported that, in the temperature range of complete ash melting, the gasification rate per unit pore surface area was even faster.
1873
E r: 5"
zn
Energy I%Fuels, Vol. 8, No. 3, 1994 605
1.0
4
Sample: Newlands char 3.589 Heating rate: 15 K lmin
0
,O*q
'\\ \ \ d,p' \ &
//
(3
0
I
I
I"
To obtain a comprehensive description of high-temperature gasification kinetics, the C02 gasification rate of char, changes of pore structure during gasification, and ash-carbon reactions were investigated. The micropore and mesopore surface areas decreased with increasing temperature, but the macropore surface area did not decrease. True gasification rate constant was evaluated based on effectivenessfactor prediction. The activation energy thus determined was 399 kJ-mol-l which is 1.88 times as large as the apparent activation energy value obtained without the correction. In high-temperature gasification, the reaction rate was reduced at high conversions showinga maximum reduction at a temperature Tx. TXwas about 150 K below the ash softening temperature ST. The ash-carbon reaction in the char occurred at around the temperature of complete ash melting. At very high temperature such as 1923K, all of the oxides were reduced by carbon and the ash-carbon reaction.
Acknowledgment. The authors deeply thank Dr. Yoshihiko Ninomiya of Department of Chemical Engineering Chubu University for advice and discussion. Appendix I The effective diffusivity for a coexisting macro- and micropore system as derived by Wakao et al.30is summarized in the following: DA,e
+ (l- tma)2DA,mi +
= tm:DA,ma
4~,(1- cma)/(l/D~,ma + l/DA,mi)
2
DA,mi= cmi / ( I - emaI2/[(1- "A,BYA)/DA,B+ ~ / D A , K , ~ ~ ] where "A,B = 1- (MA/MB)~/~, YA is mole fraction of gas A in the system, DA,Bis the molecular diffusivity, MAand MB are molecular weights for gas A and B, DA,K,ma, and D A , Kare ~ the ~ Knudsen diffusivities of gas A in the macroand micropores, respectively. Knudsen diffusivities were estimated by substituting pore radius r into the following equation: DA,K = ( 2 / 3 ) ( 8 R T / ~ M ~ ) ' / ~ r where R denotes gas constant and T the temperature.
Appendix I1 Approximating the diffusion-reaction system around a macropore wall by the uniform micropore region between a bundle of straight macropores of radius rmein a square (45)Johnson, D.B.Kinetics of Coal Gasification,John Wiley & Sons: New York, 1979; p 82. (46)Anthony, D. B.;Howard, J. B. MChE J . 1976,22,625.
(47)Nakamura, M.;Ninomiya, Y.; Fujimoto,T.; Hirato, M. R o c . 22th Coal Sci. Mtg. Jpn 1985.
Lin et al.
606 Energy & Fuels, Vol. 8, No. 3, 1994
array of center to center spacing Lma, we can describe the system by d2C/dr2+ (l/r)(dC/dr)- (kV,,,/De,,,)C = 0 Boundary conditions are written as
r = L,,/2,
dC/dr = 0; r = r,,,
C=C ,
(Al)
Let m = r(kv,mi/De,mi)1/2;eq A1 can be written as d2C/dm2+ (l/m)(dC/dm)- C = 0 (A31 The general solution for the above equation is given as
C = cllo(m) + c&o(m)
(A41 and the solution subject to the boundary conditions (A2) can be obtained as
C = (C,ma/A)[Kl(mL,)lo(m) - Il(mL,)Ko(m)l (A51 where
mL,
(A2)
- mmi= (L,a/2)(kv,,i/De,mi)1/2
IOand 11are Bessel functions,and KOand K1 are Macdonald functions. The effectiveness factor Ef,,i can be obtained as
E,,,
=R/R'
= (4rma/Lma)(l/mmi)(B/A)
where R E the real reaction rate = 2~,P,mi(dC/dr)r=r, and R' the reaction rate when the internal condition is equivalent to the external = ~(LmJ2)~kv,miCr,sand B is - Ki(m~,Vi(m~,)l. defined by B = [Ii(mL,)Ki(m,,)
