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Effective diffusion coefficients in coal chars
Proceedings of the Combustion Institute, Volume 28, 2000/pp. 2353–2359
EFFECTIVE DIFFUSION COEFFICIENTS IN COAL CHARS JAN E. JOHNSSON and ANKER JENSEN Technical University of Denmark Department of Chemical Engineering CHEC Research Centre DK-2800 Lyngby, Denmark
Knowledge of effective diffusion coefficients in char particles is important when interpreting experimental reactivity measurements and modeling char combustion or NO and N2O reduction. In this work, NO and N2O reaction with a bituminous coal char was studied in a fixed-bed quartz glass reactor. Particle sizes in the range 0.05–5 mm were tested, and the effective diffusion coefficients were estimated from measured effectiveness factors using the Thiele modulus. At 1079 K the effective diffusion coefficients were 5.5 ⳯ 10ⳮ6 m2/s and 6.8 ⳯ 10 ⳮ6 m2/s for N2O and NO, respectively. The experimental results were compared with theoretical values calculated from the mean pore radius and the cross-linked pore model. The method of mean pore radius underestimated the effective diffusion coefficient more than an order of magnitude. Using the cross-linked pore model, the bimodal pore size distribution, and a tortuosity factor of 5, a complete agreement between the experimental and the theoretical value was found. The conclusion was that for a char with a wide pore size distribution, the cross-linked pore model is a good choice for a theoretical calculation of the effective diffusion coefficient. In the case of strong pore diffusion limitations, the error in the interpretation of experimental results using the mean pore radius could be a factor of 5 on the intrinsic rate constant. For an average coal char reacting with oxygen at 1300 K, this would be the case for particle sizes larger than about 50 lm.
Introduction Mass transfer limitations may be important in the interpretation of experimental data for char reactivity in reactions with, for example, O2, N2O, and NO. In the comprehensive review by Aarna and Suuberg [1] on the nitric oxide–carbon reaction, it was concluded that there was a high probability that most of the reported data were obtained in or at least very near the intrinsic rate regime. However, very few experimental values for the effective diffusion coefficient in char particles have been reported in the literature, and in many cases the available data do not allow a theoretical estimation of it. The reaction between char and N2O and O2 are, respectively, roughly one and two orders of magnitude faster than the reaction with NO. For these reactions, mass transfer limitations may be important at combustion temperatures, except for the case of very small particles. In the review on char reactivity published by Smith [2], the intrinsic reactivity was calculated using an effectiveness factor and effective diffusion coefficient estimated from a mean pore radius. It was concluded, based on an earlier work [3], that using a more detailed bimodal pore model gave no significant differences in the calculated intrinsic reactivities. However, experimental values of effective diffusion coefficients at combustion temperatures are needed to verify the theoretical calculations.
The purpose of this work was to measure the effective diffusion coefficients of N2O and NO in porous char particles at temperatures of interest in fluidized-bed combustion, and to compare the experimental values with the results of theoretical calculations using data from physical characterization of the char particles obtained by CO2 adsorption and mercury porosimetry techniques. The results are used to evaluate the importance of mass transfer limitations at combustion conditions. Experimental Apparatus The experiments were carried out in a fixed-bed laboratory quartz reactor. It was mounted in an electrically heated oven with three independently controlled sections for good temperature control. In order to minimize homogeneous reactions and reactions catalyzed by the reactor surface, reactant gases were introduced separately through the main inlet in the bottom of the reactor and the secondary inlet in the top of the reactor and mixed just above the solids placed on a porous quartz plate. A gas of well-defined composition was mixed from pure gases or gas mixtures in a panel of precision mass flow controllers as described in Ref. [4]. The total gas flow rate was 30 cm3 sⳮ1 (273 K, 101.3 kPa). The reaction temperature was measured with a thermocouple in a quartz tube just below the porous quartz plate. The
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COMBUSTION OF SOLID FUELS TABLE 1 Char data Specific Surface (m2/g)
Porosity
Mean Pore Size (nm)
Particle Density (kg/m3)
150 4
0.05 0.46
1.5 500
— 920
BET (micropores) Mercury porosimetry (macropores)
Fig. 1. Observed rate constants for N2O and NO reduction over char particles of different size.
concentrations of N2O, NO, CO, and CO2 in the outlet gas and the bypass were measured with continuous analyzers and sampled by a PC, but only the N2O and NO measurements were applied in the data analysis. Chars The solids tested for the activity for N2O and NO reduction were obtained from a parametric study of the influence of operating conditions on the emissions from a 12 MW circulating fluidized bed (CFB) boiler at Chalmers University of Technology, Go¨teborg, Sweden. The fuel was a bituminous coal from Katowice, Poland. The chars were sampled from the dense bed of the CFB, and data for the chars are given in Table 1. Experimental Procedure A sample of 100–700 mg of char mixed with sand was placed on the porous quartz plate and inserted into the oven. The concentrations of N2O, NO, CO, and CO2 in the bypass were measured while the sample was heated to the reaction temperature. The decomposition of N2O in the empty reactor was tested at all experimental conditions and was found to be below 1% at 1130 K and about 7% at 1230 K at inert conditions. No decomposition of NO was observed. The experiments reported in this work
were carried out at a temperature of 1079 K, so no decomposition of N2O and NO occurred in the empty reactor. Observed rate constants have been calculated using a plug-flow reactor model. The layer of solids was a fixed bed 10–15 mm in height, and a separate test of the reactor showed that a plug-flow model is a valid assumption for the reaction conditions. The reduction of N2O and NO over char are gassolid reactions, and the char was consumed during an experiment. To avoid an initial char loss from reaction between air and char, the reactor was purged with N2 before the bottom section with the char was inserted into the oven. The char loss during an experiment was 1.3% as an average for all experiments. The relative standard deviation between duplicate experiments with new char samples from the same sieve fraction was below 10%. The results indicate that the chars from a given bed sample were rather uniform in activity. The fractional conversion was, within the experimental uncertainty, independent of the inlet concentrations of N2O and NO in the range from 50 to 450 ppmv, and a first-order reaction was assumed for both components. The measured activities of char, ash, sand, and bed material (sand Ⳮ ash Ⳮ partly sulfated limestone) have been reported elsewhere [4]. In this work, a special investigation of the effective diffusion coefficient in char particles is reported. Experimental Results The reduction of N2O and NO by char is a fast reaction at combustion conditions, and the reaction rate may be influenced by external mass transfer and pore diffusion for large particles. To investigate the influence of particle size in the size range of interest in fluidized-bed combustion, large char particles in the range 4.75–5.6 mm were crushed and sieved to narrow size fractions. Eight sieve fractions, the smallest 45/63 lm and the largest 4.75/5.6 mm, were tested at a temperature of 1079 K with inlet concentrations of 450 ppmv N2O and NO in nitrogen as described above. The observed rate constants for N2O and NO reduction versus mean particle size for the eight size fractions can be seen in Fig. 1. It is seen that the N2O reduction is influenced by mass transfer limitations for particle sizes above about
EFFECTIVE DIFFUSION COEFFICIENTS IN COAL CHARS
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The first-order observed rate constant was estimated and the relative influence of external mass transfer for spherical particles was calculated according to Ref. [6]: ⳮr ⳱ kobsC
(6)
1 1 d ⳱ Ⳮ char kobsqchar kgqchar 6kg
(7)
Relative external mass transfer resistance d /6k ⳱ char g 100% 1/kobs qchar Fig. 2. Relative influence of external mass transfer.
Fig. 3. Reduction of NO over char. Measured and calculated values of the effectiveness factor for different particle sizes using Deff ⳱ 6.9 ⳯ 10ⳮ6 m2/s.
100 lm, while for NO reduction the size limit is between 200 and 500 lm. The influence of external mass transfer and pore diffusion was estimated assuming spherical particles. The char particles were mixed with 0.3 mm sand particles, and so the external mass transfer coefficient was calculated according to Ref. [5] using a correlation valid for a fixed bed with inert and active particles of different size. 1.05
冢dd 冣 sand
ejDRem ⳱ 0.105 Ⳮ 1.505
(1)
char
0.5
m ⳱ 0.35 Ⳮ 0.29
冢dd 冣 sand
(2)
char
dsand uqgas l
(3)
Sc ⳱
l qgasD
(4)
kg ⳱
jDu Sc2/3
(5)
Re ⳱
(8)
The relative external mass transfer resistance is plotted versus particle size in Fig. 2. It is seen that for N2O reduction, external mass transfer resistance is dominating for particle sizes above 2–3 mm, while for NO reduction the influence of external mass transfer resistance is almost negligible for all particle sizes. For particle sizes below 0.2 mm, the external mass transfer resistance was negligible for both N2O and NO reduction at 1079 K. In Fig. 1, it can further be seen that the observed rate constant is independent of particle size for particles smaller than 0.2 mm, and so it is concluded that for this size there are no pore diffusion limitations either, and the effectiveness factor is unity. In this way, the intrinsic rate constant k was estimated directly from the experimental results. For particle sizes above 0.2 mm, the observed rate constants were corrected for external mass transfer resistance using equations 1–5 and 7, and gk was calculated for all particle sizes. The effectiveness factors for the larger particles can now be calculated as the ratio gk/k. The effective diffusion coefficient was estimated using an effectiveness factor calculation for spherical particles based on the Thiele modulus [7]:
⳱ g⳱
冢
1/2
冣
dchar kqchar 6 Deff
3 coth(3) ⳮ 1 32
(9) (10)
The value of the effective diffusion coefficient was estimated using a least squares method combined with a Newton iteration procedure. The result was 5.6 ⳯ 10ⳮ6 m2/s and 6.9 ⳯ 10ⳮ6 m2/s for N2O and NO, respectively. Figs. 3 and 4 show a good agreement between the effectiveness factors from experiment and the values calculated from the estimated effective diffusion coefficients using equations 9 and 10. This shows that the Thiele modulus approach is useful for estimation of an experimental value of the effective diffusion coefficient. For N2O, the two largest particle sizes were left out of the data treatment because of the estimated large influence of external mass transfer resistance in the experiments. However, it is seen that for N2O, even for the 1.0
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COMBUSTION OF SOLID FUELS
model, taking the pore size distribution into account, was used [12,13]. The binary bulk diffusion coefficients for N2O and NO in nitrogen at 1079 K were estimated to 1.25 ⳯ 10ⳮ4 m2/s and 1.60 ⳯ 10ⳮ4 m2/s, respectively, [11]. The Knudsen diffusion coefficient was then estimated from the BET surface measured by CO2 adsorption by the calculation of a mean pore radius [11]: Dk ⳱ 3.07 re 冪T/M ⳱ 3.07 Fig. 4. Reduction of N2O over char. Measured and calculated values of the effectiveness factor for different particle sizes using Deff ⳱ 5.6 ⳯ 10ⳮ6 m2/s.
and 1.7 mm particles, where the external mass transfer resistance is estimated to be 18% and 32% respectively, there is a deviation between the measured and the calculated effectiveness factor, indicating that the external mass transfer correction is inaccurate. For NO reduction, the relative external mass transfer resistance was below 15% for all particle sizes, and a better agreement is seen in the whole size range. There are only few experimental values in the literature to use for comparison. In Ref. [1], an effective diffusion coefficient for NO was derived from experimental results in Ref. [8]. The value was 1.4 ⳯ 10ⳮ6 m2/s for an anthracite char in the temperature range 900–1200 K, somewhat lower than the value found in this work. One reason for this difference may be that the porosity of the anthracite char was 0.29 compared to 0.51 for the bituminous coal char used in this work. Values of the effective diffusivity estimated from NO reduction experiments at lower temperatures (650–800 K) were very low, Deff ⳱ 4 ⳯ 10ⳮ9 m2/s, [9]. A transient pulse method was used in Ref. [10] at moderately elevated temperatures (300–550 K), and due to the relatively high activation energy of 26 kJ/mol, it was concluded that the mechanism was activated diffusion. The experimental values in Ref. [9] and Ref. [10] were well below the theoretical values calculated for Knudsen diffusion, and in both cases the mechanism may be activated diffusion as suggested in Ref. [10].
Theoretical Calculations To compare with the experimental values, the effective diffusion coefficients were calculated theoretically in two different ways. First, they were estimated from the BET surface and the porosity of the char particles using the methods of Satterfield [11], and then the more elaborate cross-linked pore
Deff ⳱
2h 冪T/M Sqchar
h 1 s 1/D Ⳮ 1/Dk
(11)
(12)
The tortuosity factor s is a parameter characteristic of the pore structure which corrects for the tortuous path through the pores due to the varying orientation and for deviations of the real pore structure from the ideal cylinder geometry. A theoretical value of s ⳱ 3 was derived for an ideal isotropic pore structure in Ref. [13]. For a porous char, the tortuosity may be larger, and in Ref. [9], a value of 5 was assumed. Using s ⳱ 5, a BET surface of 150 m2/g measured by CO2 adsorption and a porosity of 0.51 calculated as the sum of the porosities measured by BET and mercury porosimetry, the effective diffusion coefficient was estimated to 0.35 ⳯ 10ⳮ6 m2/s and 0.43 ⳯ 10ⳮ6 m2/s for N2O and NO, respectively, that is, an order of magnitude lower than the experimental value. The reason for this discrepancy is probably that the method of Satterfield [11] is valid for catalysts with a narrow pore size distribution, and in that case, a characteristic mean pore size can be calculated and used for estimation of the effective diffusion coefficient. In the case of chars from a combustion system, a wide pore size distribution with micropores and macropores can be expected. Another uncertainty in the calculation of the theoretical value is the tortuosity factor. However, it is difficult to imagine that the tortuosity factor can be below 3 to 5 in a porous char structure, and so it is not possible to get agreement between the theoretical and the measured value of the effective diffusion coefficient by choosing a lower value for s. To improve the theoretical approach, the effective diffusion coefficient was calculated using the pore size distribution measured by a combination of BET and mercury porosimetry and the cross-linked pore model [13]. In this model, diffusion in the pore structure is visualized as occurring through bundles of straight cylindrical capillaries. The diffusion in pores of different sizes occurs in parallel, whereas the pore orientation is assumed to be randomly distributed in space. For the char investigated in this work, the micropores have a narrow size distribution with a mean value of about 1.5 nm compared to a mean value of 500 nm for the macropores (see Table
EFFECTIVE DIFFUSION COEFFICIENTS IN COAL CHARS TABLE 2 Effective diffusion coefficients 10ⳮ6 (m2/s) for N2O and NO in char particles at 1079 K Bimodal Pore Mean Pore Volume Pore Volume Distribution Size, Distribution, Density Experimental Eq. 12 Eq. 13 Eq. 14 N2O NO
5.6 6.9
0.35 0.43
5.6 7.0
9.6 12.1
1). The effective diffusion coefficient may be calculated in a simplified way using the bimodal pore size distribution and the mean pore sizes for micro- and macropores respectively: Deff ⳱
1 hM s 1/DM Ⳮ 1/Dk,M Ⳮ
1 hm s 1/Dm Ⳮ 1/Dk,m
(13)
The values for pore volumes and mean pore radii are given in Table 1. The tortuosity factor is assumed to be 5 in both the micropores and the macropores. The result was an effective diffusion coefficient of 5.6 ⳯ 10ⳮ6 m2/s and 7.0 ⳯ 10ⳮ6 m2/s for N2O and NO, respectively (i.e., about 15 times the value calculated from a mean pore size and very close to the experimental value, see Table 2). In general, the pore volume distribution density can be used, and the effective diffusion coefficient for a given component in low concentration can be calculated by the expression Deff ⳱
qchar s
⬁
冮
0
(1/D Ⳮ 1/Dk)ⳮ1 v⬘(r)dr
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A number of other theoretical models for diffusion in porous materials are available, for example, the dusty gas model, Bethe network theory, and effective medium theory. In a recent comparison of the four models, it was found that the cross-linked pore model predicted the measured diffusion fluxes well solely on the basis of pore size distribution data and with a tortuosity factor of 3, which is the value derived theoretically for isotropic structures. For the three other models, there seems to be no method for estimating the model parameters without diffusion experiments [14]. These conclusions support the results from this work, that the cross-linked pore model is a convenient method for prediction of effective diffusion coefficients. The porous material Haugaard and Livbjerg [14] investigated was a commercial catalyst manufactured from a fine powder which was shaped into pellets and sintered. The resulting pore structure has a high porosity and a highly interconnected pore structure. In the case of burning char particles, the structure may not be isotropic, and this could be the reason for the disagreement between the measured value of the tortuosity factor and the theoretical value of 3. It should be noted that the reactivity of the char is almost an order of magnitude higher for N2O reduction than for NO reduction, so the experimental part covers a wide range of reactivities. As a consequence, the results provide a strong test of the theoretical models for diffusion. Also, the two different components have different diffusion coefficients. For bulk diffusion, the theoretical ratio between the diffusion coefficients is 0.78, and for Knudsen diffusion the ratio is 0.83. In our case, we have a combination of bulk and Knudsen diffusion, and the experimental value of the ratio should fall within this range, in good agreement with the measured ratio of 0.81.
(14)
The values of the effective diffusion coefficients calculated in this way are higher than the values obtained using the bimodal pore size distribution, but still within a factor of 2 compared to the experimental values (see Table 2). A reason for the overestimation could be that the porosity, and so the macropore volume of the char particles measured by mercury porosimetry, is not evenly distributed over the whole char particle. It is common practice to use s as a fitting parameter and calculate a value of the tortuosity factor by matching the experimental and the theoretical value of the effective diffusion coefficient. For the mean pore size model, this is not possible, because s must be above unity. When the pore size distribution is used, a tortuosity factor of about 5 and 9 can be estimated for the bimodal and the pore size distribution function cross-linked pore models, respectively.
Discussion It has been common practice in the combustion literature to calculate the intrinsic reactivities of char particles using the mean pore radius to calculate the effective diffusion coefficient and the effectiveness factor. For char particles with a wide pore size distribution, this method may result in too high intrinsic reactivities if pore diffusion effects are important. In our work, we found an underestimation of the effectiveness factor using the single pore method. Smith and Tyler [3] found that the reaction rate coefficient of a semianthracite char calculated by the use of a bimodal pore model was not significantly different from the result when a single mean pore size was used. In our case, the ratio of the effective diffusion coefficients for the bimodal and the single mean pore method is about 15, and in the regime of strong pore diffusion effects, the intrinsic activity
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COMBUSTION OF SOLID FUELS
Fig. 5. Calculated effectiveness factors for reaction between NO, N2O, and O2 and the char of the present study at 1300 K.
will be overestimated by a factor of about 4 if the single mean pore method is used. Obviously, the importance of using a detailed pore volume model in the regime of strong pore diffusion effects depends on the char type. In our work, there was a large pore volume in the macropores and a large BET surface and a small pore volume in the micropores, and in this case the single mean pore size model can give erroneous results. The experiments were performed with NO and N2O at a rather low temperature of 1079 K, and to give an idea about the influence of mass transfer also at higher temperatures and higher reactivities, Fig. 5 shows the value of the effectiveness factor versus particle size for the reaction between char and NO, N2O, and O2 at 1300 K. For char oxidation, the intrinsic reactivity and activation energy of bituminous coal char reported in literature [2,10] was used, and for NO and N2O the activation energy was from Ref. [4] for the same chars as in this study. At 1079 K, pore diffusion effects are negligible for particle sizes below 0.1 mm independent of the reactivity, and from 0.1 to about 1 mm pore diffusion becomes progressively more important (see Figs. 3 and 4). Calculations show that for char oxidation, particle sizes of 0.2 to 0.3 mm are in the regime of strong pore diffusion effects. At 1300 K, pore diffusion effects for char oxidation is seen even for the very small particles of 0.02 to 0.03 mm (20–30 lm). The influence of external mass transfer is not included in these figures, but a rough estimate indicates that when the particle effectiveness factor is about 0.5, the external mass transfer resistance is about 10% of the total resistance. For particle effectiveness factors below about 0.05, the reaction is essentially external mass transfer controlled. At 1300 K, this happens for particle sizes above 4, 2, and 0.7 mm for NO, N2O, and O2, respectively. Conclusions The effective diffusion coefficients for NO and N2O in char particles were estimated experimentally
by varying the particle size by two orders of magnitude. In this way, it was possible to find experimental values at temperatures close to combustion conditions. Different methods for theoretical calculation of the effective diffusion coefficient were compared. It was found that using a mean pore radius calculated from the BET surface and the total porosity did underestimate the effective diffusion coefficient by more than an order of magnitude. The coal char in this study had a wide pore size distribution, and the cross-linked pore model gave a good agreement between the measured and the theoretical value of the effective diffusion coefficient. The implications are that for a char with a wide pore size distribution, the cross-linked pore model is a good choice for a theoretical calculation of the effective diffusion coefficient. In the case of strong pore diffusion effects, the error in interpretation of experimental results using the mean pore radius could be a factor of 5 on the intrinsic rate constant. For an average coal char reacting with oxygen at 1300 K, this would be the case for particle sizes larger than about 50 lm.
Nomenclature C d D Deff Dk jD k kobs kg m M r re Re S T Sc u
molar concentration [mol/m3] particle diameter [m] molecular diffusion coefficient [m2/s] effective diffusion coefficient [m2/s] Knudsen diffusion coefficient [m2/s] mass transfer correlation factor intrinsic first order rate constant [m3 kgⳮ1 sⳮ1] observed first order rate constant [m3 kgⳮ1 sⳮ1] mass transfer coefficient [m/s] mass transfer correlation coefficient molecular weight [kg/mol] reaction rate [mol kgⳮ1 sⳮ1] or pore radius [m] mean pore radius [m] Reynolds number specific surface (BET) [m2/kg] temperature [K] Schmidt number superficial gas velocity [m/s]
Greek e fixed-bed porosity q density [kg/m3] l gas viscosity [Pa s] Thiele modulus h particle porosity g effectiveness factor s tortuosity factor t⬘(r) pore volume distribution density [m2/kg]
EFFECTIVE DIFFUSION COEFFICIENTS IN COAL CHARS
Subscripts char char particle gas gas property m micropores M macropores sand sand particle
5.
6. Acknowledgments 7. This project was carried out as part of the CHEC research program. The authors gratefully acknowledge financial support from ELSAM, ELKRAFT, the Danish Ministry of Energy, the Danish Technical Research Council, the Nordic Energy Research Program, the Technical University of Denmark, and ECSC project 7220-ED-087.
8. 9.
10. REFERENCES 11. 1. 2. 3. 4.
Aarna, I., and Suuberg, E. M., Fuel 76:475–491 (1997). Smith, I. W., Fuel 57:409–414 (1978). Smith, I. W., and Tyler, R. J., Fuel 51:312–321 (1972). Johnsson, J. E., Jensen, A., and Nielsen, J. S., “Kinetics of Heterogeneous NO and N2O Reduction at FBC Conditions,” paper FBC99-0099, in Proceedings of the Fifteenth International Conference on Fluidized Bed
12. 13. 14.
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Combustion (CD-ROM), Savannah, GA, May 16–19, 1999, American Society of Mechanical Engineers, New York, 1999. Prins, W., “Fluidized Bed Combustion of a Single Carbon Particle,” Ph.D. thesis, University of Twente, Enschede, The Netherlands, 1987. Levenspiel, O., Chemical Reaction Engineering, 2nd ed., John Wiley & Sons, New York, 1972, p. 481. Froment, G. F., and Bischoff, K. B., Chemical Reactor Analysis and Design, 2nd ed., John Wiley & Sons, New York, 1990, p. 159. Matos, M. A. A., Pereira, F. J. M., and Ventura, J. M. P., Fuel 69:1435–1439 (1990). Schuler, J., Baumann, H., and Klein J., Erdo¨l und Kohle, Erdgas, Petrochemie 70:296–299 (1988). Valix, M. G., Harris, D. J., Smith, I. W., and Trimm, D., Proc. Combust. Inst. 24:1217–1223 (1994). Satterfield, C. N., Mass Transfer in Heterogeneous Catalysis, MIT Press, Cambridge, MA, 1970. Feng, C., and Stewart, W. E., Ind. Eng. Chem. Fund. 12(2):143–147 (1973). Johnson, M. F. L., and Stewart, W. E., J. Catal. 4:248– 252 (1965). Haugaard, J., and Livbjerg, H., Chem. Eng. Sci. 53(16):2941–2948 (1998).
EFFECTIVE DIFFUSION COEFFICIENTS IN COAL CHARS JAN E. JOHNSSON and ANKER JENSEN Technical University of Denmark Department of Chemical Engineering CHEC Research Centre DK-2800 Lyngby, Denmark
Knowledge of effective diffusion coefficients in char particles is important when interpreting experimental reactivity measurements and modeling char combustion or NO and N2O reduction. In this work, NO and N2O reaction with a bituminous coal char was studied in a fixed-bed quartz glass reactor. Particle sizes in the range 0.05–5 mm were tested, and the effective diffusion coefficients were estimated from measured effectiveness factors using the Thiele modulus. At 1079 K the effective diffusion coefficients were 5.5 ⳯ 10ⳮ6 m2/s and 6.8 ⳯ 10 ⳮ6 m2/s for N2O and NO, respectively. The experimental results were compared with theoretical values calculated from the mean pore radius and the cross-linked pore model. The method of mean pore radius underestimated the effective diffusion coefficient more than an order of magnitude. Using the cross-linked pore model, the bimodal pore size distribution, and a tortuosity factor of 5, a complete agreement between the experimental and the theoretical value was found. The conclusion was that for a char with a wide pore size distribution, the cross-linked pore model is a good choice for a theoretical calculation of the effective diffusion coefficient. In the case of strong pore diffusion limitations, the error in the interpretation of experimental results using the mean pore radius could be a factor of 5 on the intrinsic rate constant. For an average coal char reacting with oxygen at 1300 K, this would be the case for particle sizes larger than about 50 lm.
Introduction Mass transfer limitations may be important in the interpretation of experimental data for char reactivity in reactions with, for example, O2, N2O, and NO. In the comprehensive review by Aarna and Suuberg [1] on the nitric oxide–carbon reaction, it was concluded that there was a high probability that most of the reported data were obtained in or at least very near the intrinsic rate regime. However, very few experimental values for the effective diffusion coefficient in char particles have been reported in the literature, and in many cases the available data do not allow a theoretical estimation of it. The reaction between char and N2O and O2 are, respectively, roughly one and two orders of magnitude faster than the reaction with NO. For these reactions, mass transfer limitations may be important at combustion temperatures, except for the case of very small particles. In the review on char reactivity published by Smith [2], the intrinsic reactivity was calculated using an effectiveness factor and effective diffusion coefficient estimated from a mean pore radius. It was concluded, based on an earlier work [3], that using a more detailed bimodal pore model gave no significant differences in the calculated intrinsic reactivities. However, experimental values of effective diffusion coefficients at combustion temperatures are needed to verify the theoretical calculations.
The purpose of this work was to measure the effective diffusion coefficients of N2O and NO in porous char particles at temperatures of interest in fluidized-bed combustion, and to compare the experimental values with the results of theoretical calculations using data from physical characterization of the char particles obtained by CO2 adsorption and mercury porosimetry techniques. The results are used to evaluate the importance of mass transfer limitations at combustion conditions. Experimental Apparatus The experiments were carried out in a fixed-bed laboratory quartz reactor. It was mounted in an electrically heated oven with three independently controlled sections for good temperature control. In order to minimize homogeneous reactions and reactions catalyzed by the reactor surface, reactant gases were introduced separately through the main inlet in the bottom of the reactor and the secondary inlet in the top of the reactor and mixed just above the solids placed on a porous quartz plate. A gas of well-defined composition was mixed from pure gases or gas mixtures in a panel of precision mass flow controllers as described in Ref. [4]. The total gas flow rate was 30 cm3 sⳮ1 (273 K, 101.3 kPa). The reaction temperature was measured with a thermocouple in a quartz tube just below the porous quartz plate. The
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COMBUSTION OF SOLID FUELS TABLE 1 Char data Specific Surface (m2/g)
Porosity
Mean Pore Size (nm)
Particle Density (kg/m3)
150 4
0.05 0.46
1.5 500
— 920
BET (micropores) Mercury porosimetry (macropores)
Fig. 1. Observed rate constants for N2O and NO reduction over char particles of different size.
concentrations of N2O, NO, CO, and CO2 in the outlet gas and the bypass were measured with continuous analyzers and sampled by a PC, but only the N2O and NO measurements were applied in the data analysis. Chars The solids tested for the activity for N2O and NO reduction were obtained from a parametric study of the influence of operating conditions on the emissions from a 12 MW circulating fluidized bed (CFB) boiler at Chalmers University of Technology, Go¨teborg, Sweden. The fuel was a bituminous coal from Katowice, Poland. The chars were sampled from the dense bed of the CFB, and data for the chars are given in Table 1. Experimental Procedure A sample of 100–700 mg of char mixed with sand was placed on the porous quartz plate and inserted into the oven. The concentrations of N2O, NO, CO, and CO2 in the bypass were measured while the sample was heated to the reaction temperature. The decomposition of N2O in the empty reactor was tested at all experimental conditions and was found to be below 1% at 1130 K and about 7% at 1230 K at inert conditions. No decomposition of NO was observed. The experiments reported in this work
were carried out at a temperature of 1079 K, so no decomposition of N2O and NO occurred in the empty reactor. Observed rate constants have been calculated using a plug-flow reactor model. The layer of solids was a fixed bed 10–15 mm in height, and a separate test of the reactor showed that a plug-flow model is a valid assumption for the reaction conditions. The reduction of N2O and NO over char are gassolid reactions, and the char was consumed during an experiment. To avoid an initial char loss from reaction between air and char, the reactor was purged with N2 before the bottom section with the char was inserted into the oven. The char loss during an experiment was 1.3% as an average for all experiments. The relative standard deviation between duplicate experiments with new char samples from the same sieve fraction was below 10%. The results indicate that the chars from a given bed sample were rather uniform in activity. The fractional conversion was, within the experimental uncertainty, independent of the inlet concentrations of N2O and NO in the range from 50 to 450 ppmv, and a first-order reaction was assumed for both components. The measured activities of char, ash, sand, and bed material (sand Ⳮ ash Ⳮ partly sulfated limestone) have been reported elsewhere [4]. In this work, a special investigation of the effective diffusion coefficient in char particles is reported. Experimental Results The reduction of N2O and NO by char is a fast reaction at combustion conditions, and the reaction rate may be influenced by external mass transfer and pore diffusion for large particles. To investigate the influence of particle size in the size range of interest in fluidized-bed combustion, large char particles in the range 4.75–5.6 mm were crushed and sieved to narrow size fractions. Eight sieve fractions, the smallest 45/63 lm and the largest 4.75/5.6 mm, were tested at a temperature of 1079 K with inlet concentrations of 450 ppmv N2O and NO in nitrogen as described above. The observed rate constants for N2O and NO reduction versus mean particle size for the eight size fractions can be seen in Fig. 1. It is seen that the N2O reduction is influenced by mass transfer limitations for particle sizes above about
EFFECTIVE DIFFUSION COEFFICIENTS IN COAL CHARS
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The first-order observed rate constant was estimated and the relative influence of external mass transfer for spherical particles was calculated according to Ref. [6]: ⳮr ⳱ kobsC
(6)
1 1 d ⳱ Ⳮ char kobsqchar kgqchar 6kg
(7)
Relative external mass transfer resistance d /6k ⳱ char g 100% 1/kobs qchar Fig. 2. Relative influence of external mass transfer.
Fig. 3. Reduction of NO over char. Measured and calculated values of the effectiveness factor for different particle sizes using Deff ⳱ 6.9 ⳯ 10ⳮ6 m2/s.
100 lm, while for NO reduction the size limit is between 200 and 500 lm. The influence of external mass transfer and pore diffusion was estimated assuming spherical particles. The char particles were mixed with 0.3 mm sand particles, and so the external mass transfer coefficient was calculated according to Ref. [5] using a correlation valid for a fixed bed with inert and active particles of different size. 1.05
冢dd 冣 sand
ejDRem ⳱ 0.105 Ⳮ 1.505
(1)
char
0.5
m ⳱ 0.35 Ⳮ 0.29
冢dd 冣 sand
(2)
char
dsand uqgas l
(3)
Sc ⳱
l qgasD
(4)
kg ⳱
jDu Sc2/3
(5)
Re ⳱
(8)
The relative external mass transfer resistance is plotted versus particle size in Fig. 2. It is seen that for N2O reduction, external mass transfer resistance is dominating for particle sizes above 2–3 mm, while for NO reduction the influence of external mass transfer resistance is almost negligible for all particle sizes. For particle sizes below 0.2 mm, the external mass transfer resistance was negligible for both N2O and NO reduction at 1079 K. In Fig. 1, it can further be seen that the observed rate constant is independent of particle size for particles smaller than 0.2 mm, and so it is concluded that for this size there are no pore diffusion limitations either, and the effectiveness factor is unity. In this way, the intrinsic rate constant k was estimated directly from the experimental results. For particle sizes above 0.2 mm, the observed rate constants were corrected for external mass transfer resistance using equations 1–5 and 7, and gk was calculated for all particle sizes. The effectiveness factors for the larger particles can now be calculated as the ratio gk/k. The effective diffusion coefficient was estimated using an effectiveness factor calculation for spherical particles based on the Thiele modulus [7]:
⳱ g⳱
冢
1/2
冣
dchar kqchar 6 Deff
3 coth(3) ⳮ 1 32
(9) (10)
The value of the effective diffusion coefficient was estimated using a least squares method combined with a Newton iteration procedure. The result was 5.6 ⳯ 10ⳮ6 m2/s and 6.9 ⳯ 10ⳮ6 m2/s for N2O and NO, respectively. Figs. 3 and 4 show a good agreement between the effectiveness factors from experiment and the values calculated from the estimated effective diffusion coefficients using equations 9 and 10. This shows that the Thiele modulus approach is useful for estimation of an experimental value of the effective diffusion coefficient. For N2O, the two largest particle sizes were left out of the data treatment because of the estimated large influence of external mass transfer resistance in the experiments. However, it is seen that for N2O, even for the 1.0
2356
COMBUSTION OF SOLID FUELS
model, taking the pore size distribution into account, was used [12,13]. The binary bulk diffusion coefficients for N2O and NO in nitrogen at 1079 K were estimated to 1.25 ⳯ 10ⳮ4 m2/s and 1.60 ⳯ 10ⳮ4 m2/s, respectively, [11]. The Knudsen diffusion coefficient was then estimated from the BET surface measured by CO2 adsorption by the calculation of a mean pore radius [11]: Dk ⳱ 3.07 re 冪T/M ⳱ 3.07 Fig. 4. Reduction of N2O over char. Measured and calculated values of the effectiveness factor for different particle sizes using Deff ⳱ 5.6 ⳯ 10ⳮ6 m2/s.
and 1.7 mm particles, where the external mass transfer resistance is estimated to be 18% and 32% respectively, there is a deviation between the measured and the calculated effectiveness factor, indicating that the external mass transfer correction is inaccurate. For NO reduction, the relative external mass transfer resistance was below 15% for all particle sizes, and a better agreement is seen in the whole size range. There are only few experimental values in the literature to use for comparison. In Ref. [1], an effective diffusion coefficient for NO was derived from experimental results in Ref. [8]. The value was 1.4 ⳯ 10ⳮ6 m2/s for an anthracite char in the temperature range 900–1200 K, somewhat lower than the value found in this work. One reason for this difference may be that the porosity of the anthracite char was 0.29 compared to 0.51 for the bituminous coal char used in this work. Values of the effective diffusivity estimated from NO reduction experiments at lower temperatures (650–800 K) were very low, Deff ⳱ 4 ⳯ 10ⳮ9 m2/s, [9]. A transient pulse method was used in Ref. [10] at moderately elevated temperatures (300–550 K), and due to the relatively high activation energy of 26 kJ/mol, it was concluded that the mechanism was activated diffusion. The experimental values in Ref. [9] and Ref. [10] were well below the theoretical values calculated for Knudsen diffusion, and in both cases the mechanism may be activated diffusion as suggested in Ref. [10].
Theoretical Calculations To compare with the experimental values, the effective diffusion coefficients were calculated theoretically in two different ways. First, they were estimated from the BET surface and the porosity of the char particles using the methods of Satterfield [11], and then the more elaborate cross-linked pore
Deff ⳱
2h 冪T/M Sqchar
h 1 s 1/D Ⳮ 1/Dk
(11)
(12)
The tortuosity factor s is a parameter characteristic of the pore structure which corrects for the tortuous path through the pores due to the varying orientation and for deviations of the real pore structure from the ideal cylinder geometry. A theoretical value of s ⳱ 3 was derived for an ideal isotropic pore structure in Ref. [13]. For a porous char, the tortuosity may be larger, and in Ref. [9], a value of 5 was assumed. Using s ⳱ 5, a BET surface of 150 m2/g measured by CO2 adsorption and a porosity of 0.51 calculated as the sum of the porosities measured by BET and mercury porosimetry, the effective diffusion coefficient was estimated to 0.35 ⳯ 10ⳮ6 m2/s and 0.43 ⳯ 10ⳮ6 m2/s for N2O and NO, respectively, that is, an order of magnitude lower than the experimental value. The reason for this discrepancy is probably that the method of Satterfield [11] is valid for catalysts with a narrow pore size distribution, and in that case, a characteristic mean pore size can be calculated and used for estimation of the effective diffusion coefficient. In the case of chars from a combustion system, a wide pore size distribution with micropores and macropores can be expected. Another uncertainty in the calculation of the theoretical value is the tortuosity factor. However, it is difficult to imagine that the tortuosity factor can be below 3 to 5 in a porous char structure, and so it is not possible to get agreement between the theoretical and the measured value of the effective diffusion coefficient by choosing a lower value for s. To improve the theoretical approach, the effective diffusion coefficient was calculated using the pore size distribution measured by a combination of BET and mercury porosimetry and the cross-linked pore model [13]. In this model, diffusion in the pore structure is visualized as occurring through bundles of straight cylindrical capillaries. The diffusion in pores of different sizes occurs in parallel, whereas the pore orientation is assumed to be randomly distributed in space. For the char investigated in this work, the micropores have a narrow size distribution with a mean value of about 1.5 nm compared to a mean value of 500 nm for the macropores (see Table
EFFECTIVE DIFFUSION COEFFICIENTS IN COAL CHARS TABLE 2 Effective diffusion coefficients 10ⳮ6 (m2/s) for N2O and NO in char particles at 1079 K Bimodal Pore Mean Pore Volume Pore Volume Distribution Size, Distribution, Density Experimental Eq. 12 Eq. 13 Eq. 14 N2O NO
5.6 6.9
0.35 0.43
5.6 7.0
9.6 12.1
1). The effective diffusion coefficient may be calculated in a simplified way using the bimodal pore size distribution and the mean pore sizes for micro- and macropores respectively: Deff ⳱
1 hM s 1/DM Ⳮ 1/Dk,M Ⳮ
1 hm s 1/Dm Ⳮ 1/Dk,m
(13)
The values for pore volumes and mean pore radii are given in Table 1. The tortuosity factor is assumed to be 5 in both the micropores and the macropores. The result was an effective diffusion coefficient of 5.6 ⳯ 10ⳮ6 m2/s and 7.0 ⳯ 10ⳮ6 m2/s for N2O and NO, respectively (i.e., about 15 times the value calculated from a mean pore size and very close to the experimental value, see Table 2). In general, the pore volume distribution density can be used, and the effective diffusion coefficient for a given component in low concentration can be calculated by the expression Deff ⳱
qchar s
⬁
冮
0
(1/D Ⳮ 1/Dk)ⳮ1 v⬘(r)dr
2357
A number of other theoretical models for diffusion in porous materials are available, for example, the dusty gas model, Bethe network theory, and effective medium theory. In a recent comparison of the four models, it was found that the cross-linked pore model predicted the measured diffusion fluxes well solely on the basis of pore size distribution data and with a tortuosity factor of 3, which is the value derived theoretically for isotropic structures. For the three other models, there seems to be no method for estimating the model parameters without diffusion experiments [14]. These conclusions support the results from this work, that the cross-linked pore model is a convenient method for prediction of effective diffusion coefficients. The porous material Haugaard and Livbjerg [14] investigated was a commercial catalyst manufactured from a fine powder which was shaped into pellets and sintered. The resulting pore structure has a high porosity and a highly interconnected pore structure. In the case of burning char particles, the structure may not be isotropic, and this could be the reason for the disagreement between the measured value of the tortuosity factor and the theoretical value of 3. It should be noted that the reactivity of the char is almost an order of magnitude higher for N2O reduction than for NO reduction, so the experimental part covers a wide range of reactivities. As a consequence, the results provide a strong test of the theoretical models for diffusion. Also, the two different components have different diffusion coefficients. For bulk diffusion, the theoretical ratio between the diffusion coefficients is 0.78, and for Knudsen diffusion the ratio is 0.83. In our case, we have a combination of bulk and Knudsen diffusion, and the experimental value of the ratio should fall within this range, in good agreement with the measured ratio of 0.81.
(14)
The values of the effective diffusion coefficients calculated in this way are higher than the values obtained using the bimodal pore size distribution, but still within a factor of 2 compared to the experimental values (see Table 2). A reason for the overestimation could be that the porosity, and so the macropore volume of the char particles measured by mercury porosimetry, is not evenly distributed over the whole char particle. It is common practice to use s as a fitting parameter and calculate a value of the tortuosity factor by matching the experimental and the theoretical value of the effective diffusion coefficient. For the mean pore size model, this is not possible, because s must be above unity. When the pore size distribution is used, a tortuosity factor of about 5 and 9 can be estimated for the bimodal and the pore size distribution function cross-linked pore models, respectively.
Discussion It has been common practice in the combustion literature to calculate the intrinsic reactivities of char particles using the mean pore radius to calculate the effective diffusion coefficient and the effectiveness factor. For char particles with a wide pore size distribution, this method may result in too high intrinsic reactivities if pore diffusion effects are important. In our work, we found an underestimation of the effectiveness factor using the single pore method. Smith and Tyler [3] found that the reaction rate coefficient of a semianthracite char calculated by the use of a bimodal pore model was not significantly different from the result when a single mean pore size was used. In our case, the ratio of the effective diffusion coefficients for the bimodal and the single mean pore method is about 15, and in the regime of strong pore diffusion effects, the intrinsic activity
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COMBUSTION OF SOLID FUELS
Fig. 5. Calculated effectiveness factors for reaction between NO, N2O, and O2 and the char of the present study at 1300 K.
will be overestimated by a factor of about 4 if the single mean pore method is used. Obviously, the importance of using a detailed pore volume model in the regime of strong pore diffusion effects depends on the char type. In our work, there was a large pore volume in the macropores and a large BET surface and a small pore volume in the micropores, and in this case the single mean pore size model can give erroneous results. The experiments were performed with NO and N2O at a rather low temperature of 1079 K, and to give an idea about the influence of mass transfer also at higher temperatures and higher reactivities, Fig. 5 shows the value of the effectiveness factor versus particle size for the reaction between char and NO, N2O, and O2 at 1300 K. For char oxidation, the intrinsic reactivity and activation energy of bituminous coal char reported in literature [2,10] was used, and for NO and N2O the activation energy was from Ref. [4] for the same chars as in this study. At 1079 K, pore diffusion effects are negligible for particle sizes below 0.1 mm independent of the reactivity, and from 0.1 to about 1 mm pore diffusion becomes progressively more important (see Figs. 3 and 4). Calculations show that for char oxidation, particle sizes of 0.2 to 0.3 mm are in the regime of strong pore diffusion effects. At 1300 K, pore diffusion effects for char oxidation is seen even for the very small particles of 0.02 to 0.03 mm (20–30 lm). The influence of external mass transfer is not included in these figures, but a rough estimate indicates that when the particle effectiveness factor is about 0.5, the external mass transfer resistance is about 10% of the total resistance. For particle effectiveness factors below about 0.05, the reaction is essentially external mass transfer controlled. At 1300 K, this happens for particle sizes above 4, 2, and 0.7 mm for NO, N2O, and O2, respectively. Conclusions The effective diffusion coefficients for NO and N2O in char particles were estimated experimentally
by varying the particle size by two orders of magnitude. In this way, it was possible to find experimental values at temperatures close to combustion conditions. Different methods for theoretical calculation of the effective diffusion coefficient were compared. It was found that using a mean pore radius calculated from the BET surface and the total porosity did underestimate the effective diffusion coefficient by more than an order of magnitude. The coal char in this study had a wide pore size distribution, and the cross-linked pore model gave a good agreement between the measured and the theoretical value of the effective diffusion coefficient. The implications are that for a char with a wide pore size distribution, the cross-linked pore model is a good choice for a theoretical calculation of the effective diffusion coefficient. In the case of strong pore diffusion effects, the error in interpretation of experimental results using the mean pore radius could be a factor of 5 on the intrinsic rate constant. For an average coal char reacting with oxygen at 1300 K, this would be the case for particle sizes larger than about 50 lm.
Nomenclature C d D Deff Dk jD k kobs kg m M r re Re S T Sc u
molar concentration [mol/m3] particle diameter [m] molecular diffusion coefficient [m2/s] effective diffusion coefficient [m2/s] Knudsen diffusion coefficient [m2/s] mass transfer correlation factor intrinsic first order rate constant [m3 kgⳮ1 sⳮ1] observed first order rate constant [m3 kgⳮ1 sⳮ1] mass transfer coefficient [m/s] mass transfer correlation coefficient molecular weight [kg/mol] reaction rate [mol kgⳮ1 sⳮ1] or pore radius [m] mean pore radius [m] Reynolds number specific surface (BET) [m2/kg] temperature [K] Schmidt number superficial gas velocity [m/s]
Greek e fixed-bed porosity q density [kg/m3] l gas viscosity [Pa s] Thiele modulus h particle porosity g effectiveness factor s tortuosity factor t⬘(r) pore volume distribution density [m2/kg]
EFFECTIVE DIFFUSION COEFFICIENTS IN COAL CHARS
Subscripts char char particle gas gas property m micropores M macropores sand sand particle
5.
6. Acknowledgments 7. This project was carried out as part of the CHEC research program. The authors gratefully acknowledge financial support from ELSAM, ELKRAFT, the Danish Ministry of Energy, the Danish Technical Research Council, the Nordic Energy Research Program, the Technical University of Denmark, and ECSC project 7220-ED-087.
8. 9.
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