Secondary fragmentation of char particles during combustion in a fluidized bed

Secondary fragmentation of char particles during combustion in a fluidized bed

C O M B U S T I O N A N D F L A M E 77: 79-90 (1989) 79 Secondary Fragmentation of Char Particles During Combustion In a Fluidized Bed R. CHIRONE, P...

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C O M B U S T I O N A N D F L A M E 77: 79-90 (1989)

79

Secondary Fragmentation of Char Particles During Combustion In a Fluidized Bed R. CHIRONE, P. SALATINO, and L. MASSIMILLA Dipartimento di Ingegneria Chimica, Universitb Istituto di Ricerche sulla Combustione, C.N.R. P.le V. Tecchio, 80125 Napoli, Italy

Batchwise fluidized bed experiments have studied secondary fragmentation of char particles during combustion. Two chars, one from a nonswelling and the other from a swelling coal, have been tested for different particle sizes and oxygen concentrations in inlet gas. Statistical functions expressing the probability of particle fragmentation and size distribution of fragments have been determined, Model calculations show that secondary fragmentation of chars significantly influences the performance of fluidized bed combustors only if it occurs in early stage of combustion and if the fraction of particles subject to breakage is above 75 %.

NOMENCLATURE a d*

db df dmax do Ec F(dO f,(df/db) F0 fw(de/db) k, ka m N No

bed carbon average Sauter diameter size of particles whose terminal velocity equals gas fluidizing velocity or minimum size of fragments produced by secondary fragmentation size of fragmenting particle fragment size maximum particle size in the feed nominal size of feed char particles carbon elutriation rate probability density that a particle of size db breaks into fragments fragment size distribution on numerical basis under similarity hypothesis carbon feed rate fragment size distribution on mass basis under similarity hypothesis attrition rate constants mass flux of particles which shrink across db number of char particles in the bed number of char particles initially fed to the combustor

Copyright © 1989 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010

n(ctf, db)

probability density on numerical basis of size df of fragments generated by the breaking of a particle of size db bed carbon loading wo w(df, db) probability density on mass basis of size df of fragments generated by the breaking of a particle of size db p(d) particle size distribution on mass basis of carbon in the combustor at steady state po(d) particle size distribution on mass basis of feed carbon particle shrinkage rate due to both R(db) combustion and attrition t combustion time tb0 burnout time U fluidizing gas velocity minimum fluidizing gas velocity Ume X unknown function in Eq. A9

Greek

7?

dimensionless particle size carbon combustion efficiency

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R. CHIRONE et al

INTRODUCTION A conceptual representation of particle size reduction in a fluidized bed coal combustor is reproduced in Fig. 1 [1]. Various phenomena are considered: primary fragmentation of coal due to thermal stresses and to stresses arising during devolatilization and buildup of pressure of volatiles in the pore network [2]; char combustion; secondary fragmentation, which breaks char due to the combined effects of collisions and of combustion burning out the bridges connecting the pieces of which each particle is made; char attrition, which generates by abrasion fines of size below d*, i.e., the size of particles whose terminal velocity equals the gas fluidizing velocity [3, 4]; and fragmentation by percolation, which breaks unburned char residues into a number of fines before they reach elutriable size d*. Char breakage observed by Campbell and Davidson [5] in their study on combustion of carbon in a fluidized bed, and analyzed by Essenhigh et al. [6], can be attributed to secondary fragmentation. The overall (primary plus secondary) fragmentation of nondevolatilized Kentucky No. 9 coal has been investigated by Sundback [7] and Sundback et al. [8] by means of a technique that assumes that spikes in CO2 concentration profiles at the combustor outlet are representative of fragmentation acts. A percolative model has been recently applied to secondary fragmentation by Kerstein and Edwards [9]. Finally, Salatino

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[10] has shown that secondary fragmentation can be modeled by introducing into the population balance equation of a fluidized combustor [11, 12] additional terms that account for carbon jumping from one class of particle size to another class, not contiguous (Appendix). This work is directed to compare secondary fragmentation behaviors of chars from two bituminous coals, one nonswelling (Snibston coal), the other with a relatively high swelling index (Kentucky No. 9 coal). These materials, denominated as Snibston char and Kentucky No. 9 char, are tested by means of the basket technique [13-16]. The work also assesses the effect of secondary fragmentation on the performance of fluidized bed combustors. To this end, experimental data are worked out to evaluate the probabilities of particles of a given size fragmenting and to determine particle size distributions of generated fragments. Such relationships are embodied in population balance equations, properly modified to account for primary fragmentation and attrition [3, 17], and also for secondary fragmentation [10].

EXPERIMENTS Materials and Experimental Conditions Properties of Snibston and Kentucky No. 9 coals are reported in Table 1. Char particles were prepared by devolatilizing each coal in a fluidized bed under an inert atmosphere at 1173 K, the same temperature at which fragmentation experiments have been performed. The bed was made of 0.30.4-mm silica sand, whose minimum fluidizing velocity at bed temperature is 0.02 m/s. Experiments with the Snibston char were with 21% and 5% oxygen concentration in the gas at the combustor inlet. Feed particles were in the size ranges of 6.35--4.76 mm and 4.0-3.0 ram. The Kentucky No. 9 char was tested with 5 % 02 in the fluidizing gas. Only particles in the range between 6.35 and 4.76 mm were used in this case. Sizes of 6.35-4.76 mm and 4.0-3.0 mm are indicated as 5 and 3 mm nominal feed size do, in the following. The fluidizing velocity was 0.4 m/s at bed temperature in all runs.

S E C O N D A R Y F R A G M E N T A T I O N IN A F L U I D I Z E D BED

TABLE 1

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Properties of the Coals Whose Chars Have Been Tested Kentucky No. 9

81

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Snibston

l;roximate analysis (as received) Moisture Ash Volatile matter Fixed carbon

0.033 0.067 0.415 0.485

0.146 0.040 0.352 0.462

0.783 0.056 0.017 0.033 0.111

0.813 0.052 0.016 0.010 0.109

L~ltimateanalysis (dry, ash free) Carbon Hydrogen Nitrogen Sulphur Oxygen

(;ross heating value (kcal/kg), as received 7410

7900

Free swelling index, ASTM D720 2.5

Experimental Apparatus and Technique The experimental apparatus (Fig. 2) consists of a 40-mm i.d. quartz combustor equipped with a basket made of a 0.6 m m mesh to allow collection ~f char from the bed. Details of the design and 3peration are given elsewhere [2]. The combustor is operated by means of the basket holder (1). It is moved up and down along the guide (2) and pivoted around the axis (8) in order to locate the Oasket in the bed and collect material of size :oarser than mesh openings. Single char particles are injected into the bed and left there for time intervals of about 30 s. their combustion is stopped by switching fluidizing gas from an oxygen-nitrogen mixture to nitrogen. Char in the bed at that time is collected with the basket. Nitrogen issuing from a channel between the basket holder and an outer tubing quenches the char and prevents its further burning in the atmosphere. Each particle generated by a fragmentation act and collected from the bed is subsequently tested for further secondary fragmentation in separate runs. With this procedure it is possible to see whether

Fig. 2. Fluidized bed combustor used in fragmentation experiments. A. The reactor: (1) basket holder; (2) basket holder guide; (3) air preheater; (4) preheater electric furnace; (5, 6) electric furnaces for fluidization column; (7) quartz tube; (8) spherical pivot for basket holder; (9) stack. B. Basket, front and top views. the particle breaks and weigh and size each char particle collected from the bed at fixed times during char combustion. A drawback of the basket technique is that only particles coarser than 1 m m are retrieved, due to the possible loss of carbon fragments through the mesh and through the spacing between the combustor wall and the basket when pulling it out. RESULTS

Evaluation of Fragmentation Functions Using the basket technique the combustion-secondary fragmentation histories of individual char particles were followed. Typical patterns are illustrated in Figs. 3A and 3B, for Snibston and Kentucky No. 9 chars, respectively. Solid lines represent continuous particle shrinkage due to combustion and attrition. Dotted lines are repre-

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Fig. 4. Numberof char particles actually in the bed over the number of pai'ticles initially charged N / N o as a function of dimensionless time t/tbo for the two chars tested in this work (@, &) and for the coal tested by Sundback [7] (A). T 1173 K; U = 0.4 m/s; 02% = 5; sand 0.3-0.4 ram; do = ram.

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Fig. 3. Typical fragmentation paths of Snibston (A) and KentuckyNo. 9 (B) char particles. T = 1173K; U = 0.4 m/s; 02% = 5; sand 0.3-0.4 ram. sentative of the jump of particle size due to breaking. Fragments may themselves eventually undergo further secondary fragmentation. Some particles do not fragment at all during the whole of the burnout. The secondary fragmentation pattern shown in Fig. 3 changes from one particle to another, even in the case of the same parent coal under identical test conditions.

The average ratio N / N o of total number of particles generated by one char particle through one or more fragmentation acts is reported in Fig. 4 as a function of the ratio t/tbo of the combustion time t to the burnout time tb0. For the Kentucky No. 9 char, present values of N / N o are compared with those evaluated by Sundback [7]. The latter are increasingly higher the higher is t/tbo. Two reasons are given to explain such a discrepancy. First, the data of Sundback were obtained with nondevolatilized particles, so that N / N o could be somewhat increased by the contribution of primary fragmentation. Second, the basket does not collect fragments below 1 mm, which instead could be taken into account in Sundback's work by fitting the CO2 concentration in the outlet gas. In any case, whatever the experimental technique used, Fig. 4 suggests that Kentucky No. 9 char is subject to more secondary fragmentation than is Snibston char. As shown in the Appendix, quantitative treatment of secondary fragmentation has used two statistical functions [10, 18]: the first, F(db), is the probability density that a particle of size db

83

~ECONDARY FRAGMENTATION IN A FLUIDIZED BED breaks into fragments; the second, n(df, db), gives the size distribution on a numerical basis of fragments of size dr, generated by the breaking up of a particle of size db. This latter may be equivalently expressed on a mass basis as w(df, "~tb). These functions are constitutive relationships of the char, valid for a given temperature, fluidization velocity, and bed solids size and density. A simplification is achieved if one can assume that size distributions of the fragments from particles of different sizes db can be represented by a single distribution function f , (df/db) in terms of the ratio of fragments size de to the parent particle size db (similarity hypothesis). Values of F(db) obtained for Snibston char particles of 5 mm nominal size, with 5 % and 21% 02 in the fluidizing gas, are reported in Fig. 5. F(db) for 3 mm nominal size particles of the same char with 5% O2 in the fluidizing gas are also given in Fig. 5. The largest probability of particle secondary fragmentation lies in the size range between 2 and 3.5 mm. None of the 32 original particles of 5 mm nominal size tested showed secondary fragmentation for sizes larger than 4 mm. Values of F(db) from fragmentation data

obtained with the Kentucky No. 9 char and those evaluated from Sundback data are compared in Fig. 6. Both sets of data confirm that secondary fragmentation is wider for the Kentucky No. 9 than for Snibston char. The size distribution of fragments n(df, db) is presented in Fig. 7 for the case of Snibston char of 5 mm nominal size particles burned with 21% 02 in the inlet gas. According to the definitions of n(df, db), n(de, db)Adf represents the number of fragments of sizes contained in the interval [de, de + Adf], generated by secondary fragmentation of one particle of size db. Similar results are found for different oxygen concentration (5 %) and 3 mm nominal size particles. Values of n(df, db) for Kentucky No. 9 char with 5 % 02 are given in Fig. 8. Values off,(df/db) for Snibston char are given in Fig. 9 together with the best fit curve through these points. The regression relationship is of the formf~(df/db) = a(df/db)m(1 df/db) n, satisfying the conditions thatf,(de/db) = 0 for df/db = 1 and df/db = 0. This latter is a reasonable condition because d* < db. Within this and other approximations, Fig. 9 validates the hypothesis of similarity of size distributions of fragments re-

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ab,~ Fig. 5. Probability o f particle b r e a k i n g F versus particle size db for the Snibston c h a r . T = 1173 K; sand 0 . 3 - 0 . 4 ram; --,21% O2;d0 = 5mm; .... ,5% O2;d0 = 5mm; ..... • - , 5% 02; do = 3 m m .

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, data f r o m S u n d b a c k [7].

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gardless of whether or not the fragmenting particle passed through a previous secondary fragmentation act. Data scattering indicates, however, that the significance of the statistical sample is rather limited, in spite of the fact that more than 100 I 4 2

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Fig, 8. Fragments size distribution n(df, rib) related to different values of size ranges of db for Kentucky No. 9 char, T = I 1 7 3 K ; U = 0 . 4 m / s ; O 2 % = 21; sand 0 . 3 - 0 . 4 mm; d0 = 5 mrrl.

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particles have been followed separately during their burnoff and a relatively small grid width of 0.5 mm has been chosen for de and db. The most probable value of dt/db is around 0.8. Figure 9 further confirms that the secondary fragmentation behavior of the char tested is affected neither by the oxygen concentration in the inlet gas nor by the nominal particle size do. The size distribution fn(df/db) for Kentucky No. 9 char is given in Fig. 10. In this case values offn(dt/db) at a given df/db have been averaged to give an arithmetic mean. Then, assuming similarity-hypothesis, these averaged values have been correlated to givef,(df/db) as a function of df/db. Again data obtained with this experimental technique and in the work of Sundback are compared. The comparison between curves in Figs. 9 and 10 suggests that the secondary fragmentation behaviors of Snibston and Kentucky No. 9 chars are not so different with regard to fragment size distributions n(df, du) as they are in respect to the probability of particle fragmentation F(db). Model Calculations

Secondary fragmentation data collected in batchwise combustion of Snibston and Kentucky No. 9

SECONDARY FRAGMENTATION IN A FLUIDIZED BED

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constants from Cammarota et al. [19]. Attrition of Snibston char was directly investigated in such work. Kentucky No. 9 char, on the contrary, was not subjected to attrition tests. The rate constant taken for this material is that found for char from the Marine coal of high swelling index. 5. A similarity hypothesis of fragment size distributions has been assumed. 6. The bed solids are 0.3-0.4 mm sand. A fluidizing velocity of 0.4 m/s is assumed, for which F(db) and fw (df/db) have been obtained in this and in previous [7, 8] work.

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Fig, 10. Fragments size distribution f , ( d f / d b ) for Kentucky No. 9 char. T = 1173 K; U = 0.4 m/s; 0 2 % = 5; sand 0 . 3 0.4 mm; do = 5 nun. - - , present work; . . . . , data from Sundback [711.

chars have been used to simulate the behavior of a continuously operated combustor charged with each of such chars. The model hypotheses are as follows: 1. Monosize (or Gaussian size distributed) particles of 5 and 3 mm nominal diameters are used as charge. Material formed by secondary fragmentation has a size coarser than d*. 2. There is perfect mixing of solid and gas phases, with 5% O2 concentration in the bed (by volume). Overflow of bed material is not considered. However, calculated bed carbon loading per unit carbon mass feed rate may give indications about carbon loss in the overflow, should this be required by the bed solids balance. Carbon fines entrained in the carryover (attrited material and unburned residues) have a size finer than d*. Postcombustion of entrained carbon in the freeboard is neglected. 3. The shrinking particle combustion model applies. Combustion is controlled by external diffusion at the bed temperature of 1173 K. A particle Sherwood number of 2 is assumed. 4. Attrition is taken into account using rate

Model equations are formulated in the Appendix. Details of the numerical solution technique may be found in Salatino [10]. Calculated steady-state size distributions of carbon particles in the bed, allowing for secondary fragmentation, are shown in Figs. 11 and 12 for Snibston and Kentucky No. 9 chars, respectively. Values of F(db) and fw(df/db) from the data obtained with the basket technique have been used in these calculations. Curves are compared with particle size distributions calculated for each char, neglecting secondary fragmentation. In particular, for Kentucky No. 9 char F(db) and fw(df/db) from data obtained fitting CO2 concentrations [7] have also been used to calculate bed particle size

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Fig. 11. Calculated particle size distributions of carbon in the bed for Snibston char. F(db) andfw(df/db) from data obtained with the basket technique. Gaussian distributed feed (mean = 5 mm; standard deviation = 0.5 ram). - . . . . . . . . , feed particle size distribution; - - , particle size distribution considering secondary fragmentation; . . . . . , particle size distribution neglecting this phenomenon.

86

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d [mm] Fig. 12. Calculated particle size distributions of carbon in the bed for Kentucky No. 9 char. F(db) andfw(df/db) from data obtained with the basket technique. Gaussian distributed feed (mean = 5 ram; standard deviation = 0.5 ram). - . . . . . . . . , feed particle size distribution; , particle size distribution considering secondary fragmentation; . . . . . , particle size distribution neglecting this phenomenon.

distribution, but this did not greatly change the solid line in Fig. 12. Calculated mean residence times of carbon in the bed Wc/Fo, bed carbon average Sauter diameters a, and loss of efficiencies 1 - 7/are given in Tables 2 and 3 and are compared with values calculated neglecting secondary fragmentation. TABLE

The loss of efficiency, 1 - ~/, is calculated as the ratio of the carbon elutriation rate Ec over the carbon feed rate F0. In spite of the discrepancies between F(db) and fw(df/db) from the data obtained fitting CO2 concentration [7] and with the basket technique experiments (this work), the results of the calculations for the Kentucky No. 9 char for the two sets of data are close to each other. Comparison of these calculations with those assuming negligible secondary fragmentation indicates that such a phenomenon positively affects the performance of the combustor with this char when the feed particle nominal size is 5 mm. There is in fact a reduction in the loss of elutriable fines per unit char feed rate Ec/Fo of about 10%. With a feed nominal size of 3 mm such reduction is below 2 %. Similar evaluations can be made as regards possible carbon loss with the overflow. There is in fact a reduction in the mean residence time of carbon in the bed, Wc/ F0, of about 20% for Kentucky No. 9 char when secondary fragmentation is taken into account in the calculations. Improvements of combustion efficiency due to secondary fragmentation of the Snibston char are negligible. Values of Ec/Fo calculated with and without secondary fragmentation are within 2 %. 2

Results of Model Calculations (5 mm feed particle nominal size) With secondary fragmentation: F(db) andfw(df/db) from data of

Without secondary fragmentation

This Work

1160-1190 a 4.01-4.00 0.0088-0.0091

1150-1170 4.01-3.99 0.0087-0.0089

1100-1130 4.00-3.99 0.053-0.054

901-913 3.69-3.66 0.046-0.047

Sundback [7]

Snibston char

Wc/Fo (s) d (rnrn)

E¢/F~

m

Kentucky No. 9 char

Wc/Fo (s) a (nun) Ec/F~

921-947 3.77-3.77 0.046-0.048

° Here and in the following the first value is relative to Gaussian distribution of feed particle sizes, the second to monosize particles.

SECONDARY FRAGMENTATION IN A FLUIDIZED BED

87

TABLE 3 Results of Model Calculations (3 mm feed particle nominal size) With secondary fragmentation: F(db) and fw(df/db) from data of

Without secondary fragmentation

This Work

424~31Q 2.44-2.40 0.0053-0.0055

401-401 2.40-2.34 0.0051-0.0052

411418 2.43-2.40 0.032-0.033

399-418 2.40-2.40 0.032-0.033

Sundback [7]

Snibston char

l'Vc/Fo (s) d (mm) Ec/Fc

m

Kentucky No. 9 char

W~/Fo (s) d (mm) E~/F~

383-393 2.37-2.35 0.031-0.032

Q Here and in the following the first value is relative to Gaussian distribution of feed particle sizes, the second to monosize particles,

Calculations indicate that altogether the effects of secondary fragmentation are small on the basis of the data collected under the given experimental conditions. They are, however, suggestive of the relevance that such phenomena might have for chars with gas fluidizing velocities, more realistically, higher than 0.4 m/s and with more friable chars. CONCLUSIONS Chars from both coals tested show secondary fragmentation. Kentucky No. 9 char is affected by particle breakup more than Snibston char. Within the range of variables tested, both oxygen concentration and initial particle size exert only a negligible influence upon the fragmentation patterns. Some discrepancies appear when comparing fragmentation data obtained with the Kentucky No. 9 char using the basket technique (this work) with those reported in the literature for the same char but using a technique based on analysis of CO2 concentration at the combustor out2et [7]. Size distributions of fragments from particles of different sizes can be represented by a single distribution function in terms of the ratio of the

fragment to the parent particle size regardless of whether this is the original particle charged into the combustor or the result of previous secondary fragmentation. This greatly simplifies the formulation of the particle population balance equation required in combustor design. Model calculations of performances of combustors directed to establishing to what extent secondary fragmentation affects combustion behavior are presented. Comparison of results of calculations with or without secondary fragmentation is made in terms of bed carbon particle size distributions, mean residence times of carbon in the bed and loss of efficiencies due to elutriation of carbon fines generated by attrition. It appears that the effects of secondary fragmentation are negligible for Snibston char. They are significant in the case of Kentucky No. 9 char when the feed particle size is of 5 ram, and may become relevant for the same material when more vigorously fluidized and for other chars showing a stronger tendency to fall apart during combustion.

The research was within the framework o f the Progetto Finalizzato Energetica of CNR-ENEA, Rome. The authors are indebted to ing. R.

88

A m i r a n d a a n d Mr. M . E l d e r m a n ( I A E S T E visitor f r o m University o f Delft, N L ) , w h o helped in p e r f o r m i n g experimental work.

R. C H I R O N E et al.

16.

REFERENCES 17. 1. Chirone, R., D'Amore, M., Massimilla, L., and Salatino, P., The efficiency of fluidized combustion of carbons of various characteristics. Engineering Foundation Conference, Combustion of Tomorrow's Fuels-H, Davos, Switzerland, 1984. 2. Chiroue, R., Cammarota, A., D'Amore, M., and MassimiUa, L., Fragmentation and attrition in the fluidized combustion of a coal. Proceedings of the Sixth International Conference on Fluidized Bed Combustion, Washington D.C., U.S. Department of Energy, 1982, p. 1023. 3. Arena, U., D'Amore, M., and Massimilla, L., AIChE J. 29:40 (1983). 4. Dousi, G., Massimilla, L., and Miecio, M., Combust. Flame 41:57 (1981). 5. Campbell, E. K., and Davidson, J. F., The combustion of coal in fluidized beds. Inst. Fuel Syrup. Set. No. 1 A2-1, London, 1975. 6. Essenlfigh, R. H., Basak, A., Shaw, D. W., and Gaugaram, G., Effect of back mixing on the particle size distribution in a perfectly stirred reactor. 1986 Spring Technical Meeting, Central States Section: Combustion Institute, Cleveland, OH, 1986. 7. Sundback, C. A., Fragmentation behavior of single coal particles in a fluidized bed. Ph.D. thesis, M.I.T., Cambridge, MA, 1984. 8. Sundback, C. A., Be~r, J. M., and Sarofim, A. F., Fragmentation behavior of single coal particles in a fluidized bed. Proceedings of the Twentieth Symposium (International) on Combustion, Pittsburgh, The Combustion Institute, 1984, p. 1495. 9. Kerstein, A. R., and Edwards, B. F., Chem. Eng. Sci. 46:1629 (1987). 10. Salatino, P., in Proceedings of the XVIII Conference on the Use of Computers in Chemical EngineeringCEF, Taormina, Italy, 1987, p. 249. 11. Hulburt, H. M., and Katz, S., Chem. Eng. Sci. 19:555 (1964). 12. Levenspiel, O., Kunii, D., and Fitzgerald, T., Powder TechnoL 2:87 (1969). 13. Andrei, M. A., Time resolved burnout in the combustion of coal particles in a fluidized bed. Sc.M. thesis, M.I.T., Cambridge, MA, 1978. 14. Andrei, M. A., Sarofim, A. F., and Be~r, J. M., Combust. Flame 61:17 (1985). 15. Chirone, R., La frammentazione del carbone nei corso della combustione in letto fluido. Thesis in Chemical Engineering, Universith di Napoli, Napoli, 1980; Frammentazione primaria di carboue durante la combustione

18. 19.

in letto fluido. Doctoral thesis in Chemical Engineering, Universit~ di Napoli, Napoli, 1987. La Nauze, R. D., and Jung, K., The kinetics of" combustion of petroleum coke particles in fluidized-bed combustor. Proceedings of the Nineteenth Symposium (International) on Combustion, Pittsburgh, The Combustion Institute, 1982, p. 1087. Chandran, R. R., Duqum, J. M., Perna, M. A., Jafari, H. C., and Rowley, D. R., in Proceedings of the Ninth International Conference on Fluidized Bed Combus-. tion, Boston, MA, 1987, vol. 1, p. 292. Peterson, T. W., Scotto, M. R., and Sarofim, A. F., Powder TechnoL 45:87 (1985). Cammarota, A., Chirone, R., D'Amore, M., and Massimilla, L., Carbon attrition during the fluidized combustion of coals. Proceedings of the Eighth International Conference on Fluidized Bed Combustion, Houston, TX, 1:43 (1985).

Received 15 October 1987; revised 14 June 1988

APPENDIX Fluidized combustor models are based on particle size population balance equations. Previous works [11, 12] formulate these equations in the case of negligible secondary fragmentation. Such phenomena are taken into account in a recent work by Salatino [10]. Carbon balance over a size interval [d, d + dd] at steady state gives in this case:

mass feed rate ] to the reactor l in the interval J

F

mass rate entering the ] + I interval by continuous change l of particle size due to [. combustion and attrition

J

_

_

mass rate l e a v i n g the ] interval by continuous change o f particle size due to c o m b u s t i o n and attrition rate o f c a r b o n c o n s u m p t i o n within the interval due to c o m b u s t i o n and attrition

SECONDARY FRAGMENTATION IN A FLUIDIZED BED

+

89

mass rate entering the ) interval by secondary fragmentation from sizes coarser than d

ddf ]. Therefore:

mass rate leaving the ] interval by secondary fragmentation = 0 to sizes smaller than d

is the total mass of carbon entering per unit time the given size interval [d, d + dd] from any size interval [db, d b + ddb], with d < db --< d~x. dm~x is the maximum particle size in the feed. On the other hand,

]

Is= (A1)

.

Fragmentation contributions to the terms on the left-hand side of Eq. A1 are expressed by means of statistical functions: F(db), i.e., the probability density that a particle of size db breaks into fragments; and

is the amount of carbon leaving the interval [d, d + dd] per unit time towards any size interval [dr, de + ddf], with d* -< de < d. Moreover, it is

n(df, db), or w(df, db), i.e., the probability densities of size de of fragments generated by the breaking of a particle of size db, on numerical or mass basis, respectively. = - Wd~(d)R(d)F(d)dd,

It is F(db) =

l dm m ddb

considering that (A2)

ia w(df, d)ddf= 1 (A3)

is the mass flux of particles that shrink crossing the size db. W~ and P(db) are, respectively, the overall mass and particle size distribution, on a mass basis, of carbon present in the combustor at steady state. R(db) is the particle shrinkage rate due to both combustion and attrition:

R (db)= (ddb/dt)c+ (ddb/dt).,

(A4)

with the attrition contribution expressed as

(ddb/dt)a = - ka(U- Umf),

for any d.

(A7)

d*

where

m = - W~p(db)R(db)

(A6)

Note that the right-hand side of Eq. A6 can be directly derived from Eqs. A2 and A3. Using Levenspiel et al. [12] nomenclature and embodying into Eq. A1 these expressions for fragmentation contributions, we obtain

FoPo(d)dd- [W~R (d)p(d)]a+aa + [ WcR (d)p (d)]a 4

3R(d)p(d) W~dd d

-Wc[[am'XR(db)P(db)F(db)w(d'db)ddb] d d . d

(A5) + WcR(d)p(d)F(d)dd= O,

where ka is an attrition rate constant and U U ~ is the excess of gas velocity over the minimum for fluidization [3, 4]. By definition, -Wcp(db)R(db)F(db)w(df, db)ddbddf is the mass rate of carbon leaving the size interval [db, db + ddb] and entering the size interval [dr, df +

(A8)

where Fo and po(d) are, respectively, mass rate and particle size distribution, on a mass basis, of carbon fed to the reactor. Dividing by Fodd, defining X ( d) = Wcp( d)R ( d)/ Fo, and introducing the dimensionless size ~ = d/dm~x, Eq. A8

90

R. CHIRONE et al.

becomes

relationship [3]:

- -1 dX(6) -~- S~X(6b)F(~b)W(6,

Ec = k ( U - Umf)

dmudt~

~b)d6b

- X(~i) [ d 3max~5+F(6)] =Po(~5)'

(A9)

with the boundary condition X(1) = 0. If secondary fragmentation is negligible, F(tS) = 0 for any ~ and Eq. A9 reduces to the Levenspiel et al. [12] equation. The population equation takes the form of an integrodifferential equation in the unknown X(5) when fragmentation is taken into account. Its solution gives Wcp(d)R(d)/Fo, and, according to the usual procedures, Wc, W¢/Fo, p(d), and a~. E¢/Fo is calculated from WJFo and ~ on the basis of the

wc -

-

(A10)

where k = 3ka. Solution of Eq. A9 is considerably simplified if, according to Peterson et al. [ 18], the similarity of fragments size distribution is assumed. It is

n(df, db)ddf =fn(de/db)d(df/db) ~ fn(df/db) = clbn(ae, db) for any db,

(All)

w(df, db)dde= fw(df/db)d(df/db) ~ fw(df/db) = db w(df, db) for any db,

(A12)

respectively, on a numerical and mass basis. The advantage of this simplification has been taken in present calculations.