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Ind. Eng. Chem. Res. 2003, 42, 2746-2755

A Multiscale Transient Model for Combustion of Highly Porous Chars Yingwei Cai and Kyriacos Zygourakis* Department of Chemical Engineering, Rice University, Houston, Texas 77251-1892

We develop a transient multiscale model to describe the combustion of chars derived from plastic coals. This model visualizes the pore structure as an interconnected network of large cavities separated by walls consisting of microporous grains containing carbon and ash. The diffusionreaction problems formulated for the particle and grain scales are solved numerically, and a systematic comparison of model predictions and experimental combustion data is carried out. The main conclusion from this analysis is that the porosity and surface area associated with large cavities can drastically affect the reactivity and ignition behavior of char particles in the regime of pore diffusional limitations. This effect diminishes when combustion takes place in the regimes of kinetic or external mass-transfer control. 1. Introduction Coal accounts for about 34% of all of the electricity generated in the world. Continued use of coal for electric power generation will depend on our ability to build cleaner and more efficient plants. To reduce emissions of atmospheric pollutants (SOx and NOx) and greenhouse gases (CO2), the industry has increased its use of modeling and computational techniques in order to optimize the performance of commercial burners and furnaces. Computational fluid dynamics (CFD) with pyrolysis and char combustion submodels are often used to design coal combustion chambers. Optimal design and operation of advanced technology combustors, however, requires precise knowledge of the fundamental pyrolysis and combustion mechanisms. Design engineers should know how operating conditions affect coal combustion, and they should have computational tools that can accurately predict the coal particle burnoff time under any process conditions. This can only be achieved, however, through the use of sophisticated combustion models that can accurately describe the reactiondiffusion processes occurring in the temporally evolving pore structure of burning char particles. Coal utilization processes involve two major stages: coal pyrolysis and char combustion. The pyrolysis conditions (heating rate, heat treatment temperature, and gas atmosphere), the coal rank, and the coal particle size drastically affect the chemical composition and the pore structure of chars. For example, particles from plastic coals soften, swell, and resolidify during the pyrolysis stage, leading to chars with very high porosities and open macropore structure. Such phenomena are not observed with nonplastic coals that produce chars with lower porosities and smaller pore sizes. Coal combustion takes place at elevated temperatures and is always diffusion-limited. Reactions occur mostly in the surface around the mouths of the micropores where they open into the large macropores that form the main arteries for the transport of reactants to the particle interior. At higher temperatures, diffusional limitations appear even in the larger macropores and combustion eventually proceeds under external mass* To whom correspondence should be addressed. Tel.: 713348-5208. Fax: 713-348-5478. E-mail: [email protected].

transfer control. At the same time, the carbon is consumed, changing the pore structure and, consequently, the relative magnitudes of transport and reaction rates. This important interaction between the pore structure and reactivity will be our main focus here. Earlier studies of coal combustion either considered particles that were impervious to gaseous reactants1-3 or used a reaction expression that lumped the reaction kinetics and intraparticle diffusional resistance onto the particle’s external surface.4-10 Such assumptions may provide a sufficiently accurate description of the combustion process for a limited set of operating conditions (low rank, nonplastic coals, extremely high temperatures, and excess oxygen). Models with lumped kinetic expressions, however, are inherently incapable of describing the interaction between the intraparticle reaction and diffusion. Realizing the limitations of this approach, several investigators11-19 developed models that considered the effect of the internal pore structure on the gasification process. In general, these models described the internal pore structure of a char particle as a network of randomly overlapping objects (pores or grains) with different geometries. In many cases, the predicted evolution patterns of the internal pore surface area were compared to experimental gasification data to validate the models with varying degrees of success. Sotirchos and Amundson20-23 conducted one of the most comprehensive studies in this area to illustrate the effect of intraparticle thermal gradients and different modes of internal surface area evolution on the pseudo-steady-state and transient combustion. To estimate the intraparticle diffusional resistance, they simplified the problem by neglecting the diffusion in micropores and only considered mass-transfer resistance in macropores. Macropore diffusional resistance was lumped into a parameter representing the average macropore size. Their pseudo-steady-state analysis revealed an ambient temperature range where multiple solution loci existed for the particle internal temperature. They also concluded that the inclusion of intraparticle thermal gradients significantly affected the solution structure by expanding the multiplicity region and shifting it to lower ambient temperatures. As a result, their models predicted ignition temperatures that were much lower than those of isothermal models.

10.1021/ie0205391 CCC: $25.00 © 2003 American Chemical Society Published on Web 05/17/2003

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To account for the diffusional resistance in micropores, Sotirchos and Burganos24 further refined the model by incorporating Gavalas’25 random pore model into their analysis. Feng and Stewart’s diffusion model 26 was then used to lump the diffusional resistance in macro- and micropores into a single effective diffusional coefficient. This model effectively demonstrated that the increasing diffusional resistance shrank the multiplicity region and shifted it toward higher ambient temperatures, thus making it more difficult for ignitions to occur. Even these models, however, did not explicitly account for the presence of the extensive network of large spherical cavities observed in chars derived from pyrolyzing plastic coals.27,28 Figure 1A shows the cross section of a typical char particle produced from an Illinois #6 coal. This image reveals the presence of a “lacy” macropore structure with many spherical cavities separated by thin walls. Different parent coals and pyrolysis conditions may lead to internal structures characterized by thin-walled or thick-walled “cenospheres”, large cavities occupying most of the volume of a char particle.27,28 The significance of the macropore structure became obvious after we analyzed the data from a series of experiments that studied the combustion of chars produced by pyrolyzing plastic coals under flowing oxygen/nitrogen mixtures at different pyrolysis heating rates. The following trends were observed: (a) Chars from the same coal produced in the same gas environment, but at different heating rates, exhibited the same reactivity when combusted with oxygen at low temperatures, where reactions are expected to occur in the kinetic control regime (see, for example, the experimental data presented in Figure 2). (b) The same chars exhibited significantly different combustion rates and ignition behavior at higher temperatures where intraparticle diffusional limitations must be present (see, for example, the experimental data presented in Figure 4). Further analysis of the combustion data29 revealed that the macropore structure of these chars is the primary cause for the observed reactivity patterns in the regime of significant diffusional limitations. In general, chars produced at higher heating rates have higher porosities and surface areas associated with cavities larger than 1 µm in size.30 The more open structure of these chars increases the accessibility of micropores located in the particle interior to oxygen when combustion is carried out in the regime of significant diffusional limitations. This study presents the development of a comprehensive model designed to test the previous assertion. All of the key parameters of the model (porosities, pore surface areas, and reaction kinetics) are directly obtained from independent experimental measurements performed in our laboratory. Finally, the validity of the model is checked by comparing its predictions to combustion data obtained in a thermogravimetric reactor.31 2. Model Development Our model is an extension of the well-known grain model32-36 and describes the structure of chars with two pore networks. The first network consists of large, interconnected cavities (see Figure 1A) that act as the main arteries bringing gaseous reactants to the particle interior. The second network is formed by smaller micropores that emanate from the surface of the large

Figure 1. (A) Cross section of a typical char particle derived from a plastic coal (Illinois #6). This digital image was obtained using polished cross sections of char particles and video microscopy techniques. (B) Schematic showing the microporous grains that form the walls separating the large cavities.

cavities and penetrate the walls separating the large pores. We will assume that the walls are formed by overlapping (or touching) grains as shown in Figure 1B. To simplify the grain effectiveness factor calculations, we will also assume that the grains are spherical. These grains consist of microporous carbonaceous material that contains a significant amount of inorganic material (ash). As the carbon in each grain is consumed, a porous ash layer is formed around the shrinking (and partially reacted) core (Figure 1B). The size of the grains does not change as the reaction proceeds, and the particles maintain their structural integrity. Data from our laboratory and other studies in the literature provide strong support for this assertion.18,29 Because of the large amounts of ash present in many widely used coals, the size of their char particles changed little with conversion and negligible fragmentation was observed for particles burned in our thermogravimetric reactor. Two structural parameters must first be computed: the grain size rg and the total number n of grains per unit volume of char particle. These parameters are chosen so that the model porous solid has the macroporosity a and the macropore surface Sa of the actual chars. The random overlapping grain model15,37 was used for these calculations:

{

4 a ) exp - nπrg3 3 Sa ) a × 4πrg2n

(

)

(1)

The macroporosity and macropore surface areas of chars derived from plastic coals were measured using

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Table 1. Macropore Structure Data and Grain Model Parameters (Data from Reference 30) heating macropore total number rate macroporosity surface area of grains (°C/s) a Sa (cm2/cm3) per cm3 0.1 1 10

0.592 ( 0.030 0.595 ( 0.025 0.648 ( 0.025

865 ( 73 1244 ( 98 1040 ( 85

1.01 × 108 3.00 × 108 1.94 × 108

grain radius rg (cm) 1.08 × 10-3 7.46 × 10-4 8.11 × 10-4

optical microscopy methods.30 Only pores larger than about 1 µm were considered for these measurements. Table 1 gives these values for three chars derived from Illinois #6 coal under different pyrolysis heating rates. These chars have very high macroporosities, leading to a small overlap (10-15 vol %) of the spherical grains. This observation and the fact that the particles maintain their structural integrity as the carbon is consumed (leaving behind ash) give validity to our assumption that the grains react independently. We consider that the porous char particles burn in an environment of oxygen and inert gas (N2). The oxygen reacts with the carbon, producing carbon monoxide according to the reaction

2C(s) + O2(g) f 2CO(g)

(exothermic) (R1)

The homogeneous reaction

2CO(g) + O2(g) f 2CO2(g)

(exothermic)

(R2)

is not considered here. Earlier studies20,21 have shown that the inclusion of the homogeneous reaction (R2) does not have a significant impact on the solution. Thus, only three gases (O2, CO, and N2 denoted by the subscripts 1-3, respectively) should be considered. The binary diffusion coefficients calculated from the ChapmanEnskog equation satisfy the relation D12 ≈ D23 ≈ D13 ≈ Dab over the temperature range of interest here. Because the molecular weights of O2, CO, and N2 do not differ by more than 12%, we can also consider that the Knudsen diffusion coefficients are similar, i.e., D1 ≈ D2 ≈ D3 ≈ Dk.24 Thus, our multicomponent mixture (O2, CO, and N2) can be treated as a pseudobinary mixture of oxygen (indicated subsequently by the subscript A) and a mixture of carbon monoxide and nitrogen. 2.1. Mass and Heat Transfer in Macropores. Using an equivalent continuum assumption, the mass balance of oxygen in the macropores can be written as

∂(aCA) 1 ∂ ) 2 (-r2NA) + νARsηg(1 - a)(1 - ξg) ∂t r ∂r 0 < r < R0 (2) where R0 is the char particle radius (several hundred microns in our case), Rs is the intrinsic reaction rate, and ξg is the local carbon conversion. An effectiveness factor ηg (which does not appear in other models20-24) is introduced in eq 2 to account for diffusional limitations in the micropores of the spherical grains. By decoupling of the diffusional resistance in the micropores and the large cavities, this model can provide a more realistic description of the combustion process. The reaction rate Rs is given by

(

Rs ) A(x) k0 exp -

)

E C n RTs A

(3)

where A(x) is the term describing the evolution of the

reactive surface area (RSA) of the char with conversion x. Because the RSA is proportional to the micropore surface area, our model computes A(x) using the random pore model developed by Ballal and Zygourakis.14 This model employs probabilistic arguments to describe the evolution of the pore surface area with conversion for solid reactants having pores of different shapes and arbitrary size distributions. A bimodal size distribution is used to describe the pore structure of the grains, using the data obtained in earlier studies.14 The kinetic constants of eq 3 were obtained from a series of combustion experiments performed in the kinetic control regime.38 The reaction order n was determined to be equal to 1, in agreement with earlier studies.29,39,40 The mass flux can now be written as

dCA 1 NA ) -a2 ) [1 + xA(νB - 1)]/DAB + 1/Dka dr dCA -Dea (4) dr where Dea is the effective diffusion coefficient in char particles that is computed by the Smith-Wakao model.41,42 Because of the high macroporosity of these char particles, the transport of oxygen into the particle interior proceeds mainly through the macropores. On the other hand, Knudsen diffusion is expected to dominate in the micropores because their average pore radius is usually less than 20-30 Å. Thus, the final expression for mass flux in micropores is given by

Nµ ) -µ2Dkµ

dCA dCA ) -Deµ dr dr

(5)

where Deµ is the effective diffusion coefficient in micropores. When eq 4 is substituted into eq 2 and the problem is cast in terms of the oxygen mole fraction, the following working equations are obtained:

Ca

(

)

∂xA ∂xA 1 ∂ Cr2Dea ) - (1 + xA) 2 ∂t ∂r ∂r r -(1 + xA)Rsηg(1 - a)(1 - ξg) (6)

where C is the total concentration given by

a

(

)

∂C 1 ∂ 2 e ∂xA r Da C ) Rsηg(1 - a)(1 - ξg) + 2 ∂t ∂r r ∂r

(7)

This system of partial differential equations must be solved subject to the following boundary and initial conditions:

∂xA ∂C ) 0, )0 ∂r ∂r ∂xA ) Kg(xA - xAb) r ) R0 -Dea ∂r ∂C ) Kg(C - Cb) r ) R0 -Dea ∂r t ) 0 xA ) xA0, C ) C0 r)0

(8) (9) (10) (11)

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The energy balance takes the form

Cep

∂xA ∂T ∂T 1 ∂ 2 e ∂T - 2 ) rk +C h pgCDea ∂t ∂r ∂r ∂r r ∂r (-∆HR)Rsηg(1 - a)(1 - ξg) (12)

(

)

with boundary and initial conditions

r)0

∂T )0 ∂r

r ) R0 -ke

(13)

∂T ) hf(T - Tb) + QR ∂r

(14)

t ) 0 T ) T0

(15)

The radiant interaction of the reacting particles with their environment was described by the equilibrium radiation scheme QR ) 0.24,46 This assumption is valid when a particle is surrounded by other burning particles, as in a gasifier. It should also be approximately valid for our thermogravimetric analysis (TGA) experiments, where each burning particle is part of a larger population of 10-50 particles placed on the TGA pan.29,31,38 We should note, however, that char particles burning on the surface of the TGA pan can directly view cold surfaces outside the furnace and may experience heat losses through radiation to these surfaces. 2.2. Mass Balance in Porous Grains. Two distinct zones can be distinguished in all partially reacted grains: a core that contains unreacted carbon and a layer of ash that surrounds this core. We should note here that the model considers diffusion in both the ash layer and the reacting core. With the assumptions of a first-order reaction and spherical grains that react independently, the grain reaction-diffusion problem can be solved analytically. The grain effectiveness factor ηg can be derived in a straightforward fashion and is given by

ηg )

[

]

sinh(aΦ) a 3 12 1 - a a sinh(aΦ) a(a - 1)Φ cosh(aΦ) Φ (16) where Φ is a Thiele modulus defined in the standard fashion. Once the local effectiveness factor is determined through eq 16, the rate at which the carbon core shrinks can be easily computed from

4πy2Fm

dy 4 3 ) πy ηgRs dt 3

(17)

where y is the radial coordinate in a grain. The local conversion for each grain is given by

ξg ) 1 -

y3 rg3

(18)

and the conversion of the entire char particle (global conversion) can be calculated from

ξp )

( ) 3

∫0R R2 1 - ry 3

3 R03

0

g

dR

(19)

Figure 2. Experimental and model predicted reactivity patterns for the combustion of three chars at 400 °C and 21% oxygen. The chars were produced by pyrolyzing Illinois #6 particles (28-32 mesh) at different heating rates (0.1, 1, and 10 °C/s). Experimental curves are averages from 3-5 combustion runs.

Notice that in eq 19 y ) f(R) because the local conversion is a function of the radial coordinate of the char particle. According to Table 1, the grain size is smaller for char particles prepared at higher heating rates. Because eq 17 implies that carbon is consumed faster for larger ηg, we expect that chars produced at a high pyrolysis heating rate will exhibit higher overall reactivity when combusted in the diffusional limitation regime. 2.3. Physical Properties and Numerical Solution. We used the well-known correlations found in the literature43 to estimate the mass- and heat-transfer coefficients appearing in boundary conditions of our model. The effective heat conductivity ke for the porous char consisting of carbon, ash, and gas was calculated using Russell’s equation,44 while the heat capacity Cpg of the pure gas was calculated from a polynomial expression using coefficients found in the literature.45 The coupled system of differential equations developed in the previous sections must be solved numerically to compute the following profiles inside a reacting char particle: (a) oxygen mole fraction xA ) xA(r,t), total gas concentration C ) C(r,t), and particle temperature T ) T(r,t). Several investigators32,47,48 established the validity of the pseudo-steady-state approximation for gas-solid systems. We will also use this approximation here to uncouple the system of equations by assuming that the oxygen profile is always at steady state because its transport rate is large when compared to the rate at which the reaction layer moves. At each time level, the steady-state form of eq 6 is solved to compute the oxygen profile using the temperature profile from the previous time level. The energy balance equation is then solved with an implicit CrankNicholson scheme to advance the temperature profile. The purely implicit method requires that the physical parameters be evaluated using the temperature at the current time level. A predictor-corrector scheme is employed to overcome this problem. Once the temperature profile is known, the local and total conversion, the grain effectiveness factor, and the other physical parameters can be reevaluated using the newly computed profiles. The simulation continues until the overall conversion exceeds 99.9%.

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Figure 3. Combustion of three chars in the kinetic control regime (400 °C and 21% oxygen). Average reactivity measured from 0% to 99% conversion. Temperature at the particle center as a function of conversion. Oxygen concentration at the particle center as a function of conversion. Grain effectiveness factors at the particle center as a function of conversion.

3. Results and Discussion To evaluate the accuracy of our model, theoretical predictions were compared to experimental data from combustion experiments. Chars produced by pyrolyzing Illinois #6 coal particles (28-32 mesh) were reacted with oxygen at ambient temperatures ranging from 400 to 1000 °C. According to data obtained in our laboratory,38 these chars react in the regime of kinetic control at temperatures of around 400 °C. Significant pore diffusion limitations appear at temperatures higher than 500 °C, while the process shifts to external mass-transfer control at temperatures much higher than 700 °C. To elucidate the effect of the internal pore structure on combustion rates, chars produced at different pyrolysis rates were considered. As discussed earlier, previous experimental studies in our laboratory revealed that different pyrolysis heating rates lead to chars with different macropore structures.28,30 All char structural parameters (porosities, pore surface areas, pore sizes, particle diameters, etc.) and kinetic rate constants were measured with independent experiments. The same set of model parameters was used for all temperatures studied here. There was no parameter adjustment from run to run.

3.1. Kinetic Control Regime. Figure 2 compares model predictions and experimental reactivity data for the combustion of three chars at 400 °C and 21% oxygen. These chars were produced at different pyrolysis heating rates: 0.1, 1, and 10 °C/s. Each experimental point shown in Figure 2 represents the average of about 3-4 runs. In agreement with the experimental measurements, the model predicts similar reactivity rates for all three chars. Figure 3A presents the average reactivity calculated from the following formula:

r/s ) -

1 t*

∫0t*m10

( )

(

)

dm 1 m0 - m(t*) ξ* dt ) ) dt t* m0 t* (20)

where ξ* is the reference conversion, t* is the time to achieve this conversion ξ*, m0 is the initial mass of the solid in an ash-free basis, and m(t*) is the unreacted mass at conversion ξ*. Therefore, only the burnoff time t* required to achieve a specific conversion ξ* is needed to compute the average reaction rate from the conversion interval [0, ξ*]. Because it is difficult to precisely measure the burnoff time required to reach 100% conversion, the average reactivities reported here were

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Figure 4. Experimental and model predicted reactivity patterns for combustion of three chars at 550 °C and 21% oxygen. The chars were produced by pyrolyzing Illinois #6 particles (28-32 mesh) at different heating rates (0.1, 1, and 10 °C/s). Experimental curves are averages from 3-5 combustion runs.

computed for ξ* ) 0.99. Figure 3A shows a good agreement between the model and experiments. Clearly, all three chars have similar RSAs at all conversions, which also implies that they have similar micropore and mesopore structures. Parts B and C of Figure 3 present the temperature and oxygen concentration, respectively, at the particle center as a function of conversion. Both the particle center temperature and oxygen concentration remain close to the ambient conditions (400 °C and 21% oxygen) over the entire conversion range. All of these results suggest that the reaction takes place in the kinetic control regime, all intraparticle gradients are negligible, and there are no diffusional limitations in the grain pores. This assertion is further supported by Figure 4D, which shows that all three chars exhibit identical effectiveness factors at this temperature, even though their grain sizes are different. It should be noted that the decrease of the effectiveness factor with conversion is due to the consumption and shrinkage of the carbon core and not to the presence of diffusional resistances inside the grain. 3.2. Diffusion Control Regime: Diffusion Limitations Mainly in Micropores. Figure 4 compares model predictions and experimental reactivity data for the combustion of three chars at 550 °C and 21% oxygen. Again, these three chars were produced at different pyrolysis heating rates: 0.1, 1, and 10 °C/s. Significant differences in the reactivities of the three chars are now observed. Note that both the experimental data and model predictions show that chars produced at 1 °C/s heating rate have higher reactivity than those produced at 10 °C/s heating rate. This is not an unexpected result when one considers the structural data of Table 1. The average reactivities shown in Figure 5A show a good agreement between the model and experiment. They also provide another quantitative measure of the reactivity differences between the three chars. Figure 5B shows significant rises of the temperature at the particle center, particularly for the chars produced at 1 and 10 °C/s. Cai38 calculated the overall effectiveness factor η using experimental reactivity data

and found that, despite the presence of diffusional resistance, η was greater than unity in this temperature range. Figure 5B provides the explanation for this observation. At a combustion temperature of 550 °C, the highly exothermic reaction leads to significant heat accumulation inside the particle and large temperature excursions above the ambient. Consequently, the overall effectiveness factor takes values larger than unity. Figure 5C shows the local reactivity inside the particle at an overall conversion level equal to 40%, while Figure 5D presents the grain effectiveness factors at the particle center. All of these results suggest the presence of diffusional limitations at this temperature. The diffusional resistance, however, comes primarily from the grain micropores. Thus, char particles with more open macropore structure and larger macropore surface area will have higher reaction rate because of the enhanced accessibility of the micropores. Table 1 shows that the grain model parameters strongly depend on the char macropore structure: chars produced at 1 °C/s have the largest surface area and, thus, the smallest grain size. As shown in Figure 5D, these chars have the largest effectiveness factor. Because their grain micropores are more extensively utilized, these chars will exhibit higher reactivity when combustion takes place in the regime of diffusional limitations. 3.3. Diffusion Control Regime: Diffusion Limitations in All Pores. At even higher combustion temperatures, diffusional limitations also appear in the larger macropores (cavities). As a result, the effect of the macropore structure on the overall reactivity becomes less pronounced. This hypothesis is supported by the results presented in Figures 6 and 7A. The model predicts that chars produced at faster pyrolysis heating rates still have higher reactivity when combusted at 700 °C. In agreement with the experimental measurements, however, reactivity differences among chars are much smaller than those observed at 550 °C. The model predictions shown in Figure 6 exhibit an initial rise in reactivity that is much steeper than that computed from the experimental data. A possible explanation for this discrepancy is that the model overpredicts the initial rapid temperature rise because our energy balance ignores heat losses (through radiation or other modes) from our thermogravimetric reactor. Although such losses are negligible at low combustion temperatures, they may become significant as the reaction temperature increases. Figure 7B shows that the center of char particles combusted at 700 °C reaches temperatures much higher than the ambient, revealing the presence of sharp intraparticle temperature gradients. Experimental measurements suggest that at 700 °C the overall effectiveness factors are much smaller than unity. This implies that diffusional limitations in the grain pores and the cavities are so severe as to counterbalance the effect of high intraparticle temperatures. Parts C and D of Figure 7 show the local reactivity profile at 30% overall conversion (this is the conversion where the overall reactivity exhibits a maximum) and snapshots of the moving reaction front as the overall conversion increases from 30% to 45%. Because significant intraparticle oxygen concentration gradients are now present, the local reactivity near the particle surface is much higher than that at the center. As a result, the reaction does not proceed uniformly throughout the particle. Carbon near the particle surface is consumed faster because of

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Figure 5. Combustion of three chars in the regime of diffusional limitations (550 °C and 21% oxygen). (A) Average reactivity measured from 0% to 99% conversion. (B) Temperature at the particle center as a function of conversion. (C) Oxygen concentration at the particle center as a function of conversion. (D) Grain effectiveness factors at the particle center as a function of conversion.

the higher oxygen concentration in that region, and the local conversion profile moves as a front toward the particle center. 3.4. External Mass-Transfer Control Regime. External mass transfer began to control combustion at temperatures above 850 °C. Char reactivities increased very little when the reaction temperature was raised from 850 to 1000 °C, and differences among chars produced at different pyrolysis heating rates diminished even more. Clearly, the macropore structure played a smaller role in determining the reactivity when the reaction was restricted to a thin layer near the particle surface. While the magnitude of the overall reactivity predicted by the model was comparable to the experimental value, we found significant differences between predicted and measured reactivity patterns. While the model predicted that the reactivity reached a maximum at 10-15% conversion, experimentally measured reactivities did not peak until 60% conversion. A possible explanation for this discrepancy is the pore opening effect observed by Perkins.29 At high ambient combustion temperatures, solid temperatures may rise well above the ash fusion temperature, which is approxi-

Figure 6. Experimental and model predicted reactivity patterns for combustion of three chars at 700 °C and 21% oxygen. The chars were produced by pyrolyzing Illinois #6 particles (28-32 mesh) at different heating rates (0.1, 1, and 10 °C/s). Experimental curves are averages from 3-5 combustion runs.

Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2753

Figure 7. Combustion of three chars in the regime of diffusional limitations (700 °C and 21% oxygen). (A) Average reactivity measured from 0% to 99% conversion. (B) Temperature at the particle center as a function of conversion. (C) Reactivity profile at 30% total conversion. (D) Movement of the reaction front as the total conversion increases.

mately 1000 °C. Fused ash can then peel off the particle exterior and open “holes” that expose previously inaccessible pores to oxygen. While distinct from the fragmentation described by several investigators, this process can shift the reactivity maximum to higher conversion levels. To accurately describe this process, we must relax the key model assumption stating that char particles maintain their structural integrity during combustion. 4. Conclusions The model presented in this study provided us with some new insights into the combustion mechanism of chars with open macropore structures. By analyzing the temperature, concentration, and solid conversion profiles predicted by the model, we were able to explain experimentally observed changes in the char reactivity behavior as increasing temperatures shifted the combustion from the kinetic control regime to the regime of significant diffusional limitations. The main conclusion from this analysis is that the porosity and surface area associated with large cavities (larger than 1 µm in size) can affect the char reactivity when pore diffusion

resistances become important. There are no such differences when reaction takes place in the kinetic control regime. Also, these differences diminish as the reaction shifts to the high-temperature regime where external mass transfer controls. The model predicts significant increases in particle temperatures, raising the possibility of thermal ignitions. Figure 5B, for example, strongly suggests that char particles produced at high pyrolysis heating rates (1 and 10 °C/s) are much more likely to ignite than particles produced at the low heating rate (0.1 °C/s). This is exactly what we observed in our laboratory using a video microscopy technique to detect thermal ignitions of burning particles.29 In general, our model predictions are in good qualitative agreement with the dynamic analysis conducted by Sotirchos and Amundson.22,23 Their computations demonstrate that, at high combustion temperatures, the reaction can shift from the unignited solution branch (where the particle temperature is about the same as the ambient temperature) at low conversion to the ignited solution branch (where the particle temperature is much higher than the ambient temperature) at higher conversion as more heat is accumulated inside the particle.

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We should emphasize again that the same set of all model parameters was used for all temperatures studied here. There was no parameter adjustment from run to run. In particular, all char structural parameters (porosities, pore surface areas, pore sizes, particle diameters, etc.) and kinetic rate constants were measured with independent experiments. This allowed us to directly compare model predictions with experimental reactivity data and provided a rigorous test of the predictive capabilities of our model. Acknowledgment This work was supported by the U.S. Department of Energy (University Coal Research program) under Grant DE-FG22-97PC96214. Notation a ) radius of the solid carbon core and the grain, m CA ) molar concentration of the oxygen inside the particle, kgmol/m3 Cb ) ambient oxygen concentration, kgmol/m3 Dea ) effective diffusion coefficient in the char particle, m2 Deµ ) effective diffusion coefficient in micropores, m2/s kg ) heat conductivity of the pure gas, kJ/m‚s‚K kes ) effective heat conductivity of the solid, kJ/m‚s‚K Kg ) external mass-transfer coefficient, m/s m0 ) initial mass of solid (on an ash-free basis), kg m(t*) ) unreacted mass of solid at time t*, kg NA ) molar flux of species A, kgmol/m2‚s QR ) radial heat flux, kJ/m2‚s r ) radius of the carbon core at the current time, m rg ) initial radius of the grain, m R ) radial coordinate of the char particle, m R0 ) radius of the char particle, m Rs ) reaction rate per unit volume of the carbon solid Sµ ) micropore surface area, m2/kg Sa ) macropore surface area, m2/kg T0 ) initial char particle temperature, K Tb ) ambient temperature, K t ) time, s t* ) time to achieve conversion ξ*, s Vp ) char particle volume, m3 V∞ ) gas flow velocity, 240 cm3/min in our TGA system xA ) oxygen mole fraction xA0 ) initial oxygen mole fraction xAb ) ambient oxygen mole fraction Greek Letters R ) ratio of the heat conductivity of gas over solid -∆H ) heat of the reaction, kJ/kgmol  ) user-specified error tolerance a ) macroporosity of the char particle µ ) microporosity of the char particle p ) total porosity of the char particle η ) effectiveness factor of the particle ηg ) effectiveness factor of the grain νA ) stoichiometric coefficient of species A Fm ) density of the carbon core, kg/m3 Fp ) density of the char particle, kg/m3 Fgas ) gas density, kg/m3 ξ* ) reference conversion for computing the average reactivity ξg ) conversion of the individual grain (local conversion) ξp ) conversion of the char particle (global conversion) Φ ) Thiele modulus

Subscripts a ) macropores µ ) micropores A ) oxygen

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Received for review July 11, 2002 Revised manuscript received March 12, 2003 Accepted March 17, 2003 IE0205391