On the validity of thermogravimetric determination of carbon gasification kinetics

Chemical Engineering Science 57 (2002) 2907 – 2920

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On the validity of thermogravimetric determination of carbon gasi"cation kinetics Bo Feng, Suresh K. Bhatia ∗ Department of Chemical Engineering, The University of Queensland, St Lucia, Qld. 4072, Australia Received 4 January 2002; received in revised form 11 March 2002; accepted 23 April 2002

Abstract Thermogravimetric analysis has been widely applied in kinetic studies of carbon gasi"cation, with the associated temporal weight change pro"les being used to extract kinetic information and to validate gasi"cation models. However the weight change pro"les are not always governed by the intrinsic gasi"cation activity because of the e5ect of chemisorption and its dynamics. In the present work we theoretically determine the criteria under which weight change pro"les can be used to determine intrinsic kinetics for CO2 and O2 gasi"cation by examining the region in which the chemisorption dynamics can be assumed pseudo-steady. It is found that the validity of the pseudo-steady assumption depends on the experimental conditions as well as on the initial surface area of carbon. Based on known mechanisms and rate constants an active surface area region is identi"ed within which the steady state assumption is valid and the e5ect of chemisorption dynamics is negligible. The size of the permissible region is sensitive to the reaction temperature and gas pressure. The results indicate that in some cases the thermogravimetric data should be used with caution in kinetic studies. A large amount of literature on thermogravimetric analyzer determined char gasi"cation kinetics is examined and the importance of chemisorption dynamics for the data assessed. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Adsorption; Carbon; Energy; Gasi"cation; Kinetics; Reaction engineering

1. Introduction Thermogravimetry provides one of the most convenient and widely used methods for analyzing the kinetics of gas–solid reactions, and distinguishing between competing models. The method relies on the measurement of temporal variation of sample mass and of the rate of change of mass, as reaction occurs, using a thermogravimetric analyzer (TGA). The resulting curve of variation of sample weight with time, and of its derivative (the rate curve), is then represented in terms of a rate-conversion curve that is interpreted by means of a suitable model. In doing so it is commonly assumed that the rate is proportional to the geometric area of the gas–solid interface, so that the rate-conversion curve is then linearly related to the variation of the geometric surface area with conversion. Suitable structural models are available for prediction of the evolution of surface area with conversion (Sahimi, Gavalas, & Tsotsis, 1990; Bhatia & Gupta, 1994) and by this means ∗ Corresponding author. Tel.: +61-7-3365-4263; fax: +61-7-33654199. E-mail address: [email protected] (S. K. Bhatia).

the TGA pro"les are interpreted and the reaction kinetics analyzed. Carbon and char gasi"cation perhaps provides the most illustrative and interesting application of this method, in which the often-observed maximum in reaction rate with increase in conversion (Hashimoto, Miura, Yoshikawa, & Imai, 1979; Ge, Kimura, Tone, & Otake, 1981; Su & Perlmutter, 1985) is explained by a variety of structural models (Petersen, 1957; Bhatia & Perlmutter, 1980; Gavalas, 1980; Miura & Hashimoto, 1984; Ballal & Zygourakis, 1987; Bhatia, 1998; Kantorovich & Bar-ziv, 1994). Perhaps the most popular of these is the random pore model (Bhatia & Perlmutter, 1980; Gavalas, 1980). Although the above approach relating the measured rate with surface area is long established and widely used, there have been observations in the literature (Lizzio, Piotrowski, & Radovic, 1988) suggesting that thermogravimetrically determined rate-conversion curves must be corrected for chemisorbed complexes on the carbon surface, and that the rate maximum may be an artifact of the chemisorption dynamics. Thus the observed rate of weight change actually represents the di5erence between the rate of chemisorption and desorption, and is not necessarily representative of the intrinsic surface reaction rate (Lizzio et al., 1988).

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 1 8 9 - 6

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B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

Lizzio et al. (1988) measured the gasi"cation reactivity of a bituminous coal char in oxygen and carbon dioxide using TGA and product gas analysis (non-dispersive infrared spectroscopy, i.r.). Di5erences between reactivity pro"les obtained by these two methods were observed and attributed to signi"cant amounts of stable complex being formed during the initial stages of reaction. The TGA reactivity pro"les become equivalent to i.r. reactivity pro"les, when corrected to account for stable complex formation. This suggests that the former may not provide accurate representation of the variation in intrinsic reaction rate in this case. Guerin, Siemieniewska, Grillet, and Francois (1970) have also observed the signi"cant e5ect of chemisorbed oxygen on TGA reaction rates. The reactivities of a lignite char, gasi"ed at various partial pressures of oxygen at temperatures between 573 and 673 K, were determined by TGA and i.r. analysis. Their comparison showed considerable disagreement, particularly during the initial stages of gasi"cation. They attributed this to the ability of the char to adsorb large amounts of oxygen at the reaction temperature. During the initial stages of gasi"cation, even ‘negative’ reactivities were measured and interpreted to imply that the mass gain due to chemisorption exceeded the mass loss due to char gasi"cation. However gasi"cation kinetics have also often been well "tted by the random pore model with parameters corresponding to experimentally determined ones (Su & Perlmutter, 1985; Chi & Perlmutter, 1989; Ge et al., 1981; Lua & Guo, 2001). In these cases the TGA reactivity pro"les were "tted with chemisorption e5ects being neglected, and the structural parameters obtained by the model were close to those measured by gas adsorption techniques. The location of the maximum was also predictable by the model. This suggests that chemisorption dynamics is not always important, and in such cases TGA determination of reactivity can be adequate. The contradiction between the above investigations does suggest that there exists a region of parameter values and operating condition in which the e5ect of chemisorption dynamics is not important but outside which the latter is important. The present paper attempts to locate this region for carbon gasi"cation in oxygen and carbon dioxide, by studying the criteria of validity of the steady state assumption, i.e. the concentration of the oxygen complex remaining in an apparent pseudo-steady state during reaction. At such a pseudo-steady state, the rate of weight change due to accumulation of surface complexes is negligible in comparison to the total rate of weight change, so that the latter is essentially determined by the intrinsic surface reaction rate. Under this circumstance, TGA rate data will be adequate for kinetic studies, and correction to account for chemisorption dynamics is unnecessary. To perform the analysis we use published mechanisms and rate constant values for gasi"cation by carbon dioxide (Huttinger & Nill, 1990) and oxygen (Hurt & Calo, 2001). Subsequently, a large amount of

literature on TGA-determined char gasi"cation kinetics is examined and the importance of chemisorption dynamics for the data assessed.

2. Theoretical approach The analysis method adopted involves normalization of the rate equations and examination of the terms in the scaled equations. The equations for the weight change of carbon as well as the concentration of oxygen complex are "rst written based on the reaction mechanisms available in the literature and form the starting point of the analysis. Each variable in the equations is scaled to unit order of magnitude as is common to applications of the perturbation technique (Lin & Segel, 1974; Nayfeh, 1981), in which the zeroth order and successively improved solutions are obtained in terms of a small parameter. The scaled equation for the weight change of carbon is then analyzed for the necessary criteria by comparing magnitudes of the various terms in the equation. This approach has been successfully used by Bhatia (1987) in analysis of pseudo-steady behavior of solid-catalyzed reactions. Two gasi"cation models are studied here using this approach: a well-known CO2 gasi"cation model (Ergun, 1956) and a recently proposed oxidation model (Hurt & Calo, 2001), which are discussed in the following sections. 3. Gasication in carbon dioxide 3.1. Model formulation The CO2 gasi"cation model is formulated based on the following assumptions: 1. The initial surface is fully accessible. In other words, there is no blocked porosity that opens during the reaction, although there is experimental evidence (Buiel, George, & Dahn, 1999) that some micropores can be blocked and become inaccessible to the gasifying gases until after some conversion level. Thus all the initial surface sites are available for gasi"cation reactions. However, not all of these can actually react, as some may be very stable basal plane sites. On the other hand, edge sites and defective basal plane sites may participate in reaction. K2 2. The area of a chemisorption site is approximately 8:3 A (Gregg & Sing, 1982). Therefore the initial active surface area per unit mass of carbon, Sg0 , can be related to the initial amount of active sites, [Ct ]0 (expressed in mol=g), as follows: Sg0 = [Ct ]0 × 8:3 × 10−20 × 6:023 × 1023 = 5:0 × 104 × [Ct ]0 m2 =g:

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

3. The mechanism for CO2 gasi"cation is as follows (Ergun, 1956): k1

Cf + CO2
C(O) + CO k−1

k

2 + Cf ; C(O)→CO

(1)

where Cf is the empty site ready for reaction and C(O) is the oxygen complex formed by chemisorption of CO2 on carbon surface. This two-step mechanism can explain most of the experimental observations while remaining simple in form. More elaborate mechanisms are available in literature (Key, 1948; Koenig, Squires, & Laurendeau, 1985; Adschiri, Zhu, & Furusawa, 1987; Gadsby, Long, Sleightholm, & Sykes, 1948; McCarthy, 1986; Radovic, Jiang, & Lizzio, 1991; Kapteijn, Meijer, & Moulijn, 1992), but without detailed kinetic data provided. The above Ergun mechanism has been studied by many researchers and the kinetic constants are available (Huttinger & Nill, 1990). Therefore it is used for this analysis. 4. Di5usional limitations are absent, and do not inPuence the kinetics. Based on the above assumptions and mechanism given in Eq. (1), the following equations can be written for the weight change and the concentration of the intermediate oxygen complex, C(O): 1 dw = (k1 [Cf ]PCO2 − k−1 [C(O)]PCO )MO w0 dt −k2 [C(O)]MCO ;

3.2. Model scaling When thermogravimetric pro"les, i.e. sample mass changes, alone suRce to determine the intrinsic rate, the rate of weight change will be proportional to the total number of surface sites, i.e. 1 dw ˙ [Ct ]; − (4) w0 dt where Ct is the total number of sites per unit initial mass, at any time. The criterion for suRciency of the TGA pro"les for determining the intrinsic kinetics may be determined by examining the conditions under which the combination of Eqs. (2) and (3) reduces to a form similar to Eq. (4). To this end we utilize the site balance (5)

(6)

where ke = k1 PCO2 + k−1 PCO + k2 :

(7)

Eqs. (2) and (6) now combine to provide −

dx k1 k 2 M C [Ct ]PCO2 =− dt ke +

(ke MO + k2 MC ) d[C(O)] ; ke dt

(8)

where x=1−

w w0

(9)

is the carbon conversion and MC (=MCO −MO ) is the atomic weight of carbon. Reduction of Eq. (8) to a form similar to Eq. (4) now rests on the negligibility of the second term on the right-hand side of Eq. (8) in comparison to the "rst. To obtain the associated criteria it is necessary to appropriately scale the various terms and assess their relative signi"cance. From the "rst term on the right-hand side in Eq. (8), it is readily seen that the process time scale is given by tc =

(2)

d[C(O)] = k1 [Cf ]PCO2 − k−1 [C(O)]PCO − k2 [C(O)]: (3) dt Here Cf is the amount of vacant or free sites per unit initial mass (mol=g), at any time, w0 is the initial sample mass, MO is the atomic weight of oxygen (=16) and MCO is the molecular weight of carbon monoxide (=28).

[Ct ] = [Cf ] + [C(O)]

along with Eq. (3) to obtain 
d[C(O)] 1 ; k1 [Ct ]PCO2 − [C(O)] = ke dt

2909

ke [Ct ]0 k1 k2 MC PCO2

(10)

which is the appropriate characteristic value for scaling time. A suitable scaling value for the amount of surface complex per initial mass [C(O)] is given by its initial pseudo-steady state value, obtained upon setting d[C(O)]=dt = 0 at t = 0. Use of this condition in conjunction with Eqs. (2) and (5) provides [C(O)]SS 0 =

k1 PCO2 [Ct ]0 : ke

(11)

The scaled form of Eq. (8) now follows ∗ dx [Ct ]0 PCO2 k1 (MO ke + k2 MC ) dCC(O) = Ct∗ − ; d ke2 d

(12)

where Ct∗ =

[Ct ] [Ct ]0

(13)

and C∗C(O) =

[C(O)] [C(O)]SS 0

(14)

are the scaled values of [Ct ] and [C(O)] respectively and  = t=tc is the scaled time.

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B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

3.3. Criteria for validity of the apparent pseudo-steady state assumption in CO2 gasi2cation It is evident that all the terms in the above equation are scaled to order unity. An apparent steady state concentration of the surface complex exists if the second term on the right-hand side is much less than the "rst term, in which case thermogravimetric pro"les suRce to determine reactivity. This requires that [Ct ]0 [CO2 ]k1 (MO ke + k2 MC ) 6 0:1 (15) ke2 or ke2 − 10[Ct ]0 [CO2 ]k1 MO ke − 10[Ct ]0 [CO2 ]k1 k2 MC ¿ 0: (16) In interpreting the above criterion it should be noted that the negligibility of the second term on the right-hand side in Eq. (12) does not imply that d[C(O)]=dt = 0 at all times. Indeed Eq. (3) may be rewritten as ∗ dCC(O) ∗ = ke tc (Ct∗ − CC(O) ) (17) d so that true steady state on the surface is never achieved ex∗ cept when Ct∗ = CC(O) = 0. The criterion in Eq. (15) merely implies that the rate of weight change due to accumulation of surface complex is negligible in comparison to that measured. Proceeding with the analysis of the associated inequality in Eq. (16) provides the solution √ b − b2 − 4c ; (18) ke ¿ − 2 where b=−10[Ct ]0 PCO2 k1 MO and c=−10[Ct ]0 PCO2 k1 k2 MC . Equation (18) may be rewritten as  q2 ¿ 80[Ct ]0 q1 + 0:5 25 600[Ct ]20 q12 + 480[Ct ]0 q1 ; (19) where q1 = k1 PCO2 =k2 and q2 = 1 + q1 + k−1 PCO =k2 . Under gasi"cation conditions, the environment is expected to have a low CO concentration so that k−1 PCO =k2 1, and inequality (19) reduces to q12 {1 − 160[Ct ]0 } + q1 {2 − 280[Ct ]0 } + 1 ¿ 0:

(20)

Upon de"ning a1 = 1 − 160[Ct ]0 , b1 = 2 − 280[Ct ]0 , the solution of inequality (20) is obtained as a1 ¿ 0; a1 6 0;

(21) q1 6

−b1 −



b21 − 4a1 = q3 : 2a1

(22)

3.4. Valid region in CO2 gasi2cation From inequalities (21) and (22) it is clear that there are two regions in which the steady state assumption is valid, depending on whether a1 is larger or less than zero. Since a1 is only a function of [Ct ]0 , there will be a critical value

of [Ct ]0 separating the two regions. This critical value is [Ct ]cr = 1=160 according to the de"nition of a1 , which corresponds to the critical active surface area of Scr = 312 m2 =g following assumption 2. This critical active surface area is readily seen to be related to the speci"c monolayer adsorption capacity. If the weight of the adsorbed oxygen at complete coverage is less than 10% of the carbon weight, we have [Ct ]0 × 16 6 0:1, which after rearrangement is the "rst criteria found for CO2 gasi"cation. Therefore the two valid regions are as follows for carbon gasi"cation in CO2 : Region 1: For carbons with initial active surface area of less than Scr = 312 m2 =g, the pseudo-steady state assumption is always valid, independent of the experimental conditions. For such carbons, negligible (¡ 10%) weight change occurs even on complete monolayer coverage, and weight change dynamics then suRces in studying gasi"cation kinetics. Region 2: For carbons with initial active surface area larger than Scr = 312 m2 =g, an apparent steady state exists when q1 6 q3 . From its de"nition it is evident that q1 is a function of rate constants k1 , k2 and gas pressure, while q3 is only a function of active surface area. The rate constants of Huttinger and Nill (1990): k1 =k2 = 20:9 exp(−25 000=RT ) bar −1 , obtained for a carbon with an initial total surface area of 1 m2 =g, were used to identify the valid region. Fig. 1 shows the variation of q3 with [Ct ]0 , as well as of q1 at various conditions for carbons with a site density higher than [Ct ]cr , and negligible CO in the gas (i.e. PCO ≈ 0). The value of q3 decreases quickly with increase of [Ct ]0 while q1 is independent of [Ct ]0 . The horizontal lines in Figs. 1(a) – (d) represent the q1 values at four gasi"cation conditions: at 973 K in 10 bar CO2 , at 1500 K in 1 bar CO2 , at 973 K in 1 bar CO2 and at 973 K in 0.5 bar CO2 . Clearly the valid region, in which q1 6 q3 , is the hatched area in each "gure, which is the active surface area region between the critical active surface area, Scr (312 m2 =g), and a certain active surface area, Svalid , at which q1 and q3 intersect. The value of the latter, Svalid , depends on the experimental conditions, being 320; 354; 463 and 643 m2 =g, respectively, in Figs. 1(a), (b), (c) and (d). It is also evident that the valid active surface area region enlarges with decrease of temperature and CO2 pressure. Fig. 2 shows the variation of Svalid with temperature at various CO2 pressures. The region between 312 m2 =g and the Svalid curve is the area in which the steady state assumption is valid at that CO2 pressure. It is clear for any carbon with an active surface area larger than 312 m2 =g, the CO2 pressure and reaction temperature should be low enough to avoid signi"cance of the chemisorption e5ect in kinetic studies in a TGA. In the cases of high-pressure gasi"cation, the valid active surface region is very narrow from 312 m2 =g to only slightly higher, suggesting that the steady state assumption will generally be invalid for carbons with initial active surface area higher than 312 m2 =g when gasi"ed at high pressures of CO2 .

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

active surface area (m2/g)

active surface area (m2/g) 0

250

500

750

1000

0

12

250

9

6

q1, q3

q1 q3

T=1500 K PCO2= 1 bar

6

q3 q

T= 973 K

1

PCO2 = 10 bar

3

3

0 0.000 0.005 0.010 0.015 0.020 [Ct]0 (moles surface sites/mg solid)

0 0.000 0.005 0.010 0.015 0.020 (b)

[Ct]0 (moles surface sites/mg solid)

active surface area (m2/g) 0

250

500

active surface area (m2/g)

1000

750

0

12

250

500

750

9

Svalid

PCO2 = 1 bar

6

q3 q1

3

T= 973 K

9

T= 973 K

q1, q3

q1, q3

1000

12 Svalid

PCO2= 0.5 bar

6

q3

3

0 0.000 0.005 0.010 0.015 0.020 (c)

1000

750

Svalid

9 q1, q3

500

12 Svalid

(a)

2911

[Ct]0 (moles surface sites/mg solid)

q1

0 0.000 0.005 0.010 0.015 0.020 (d)

[Ct]0 (moles surface sites/mg solid)

Fig. 1. Variation of q1 (the horizontal line) and q3 (the solid curve) with the initial density of active sites, [Ct ]0 , for CO2 gasi"cation of carbon at various conditions. The region in which the steady state assumption is valid (q1 6 q3 ) is the (hatched) area between the left solid vertical line ([Ct ]cr ) and the dashed vertical line. The valid region corresponds to initial active surface area between (a) 312 and 320, (b) 312 and 354, (c) 312 and 463 and (d) 312 and 643 m2 =g.

active surface area (m2/g)

700

Most of the carbons and chars used in kinetic studies and in actual gasi"cation have an active surface area less than the critical value of 312 m2 =g. Therefore, in most cases of CO2 gasi"cation the steady state assumption will be valid. The carbon used by Huttinger and Nill (1990) has a total surface area of 1 m2 =g. Consequently, the kinetic data obtained by them are unlikely to be inPuenced by chemisorption dynamics. Thus, they could extract the rate constants of k1 and k2 from the gasi"cation kinetics with the surface under pseudo-steady state conditions. To compare with the above results, we also studied an oxidation model as below.

PCO2 = 0.5 bar

600

PCO2 =1.0 bar

500 PCO2 = 5.0 bar PCO2 = 10.0 bar

400

PCO2 = 20.0 bar

4. Gasication in oxygen

300 1000

1100

1200

1300

1400

1500

temperature (K) Fig. 2. Variation of maximum permissible initial active surface area, Svalid , with reaction temperature at various CO2 pressures, in the case of q1 = q3 for carbons with active surface area of larger than 312 m2 =g. The region below each curve is the area in which the steady state assumption is valid at that CO2 pressure.

4.1. Model formulation The oxidation model is formulated based on the following assumptions: 1. The initial surface is fully accessible and all the surface sites are available for reaction, though only the active ones can actually participate in the reaction.

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B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

2. The active surface area is related to the site density following the relation given earlier: Sg0 = 5:0 × 104 × [Ct ]0 m2 =g. 3. The mechanism for carbon oxidation is as follows (Hurt & Calo, 2001):

Eqs. (24) and (26) combine to provide −

k1 PO2 [Ct ]MC dx =− {k2 PO2 + k3 } dt k1 P O 2 + k 3 +

k1

2Cf + O2 → 2C(O);

k

(27)

where

k

C(O) + O2 →2 CO2 + C(O); C(O)→3 CO + Cf :

d[C(O)] ke ; k 1 PO 2 + k 3 dt

(23)

In this mechanism, step 1 is similar to that in the Ergun mechanism, except that here the reaction involves two empty sites. Step 3 is exactly the same reaction as in the CO2 gasi"cation mechanism, and step 2 is a surface reaction producing CO2 . Any of the steps, particularly step 2, may be a lumped description of several elementary steps and probably involves two sites. For simplicity the reactions are assumed "rst order in site density. This mechanism is capable of describing the major trends in reaction order, activation energy and CO=CO2 ratio over a wide range of temperature, and is supported recently by Haynes (2001). The kinetic data were also given by Hurt and Calo (2001) by "tting the experimental data. Therefore this mechanism for carbon oxidation is used for analysis rather than the other mechanisms available in literature (e.g. Essenhigh, 1981; Walker, Rusinko, & Austin, 1959; Laurendeau, 1978; Zhuang, Kyotani, & Tomita, 1995; Back, 1997; Walker, Taylor, & Ranish, 1991; Moulijn & Kapteijn, 1995; Chen, Yang, Kapteijn, & Moulijn, 1993). 4. The kinetics is not inPuenced by di5usional limitations. Based on the above assumptions and mechanism given in Eq. (23), the following equations can be written for the temporal variation of sample weight and the concentration of the oxygen complex, C(O): 1 dw = k1 [Cf ]PO2 MO − k2 [C(O)]PO2 MC w0 dt −k3 [C(O)]MCO ;

(24)

d[C(O)] = k1 [Cf ]PO2 − k3 [C(O)]; dt

(25)

where, w0 and w are the initial weight and weight at any time, while k1 , k2 and k3 are the reaction rate constants for the three steps in Eq. (23).

ke = k1 PO2 MO + k2 PO2 MC + k3 MCO :

(28)

It is readily seen from the "rst term on the right-hand side in Eq. (27) that the process time scale is given by tc =

k1 PO2 + k3 k1 PO2 [Ct ]0 MC {k2 PO2 + k3 }

(29)

which provides the appropriate characteristic value for scaling time. Again a suitable scaling value for the amount of surface complex per unit mass [C(O)] is given by its initial pseudo-steady state value, obtained upon setting d[C(O)]=dt = 0 at t = 0. Use of this condition in conjunction with Eqs. (24) and (5) provides [C(O)]SS 0 =

k1 PO2 [Ct ]0 : k 1 PO 2 + k 3

(30)

The scaled form of Eq. (27) now follows −

∗ dx ke k1 PO2 [Ct ]0 dCC(O) = −Ct∗ + ; d {k1 PO2 + k3 }2 d

(31)

∗ have the de"nitions as in Eqs. (13) and where Ct∗ and CC(O) (14), and  = t=tc is the scaled time.

4.3. Criteria for validity of the apparent pseudo-steady state assumption in oxidation Eq. (31) indicates that for gasi"cation in oxygen the steady state assumption is valid when ke k1 [O2 ][Ct ]0 1 {k1 [O2 ] + k3 }2

(32)

and as for the earlier case the inequality (•)1 may be replaced by (•) 6 0:1 for all practical purposes. Upon de"ning q1 = k1 [O2 ]=k3 and q2 = k2 [O2 ]=k3 , inequality (32) then becomes q12 {1 − 160[Ct ]0 } + q1 {2 − 280[Ct ]0 −120[Ct ]0 q2 } + 1 ¿ 0

4.2. Model scaling

which can be rewritten as

The procedure for the normalization and analysis of the above equation is similar to that in the analysis of the CO2 gasi"cation model. First we use the site balance, Eq. (5), along with Eq. (25) to obtain 
d[C(O)] 1 (26) k1 [Ct ]PO2 − [C(O)] = k1 P O 2 + k 3 dt

a1 q12 + b1 q1 + 1 ¿ 0;

(33)

(34)

where a1 = 1 − 160[Ct ]0 ;

(35)

b1 = 2 − 280[Ct ]0 − 120[Ct ]0 q2 :

(36)

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

The solution of inequality (34) is obtained as follows: b1 ¿ 0;

a1 6 0;

q1 6 −

a1 ¿ 0;

b1 6 0;

or

0.6

(37) 

b21 − 4a1 = q3 ; (38) 2a1  b1 − b21 − 4a1 q1 ¿ − = q4 ; 2a1

b1 +

PO2 = 0.05 bar

0.5

q1 6 q3 :

(39)

PO2 = 0.2 bar

PO2 = 5.0 bar

Upon substituting a1 and b1 in Eqs. (35) and (36) into the above inequalities, the criteria become [Ct ]0 6 1=160;

q2 6 0:333;

(40)

0.1

[Ct ]0 ¿ 1=160;

q1 6 q3 ;

(41)

0.0

[Ct ]0 6 1=160;

q2 ¿ 0:333;

4.4. Valid region in oxidation The rate constants of the reaction steps in Eq. (23) have been given by Hurt and Calo (2001) as follows: k1 = 3:3 × 10−4 exp(−35 000=RT )bar −1 , k2 = 5:7 × 10−4 exp(−130 000=RT ) bar −1 and k3 =exp(−180 000=RT ). The rate constants were used here in the above equations to determine the experimental conditions in which the steady state assumption is valid. Eqs. (40) – (42) suggest that for oxidation, there are three regions in which the steady state assumption is valid. However calculation results using the rate constants show that when [Ct ]0 6 1=160 and q2 ¿ 0:333, q1 ¿ q4 is always true while q1 6 q3 is always false. Therefore the criterion in Eq. (42) is never satis"ed. As a result, as for the C–CO2 reaction, there are only two regions in which the steady state assumption is valid, separately for carbons with initial site density larger and less than [Ct ]cr = 1=160: Region 1: For carbons with active surface area less than 312 m2 =g, the steady state assumption is valid when q2 6 0:333. Fig. 3 shows the variation of q2 with temperature at various O2 pressures. Clearly the size of the valid region depends on both temperature and O2 pressure. At low O2 pressure (0:05 bar) the steady state assumption is always valid if the reaction temperature is higher than Tvalid = 650 K. However at higher O2 pressure (1.0 bar) the steady state assumption is valid only if the temperature is higher than 950 K. The value of Tvalid rises with increase of O2 pressure. Fig. 4 shows the region (hatched area) in which the steady state assumption is valid for carbons with active surface area less than 312 m2 =g. It is clear that the invalid region is larger than the valid one. The typical temperature range in TGA studies varies from 600 to 1000 K. In this temperature range only when the O2 pressure is very low can we keep the steady state assumption valid. High-pressure TGA experiments may always be expected

PO2 = 10.0 bar

500

1000

1500

2000

2500

temperature (K) Fig. 3. Variation of q2 with temperature and oxygen partial pressure in carbon oxidation. The region below q2 = 0:333 is the area within which the steady state assumption is valid for carbons with initial active surface area less than 312 m2 =g.

20 1.0 O2 pressure (bar)

q1 6 q3 : (42)

16 O2 pressure (bar)

or

q2= 0.333

0.3

0.2

q1 ¿ q4

PO2 = 1.0 bar

0.4 q2=k2[O2]/k3

a1 ¿ 0;

2913

12

0.8 0.6 0.4 0.2 0.0 500 600 700 800 900 1000 temperature (K)

8

4

0 500

1000

1500

2000

temperature (K) Fig. 4. The region (hatched area) in which the steady state assumption is valid for oxidation of carbons with active surface area less than 312 m2 =g. Inset shows the region for oxidation at O2 pressure from 0 to 1 bar.

to su5er from the e5ect of chemisorption, and product gas analysis has to be performed in addition to monitoring the weight change, for obtaining gasi"cation kinetics. Region 2: For carbons with initial active surface area larger than 312 m2 =g, inequality (41) speci"es the pseudo-steady state region. q3 varies strongly with [Ct ]0 , and weakly with temperature and PO2 , especially at higher temperatures (¿ 850 K), while q1 depends on temperature

2914

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

0

active surface area (m2/g) 500 1000 1500 2000

0

3.0

3.0

2.0 1.5 1.0 q3

0.5 0.0 0.00

0.01

1.5

0.02

0.03

0.0 0.00

0.04

0.01

0.02

0.03

0.04

[Ct]0 (moles surface sites/gm solid)

(b)

active surface area (m2/g) 0 500 1000 1500 2000 3.0

3.0 T=1573 K PO2= 0.06 bar

2.0 1.5 q3

1.0

q1 and q3

2.5 q1 and q3

q3

1.0 0.5

[Ct]0 (moles surface sites/gm solid)

q1

2.5

T=1873 K

2.0

PO2= 0.06 bar

1.5 1.0

q3 q1

0.5

0.5 0.0 0.00

T=1573 K PO2= 0.12 bar

2.0

active surface area (m2/g) 0 500 1000 1500 2000

(c)

q1

2.5 T= 673 K PO2= 1.0 bar

q1 and q3

q1 and q3

2.5

(a)

active surface area (m2/g) 500 1000 1500 2000

0.01

0.02

0.0 0.00

0.04

0.03

[Ct]0 (moles surface sites/gm solid)

(d)

0.01

0.02

0.03

0.04

[Ct]0 (moles surface sites/gm solid)

Fig. 5. Variation of q1 (the horizontal lines) and q3 (the solid curves) with the initial density of active sites, [Ct ]0 , for oxidation of carbons with initial active surface area larger than 312 m2 =g at various conditions. The region in which the steady state assumption is valid (q1 6 q3 ) is the area between the left solid vertical line ([Ct ]cr ) and the dashed vertical line. The valid region corresponds to initial active surface area between (a) 312 and 312, (b) 312 and 359, (c) 312 and 421 and (d) 312 and 1089 m2 =g.

2500 O2 pressure

active surface area (m2/g)

2000

1500

1000

1

1 - 0.06 bar 2 - 0.12 bar 3 - 0.15 bar 4 - 0.21 bar 5 - 0.50 bar 6 - 1.0 bar 7 - 2.0 bar 8 - 5.0 bar

2 3 4 5 6 7 8

500

0 800

1200

1600

2000

temperature (K) Fig. 6. Initial active surface area as a function of temperature and O2 pressure, in the case of q1 = q3 , for carbons with active surface area larger than 312 m2 =g. The region below each curve is the area in which the steady state assumption is valid at that O2 pressure.

and PO2 and is independent of [Ct ]0 . Fig. 5 shows the variation of q1 (horizontal lines) and q3 (solid curves) with the initial site density. As in Fig. 1 the active surface area range in which the steady state assumption is valid can be identi"ed in Fig. 5 as the region between the left solid vertical line ([Ct ]cr ) and the dashed vertical line. At low reaction temperatures, q1 is very large and consequently this region is very small (Fig. 5(a)). The region becomes larger when temperature is higher and oxygen pressure lower. As shown in Fig. 5(b), at 1573 K in 0:12 bar O2 , the valid active surface area region is between 312 and 359 m2 =g. At 1573 K and 0.06 bar O2 , the region expands to from 312 to 421 m2 =g (Fig. 5(c)). When temperature is even higher at 1873 K in 0:06 bar O2 , the region is rather large between 312 and 1089 m2 =g (Fig. 5(d)). Evidently high temperature and low oxygen pressure should be used for kinetic studies of carbons with initial active surface area larger than 312 m2 =g. This is similar to that for the carbons in region 1. Fig. 6 shows the region in which the steady state assumption is valid in carbon oxidation for carbons in region 2. The region is that between 312 m2 =g and each curve for

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920 Table 1 Summary of the analytical results Reaction

Valid regions

Key features of valid region

CO2 (i) Sg ¡ 312 m2 =g or gasi"cation

Always valid for carbon with active surface area less than 312 m2 =g (ii) 312 m2 =g ¡ Sg ¡ Low temperature and low CO2 partial pressure for carSvalid (T; PCO2 ), Svalid (T; PCO2 ) depicted bons with active surface area in Fig. 2 larger than 312 m2 =g

O2 (i) Sg ¡ 312 m2 =g, T ¿ Low O2 pressure and high gasi"cation Tvalid (PO2 ), Tvalid (PO2 ) temperature for carbons with depicted in Fig. 4 or active surface area less than 312 m2 =g (ii) 312 m2 =g ¡ Sg ¡ Low O2 pressure and high temperature for carbons with Svalid (T; PO2 ), Svalid (T; PO2 ) depicted in active surface area larger than 312 m2 =g Fig. 6

that O2 pressure. Clearly the region shrinks with increase of O2 pressure. Also in the typical temperature range of TGA studies, the region is very small, suggesting that the steady state assumption is unlikely to be valid in the oxidation of carbons with active surface area larger than 312 m2 =g. For most carbons the active surface area is expected to be smaller than 312 m2 =g, and the "rst criterion will apply. As discussed above, this provides a limited region in O2 pressure–temperature space in which chemisorption dynamics are unimportant. Therefore unlike in CO2 gasi"cation, the valid region for the steady state assumption in oxidation is very limited, and it appears that the steady state assumption is generally invalid in the typical conditions of TGA studies unless very low oxygen pressure (¡ 0:1 bar) is used. 5. Discussion The above results show that the steady state assumption, and TGA determination of kinetics, can be inadequate in some cases, and this depends on the chemical reaction involved (gasi"cation by CO2 or oxygen), the reaction rate constants, the experimental conditions (temperature and pressure) and the physical properties of the carbon used (initial active surface area). The results are summarized in Table 1. It is interesting to note that the initial active surface area is very important in determination of the validity of the steady state assumption. Also a critical active surface area of 312 m2 =g was found for both CO2 gasi"cation and oxidation. For kinetic studies of CO2 gasi"cation, low temperature and low pressure should be used for carbons with initial active surface area larger than 312 m2 =g. This critical active surface area has been shown previously to be related to the speci"c monolayer adsorption capacity. Of course there is a limit to which the temperature

2915

can be raised before di5usional e5ects are signi"cant. However for kinetic studies of carbon oxidation, low oxygen pressure and high temperature should be used for all carbons. Also the validity of the steady state assumption is determined by the value of q1 which is the multiplication of the gas pressure and the ratio of rate of the chemisorption reaction to that of the desorption reaction, for a carbon with a given initial site density, as suggested in Eqs. (22) and (41). The steady state assumption is valid when the value of q1 is small enough. In CO2 gasi"cation this value decreases with decrease of temperature and pressure, while in oxidation it decreases with increase of temperature and decrease of pressure. Therefore, low temperature and low pressure in CO2 gasi"cation, and high temperature and low pressure in oxidation, are favorable for the validity of the steady state assumption. Most of the carbons used in kinetic studies have initial total surface area less than 312 m2 =g. Since the active surface area is only part of the total surface area, the initial active surface area of the carbons is also less than 312 m2 =g. Therefore, chemisorption dynamics is not important in CO2 gasi"cation and the TGA weight change pro"les are accurate for kinetic studies. However it can be important in carbon oxidation in the typical TGA experimental conditions as discussed above. Indeed the e5ect of chemisorption dynamics in carbon oxidation has been apparently overlooked in many of the TGA studies in literature, as shown in Fig. 7, which is a compilation of the experimental conditions for carbon oxidation used by various investigators, showing that many conditions are outside of the valid region. The fact that the TGA weight change pro"le can be inPuenced by chemisorption dynamics may explain several discrepancies reported in the literature, assuming that the reported TGA pro"les are free from di5usional limitations. Such limitations can arise due to transport resistance in the sample holder, in the particle bed and in the particles themselves. However, it is usual to conduct control studies with various Pow rates, sample and particle sizes to ensure the absence of di5usional limitations. Lizzio et al. (1988) and Guerin et al. (1970) found the importance of oxygen chemisorption while Su and Perlmutter (1985) did not. Fig. 8 shows the experimental conditions of Lizzio et al. (1988) and Su and Perlmutter (1985). The region below the solid curve is the valid region. It is easily seen that Lizzio et al.’s experimental conditions are far from the valid region while Su and Perlmutter’s conditions are partly in the valid region. Lizzio et al. (1988) attributed the di5erences in the reactivity pro"les of the Illinois coal char reacted in oxygen and carbon dioxide to the extent to which the stable oxygen complex forms during char gasi"cation. Much less oxygen complex was formed during CO2 gasi"cation than during oxidation, and therefore the e5ect of chemisorption was considered less important in CO2 gasi"cation. However this may be also explained by the fact that in CO2 gasi"cation, the steady state assumption is valid while in oxidation it is not. The carbon used in Guerin

2916

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

Fig. 7. Experimental conditions used in some TGA investigations in literature, and the critical curve below which the chemisorption dynamics is unimportant.

1.2

0.9 O2 pressure (bar)

Lizzio et al. (1988)

0.6

Su & Perlmutter (1985) 0.3

0.0 500

600

700

800

900

temperature (K) Fig. 8. Experimental conditions used in Lizzio et al. (1988) and Su and Perlmutter (1985), and the critical curve below which chemisorption dynamics is unimportant.

et al.’s experiments has a total surface area of 520 m2 =g. However the active surface area is not known and thus whether the active surface area is higher or lower than 312 m2 =g is unknown. Therefore the experimental conditions are shown in Fig. 9 considering both possibilities. In the latter case (Fig. 9(a)), the experimental conditions are in the invalid region in which chemisorption dynamics is important, particularly for condition 1. In the former case (Fig. 9(b)), the experimental conditions are also in the invalid region. Here it is assumed that the active surface area is equal to the total surface area. Also it is clear in Fig. 9(b), even if the active surface area is not equal to the total surface area, as long as the active surface area is larger than 312m2 =g, the experimental conditions will be very likely in the invalid region because the valid region is extremely narrow. The e5ect of chemisorption dynamics would then be important in their case, consistent with their "ndings. Tseng and Edgar (1984) found the characteristic curve, df=d versus conversion f( = t=t0:5 , where t0:5 is the time to reach 50% conversion), of a lignite is di5erent in di5er◦ ent oxidation temperature ranges (below 400 C and above ◦ 400 C). In both of the temperature ranges the reaction is not

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

1.2

1.2

0.9

0.9 3 O2 pressure (bar)

O2 pressure (bar)

2917

1 0.6

0.3 2 3 4

0.6 1 0.3

0.0 500

600

700

800

500

active surface area (m2/g)

600

700

800

900

temperature (K)

800

1 2 3

600

Fig. 10. Experimental conditions used in Tseng and Edgar (1984, 1985), and the critical curve below which the chemisorption dynamics is unimportant. Areas 1 and 2 correspond to conditions for lignite oxidation and 3 and 4 for oxidation of a bituminous coal and an anthracite.

4

400

200 500 (b)

4

0.0

temperature (K)

(a)

2

900

1000

1500

2000

temperature (K)

Fig. 9. Experimental conditions used in Guerin et al. (1970), and the critical curves below which the chemisorption dynamics is unimportant, assuming (a) the active surface area is less than 312 m2 =g and (b) the active surface area is larger than 312 m2 =g but equal to the total surface area. Numbers 1– 4 correspond to four experimental conditions: ◦ ◦ ◦ 1: 300 C, 0:6 bar O2 ; 2: 350 C, 0:1 bar O2 ; 3: 375 C, 0:1 bar O2 ; 4: ◦ 400 C, 0:1 bar O2 .

inPuenced by particle scale di5usional resistances, so that the characteristic curves should have been the same. They attributed the di5erence in the combustion behavior in these two temperature ranges to micropore di5usional e5ects. At lower temperature, the reaction rate is slow enough that the reactant gas has enough time to di5use into the ultramicropores, while at higher temperature the ultramicropores cannot be reached by the reactant gas. However this might be also explained by the di5erence in the e5ect of chemisorption dynamics in the two temperature ranges. Fig. 10 shows the experimental conditions they have used, as well as the

region in which chemisorption dynamics is not important (below the solid curve). Areas 1 and 2 correspond to the conditions in the two temperature ranges for lignite oxidation. In area 1 the e5ect of chemisorption dynamics is more important than that in area 2. Thus it will certainly inPuence the characteristic curve in area 1 more than that in area 2. Areas 3 and 4 in Fig. 10 correspond to the experimental conditions that Tseng and Edgar (1985) used for the study of the combustion behavior of a bituminous and an anthracite coal char. Again the e5ect of chemisorption in area 3 is more important than that in area 4. This may partly explain their ◦ observation that the characteristic curve at 600 C is di5erent ◦ from that below 600 C, in addition to the di5usional e5ects ◦ at 600 C observed by them. Miura and Silveston (1989) and Miura, Makino, and Silveston (1990) measured the gasi"cation reactivity of many Canadian coals and used the random pore model (Bhatia & Perlmutter, 1980) to analyze the data. They found that although the random pore model "tted the experimental data very well, the "tted structural parameter did not agree with the value estimated from gas adsorption for some coal chars. They attributed the discrepancy to the inaccuracy of the techniques for pore structure characterization, and=or the unrealistic assumptions in the random pore model. Here the possibility that the e5ect of chemisorption dynamics is important for those coals is explored. As discussed above and shown in Figs. 4 and 6, the region in which chemisorption dynamics is unimportant is di5erent for carbons with initial active surface area below and above 312 m2 =g. Thus we divide the coal chars in Miura et al. (1990) into two groups: total surface area below and above 312 m2 =g. Again

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

1.2

0.9 O2 pressure (bar)

because the active surface area is unknown, the active surface area of the coal chars with total surface area larger than 312 m2 =g is assumed to be less than 312 m2 =g, or larger than 312 m2 =g but equal to the total surface area. Figs. 11 and 12 show their experimental conditions as well as the valid region (below the solid curve) for carbons in groups 1 and 2, respectively. The experimental conditions for carbons in group 1 are partly in the region in which chemisorption dynamics is unimportant, while those for carbons in group 2 are completely outside the region. In other words, the e5ect of chemisorption dynamics is more important for carbons in group 2 with surface area higher than 312 m2 =g. Upon examining their experimental data we can see that more chars in group 1 have "tted structural parameter close to the measured one. This implies that the e5ect of chemisorption dynamics can be another reason for the discrepancy between the "tted and measured structural parameter. The coals in group 2 are generally low-rank coals, which are known to be very disordered and reactive. It is possible that there are many defects on the basal planes of the crystallites in these coals so that the active surface area is close to the total surface area for these coals. However, this needs to be veri"ed experimentally although there are some related discussions in literature (Walker et al., 1991). The steady state region identi"ed in the present study is admittedly dependent on the mechanism used, and the published kinetics of the reactions in the mechanism. However, the approach can be used for any mechanism provided the rate constants of the reactions in the mechanism are available. Further, in the analysis we have used, the admittedly subjective criterion that 1 can be replaced by  6 0:1. Even smaller values of the upper limit (e.g. 0.05 or 0.01) may be considered, and will yield more conservative criteria. The choice of this value depends on the accuracy desired in evaluating rate constants and reaction kinetics. However, given the accuracy with which process variables such as gas concentration, Pow rate and temperature are known, it is unlikely that it is justi"able to adopt a more conservative approach. Nevertheless, regardless of the degree of conservatism adopted, the essential and important result of the analysis is that there is a critical active surface area that governs the adequacy of TGA pro"les in determining the reaction kinetics. It appears that the TGA is suRcient for studies of CO2 gasi"cation as long as the active surface area of the carbon studied is less than 312 m2 =g, because the steady state assumption is then always valid. However for studies of gasi"cation by oxygen, there is strong possibility that the TGA reactivity pro"le is inPuence by oxygen chemisorption. In particular, at high O2 partial pressure, the TGA reactivity pro"le alone is almost always inadequate for kinetic studies because the steady state assumption is unlikely to be valid at low temperatures. The product gases must be analyzed to obtain the true reactivity in this case. Oxygen partial pressure of less than 0.1 bar is recommended for use to remain in the valid region of the steady state assumption, so that the

0.6

0.3

0.0 500

600

700

800

900

temperature (K) Fig. 11. Experimental conditions used in Miura et al. (1990) for carbons with initial active surface area less than 312 m2 =g, and the critical curve below which the chemisorption dynamics is unimportant.

800 PO2= 0.21 bar active surface area (m2/g)

2918

600

400

200 500

1000

1500

2000

temperature (K) Fig. 12. Experimental condition area (hatched) used in Miura et al. (1990) for carbons with initial active surface area larger than 312 m2 =g, based on the assumption that the active surface area is equal to the total surface area, and the critical curve below which the chemisorption dynamics is unimportant.

TGA reactivity pro"les can be used directly without correction for chemisorption. In practical combustors in which the temperature is very high (¿ 1373 K) and oxygen pressure is low (0.05 – 0.21 bar), the steady state assumption is expected to be valid. This implies that if the TGA reactivity pro"les are not corrected, the kinetic parameters obtained from the data are not representative of those at high temperature.

B. Feng, S. K. Bhatia / Chemical Engineering Science 57 (2002) 2907–2920

Actually, even after the TGA reactivity pro"les are corrected, the kinetic data obtained at low temperature cannot be used at high temperature because in di5erent temperature range di5erent reaction steps are in control (Hurt & Calo, 2001). 6. Conclusions The validity of the steady state assumption, or the negligibility of chemisorption dynamics in thermogravimetry, in carbon gasi"cation in CO2 and O2 was studied. A critical initial active surface area of 312 m2 =g, which corresponds to the monolayer adsorption amount being 10% of the initial carbon weight, was found to be crucial to the validity of the steady state assumption. There are two regions in which the steady state assumption is apparently valid, and the chemisorption dynamics does not signi"cantly inPuence thermogravimetric data, for gasi"cation in CO2 and in oxygen. Regions correspond to carbons with active surface area below and above the critical value. These two regions are summarized in Table 1. In the typical conditions of TGA studies, the steady state assumption seems always valid in CO2 gasi"cation, while it is generally invalid in oxidation. The results suggest that low oxygen pressure (¡ 0:1 bar) should be used in kinetic studies of oxidation using a TGA. Although the above results depend on the validity of the mechanisms used and the accuracy of the rate constant of the reactions in the mechanisms, the approach utilized in the present work can be used for any mechanism provided the rate constants in the mechanism are available. Notation a1 b b1 c C(O) [Cf ] [Ct ] [Ct ]0 Ccr Cf C(O) ∗ CC(O) Ct∗ k1 k−1 k2 k3

parameter, 1 − 160[Ct ]0 parameter in Eq. (18) parameter, 2 − 280[Ct ]0 or 2 − 280[Ct ]0 − 120[Ct ]0 q2 parameter in Eq. (18) density of C(O); mol=g density of Cf ; mol=g site density, mol surface sites=g solid initial site density, mol surface sites=g solid critical site density, 1=160 mol=g empty site ready for reaction oxygen complex dimensionless term, [C(O)]=[C(O)]0 dimensionless term, [Ct ]=[Ct ]0 rate constant of the "rst reaction in Eqs. (1) and (23), bar −1 rate constant of the backward reaction of the "rst reaction in Eq. (1), bar −1 rate constant of the second reaction in Eq. (1), and in Eq. (23), bar −1 rate constant of the third reaction in Eq. (23)

ke MC MCO MO PCO PCO2 q1 q2 q3 q4 Sg0 t tc w w0 x

2919

rate constant (=k1 PCO2 + k−1 PCO + k2 or k1 PO2 MO + k2 PO2 MC + k3 MCO ) g=mol atomic weight of carbon, 12 g=mol molecular weight of CO, 28 g=mol atomic weight of oxygen, 16 g=mol pressure of CO, bar pressure of CO2 , bar dimensionless term, k1 PCO2 =k2 for CO2 gasi"cation, or k1 PO2 =k3 for oxidation dimensionless term, 1+q1 +k−1 PCO =k2 for CO2 gasi"cation or k2 PO 2 =k3 for oxidation parameter, (−b1 − b21 − 41 )=2a1 parameter, (−b1 + b21 − 4a1 )=2a1 initial active surface area, m2 =g time, s characteristic time in Eqs. (10) and (29), s weight of carbon, g initial weight of carbon, g conversion level, 1 − w=w0

Greek letters 

scaled time, t=tc

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