Fluidized-bed combustion of a biomass char: The influence of carbon attrition and fines postocombustion on fixed carbon conversion

Fluidized-bed combustion of a biomass char: The influence of carbon attrition and fines postocombustion on fixed carbon conversion

Twenty-Seventh Symposium (International) on Combustion/The Combustion Institute, 1998/pp. 3103–3110 FLUIDIZED-BED COMBUSTION OF A BIOMASS CHAR: THE I...

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Twenty-Seventh Symposium (International) on Combustion/The Combustion Institute, 1998/pp. 3103–3110


di Ingegneria Chimica di Ricerche sulla Combustione—CNR Universita degli Studi “Federico II” di Napoli Piazzale Tecchio 80125 Napoli, Italy


The fluidized-bed combustion of char from a biomass fuel (Robinia Pseudoacacia) has been studied with a focus on the fate of fixed carbon under the combined effects of coarse char combustion and attrition, of attrited carbon fines postcombustion and elutriation. Extensive postcombustion under oxidizing conditions prevents the measurement of the rates of carbon fines generation by attrition from collection of elutriated carbon. A novel technique is hereby presented to overcome this problem: the rates of coarse char attrition are evaluated from the comparison of the apparent carbon conversion rate of the actual test fuel, with that of a reference fuel, selected among those which have a negligible propensity to attrition. Results from the application of the proposed technique indicate that extensive fines generation by attrition, followed by their almost complete postcombustion, occurs during fluidized-bed combustion of Robinia char. About half of the fixed carbon burns along this pathway, the remainder being directly burnt as coarse char carbon. The occurrence of extensive attrition-postcombustion leads to enhancement of the coarse char particles’ apparent Sherwood numbers by a factor of about two. Analysis of the dependence of the ratio between the fines generation rate and the parallel coarse char combustion rate on operating variables of the reactor is directed to shed light on the prevailing mechanism of carbon fines generation by attrition. It appears that, differently from low-volatile solid fuels, the percolative fragmentation mechanism plays a leading role on carbon attrition.

Introduction Carbon attrition during the fluidized-bed combustion and gasification of coal has long been recognized as the prime source of unburnt carbon at the reactor outlet [1] and has been the subject of extensive investigation [2]. The mutual interaction of carbon attrition with the progress of reaction has been addressed in a number of papers [3–5]. These have provided the mechanistic framework for the enhancement of attrition associated with carbon conversion (combustion- or gasification-assisted attrition): internal burning and uneven progress of reaction promote the increase of particle voidage and, in turn, the decay of the mechanical strength of the carbon material. The relative importance of mechanical attrition versus peripheral percolative fragmentation as sources of fines during fluidized-bed combustion has been considered by Walsh and coworkers [6–8] and Salatino and Massimilla [9]. The latter authors pointed out that the loss of particle connectivity associated with percolative fragmentation could not explain alone the extent of carbon

elutriation during the fluidized-bed combustion of a coal char and of a graphite. A framework for the quantitative assessment of the fate of fixed carbon in a fluidized-bed combustor has been recently provided [10]. The proposed series-parallel network of attrition–combustion–elutriation processes was based on the hypothesis that fixed carbon in the bed could be divided into a coarse phase and a fine phase made of, respectively, nonelutriable and elutriable particles of fixed size. This framework has been applied to the fluidizedbed combustion of a coal and two waste-derived fuels [10] and of a biomass [11]. With reference to typical operation of coal-fueled fluidized-bed combustors, it can be generally stated that generation of carbon fines by attrition is relevant to combustion efficiency and to the environmental impact of the operation but that it provides only a minor route to overall carbon conversion. On the other hand, analysis of data relating to high-volatile solid fuels suggests that their conversion takes place via the generation of fines by fragmentation or attrition followed by their postcombustion to an extent that is comparable with that of direct combustion of




2. The relatively large intrinsic combustion reactivity of fines from these chars [12] that makes their postcombustion more effective.

TABLE 1 Properties of the fuels tested Fuel


Proximate analysis, % (as received) Moisture 7.1 Ash 1.2 Volatile Matter 75.1 Fixed Carbon 16.6 Ultimate analysis, % (dry and ash free basis) Carbon 44.0 Hydrogen 7.8 Nitrogen 0.02 Oxygen 48.2 Sulphur — Char density, kg/m3 100


14.6 4.0 35.2 46.2 81.3 5.3 1.6 10.8 1.0 1040

The characterization of carbon attrition during fluidized-bed combustion has been typically accomplished under the assumption that fines postcombustion can be neglected and that, accordingly, the attrition rate can be assumed equal to the elutriation rate. This assumption does not apply to high-volatile fuels, on account of the preceding point 2. The aim of the present work is the setup of a method for quantitative assessment of attrition rate of a highvolatile fuel under conditions of nonnegligible fines postcombustion. It is based on the comparison of combustion behavior of the actual test fuel (Robinia) with that of a low-attrition reference fuel (Snibston coal). Experimental carbon elutriation rates obtained using the biomass are further compared with results of previous investigation with South African bituminous coal under a variety of operating conditions. The comparison is useful to shed light on the mechanisms of carbon attrition in the combustion of high- versus low-volatile fuels as well as on the different importance of the fines generation-postcombustion versus direct coarse char combustion routes in the overall carbon conversion. Experimental Materials

Fig. 1. Experimental apparatus. (1) Gas preheater; (2) electrical furnaces; (3) ceramic insulator; (4) gas distributor; (5) thermocouple; (6) fluidization column; (7) head with three-way valve; (8) sintered brass filters; (9) hopper; (10) stack; (11) filter; (12) membrane pump; (13) gas analyzers; (14) personal computer; (15) manometer; (16) gas flowmeters; (17) air dehumidifier (silica gel).

Robinia Pseudoacacia, a ligneous biomass common in the Mediterranean area, has been used as the test fuel. A Snibston bituminous coal has been selected as a reference fuel. Their properties are given in Table 1. Robinia was cut into particles of cylindrical shape with both diameter and length of 10 mm. Snibston was reduced into approximately spherical particles with an average size of 10 mm. The fuel particles were first devolatilized by dropping them in the bed fluidized with nitrogen at 850 8C. After about 3 min, char particles were retrieved from the bed. Particles that had undergone primary fragmentation during pyrolysis were not considered for further experimentation. The bed material consisted of 180 g of silica sand sieved in the two size ranges 0.3–0.4 mm and 0.425– 0.6 mm with Sauter mean diameters of 0.36 mm and 0.57 mm. Apparatus and Experimental Procedure

coarse particles. This route is considerably enhanced by two factors: 1. The tendency of high-volatile fuels to give rise to highly porous, or even incoherent, chars that easily yield elutriable fines upon impact or surface wear.

The experimental apparatus consisted of an atmospheric bubbling fluidized-bed combustor, 1 m high and 40 mm i.d. (Fig. 1). The stainless steel fluidization column was heated by two 2.2-kW electric furnaces driven by a PID controller. The gas distributor was a perforated plate with 55 holes 0.5 mm in



Fig. 2. Fixed carbon elutriation rate, divided by the initial carbon mass, as a function of time from batchwise experiments under inert and oxidizing conditions. U 4 0.8 m/ s; ds 4 0.36 mm. (a) Robinia [11]; (b) South African coal [4,13].

diameter in a triangular pitch. A paramagnetic analyzer and two NDIR analyzers were used for continuous monitoring of O2, CO, and CO2 concentrations, respectively, at the exhaust. Data from the analyzers were logged and further processed on a PC. Experiments were carried out by injecting singlefuel char particles of given mass of either Robinia or Snibston coal into the bed kept at 850 8C. The bed was fluidized with nitrogen–oxygen mixtures at the desired oxygen concentration. Gas superficial velocity was 0.4 or 0.8 m/s. Continuous analysis of the exhaust gases allowed us to measure the carbon combustion rate as a function of time. Experimental Results Figure 2a reports carbon elutriation rates measured during batch fluidized bed experiments using char from Robinia [11] at 850 8C in inert and oxidizing conditions (1% and 3% inlet oxygen concentration). Fig. 2b reports similar profiles obtained during batch experiments using a char from a bituminous South African (SA) coal [4,13]. The carbon elutriation rate under inert conditions (0% O2) is far larger for Robinia than for SA coal char. Notably, under oxidizing conditions, the elutriation rate increases significantly with respect to that measured under inert conditions for SA coal char, whereas it decreases, nearly dropping to zero, for Robinia. The opposite behaviors of these fuels are analyzed in the framework of networks in Fig. 3a and 3b, relative to Robinia and SA coal, respectively. Fixed carbon is divided into two phases: a coarse phase, made

of relatively large, substantially nonelutriable char particles; a fine phase, made of char particles of elutriable size. The fate of fixed carbon is followed from feeding (FO) to its final destiny, through primary fragmentation (FOC, FOF), coarse char combustion (FCP) and attrition (FCF) and fine char postcombustion (FFP) and elutriation (Ec). Fines elutriation rate under oxidizing conditions can be expressed as: Ec 4 FCF (1 1 n) 4 FCF,ib(1 1 n)


where FCF,i is the rate of fines generation by attrition under inert conditions, that is, due to purely mechanical attrition. b 4 FCF/FCF,i represents a factor, larger than unity, expressing the enhancement of fines generation associated with combustion-assisted attrition. n 4 FFP/FCF represents the degree of carbon fines postcombustion. Under oxidizing conditions, elutriation rates can be either smaller (Fig. 2a) or larger (Fig. 2b) than FCF,i depending on whether the product b(1 1 n), is smaller or larger than unity, respectively. The consideration [12] that the intrinsic combustion reactivity of char from Robinia is much larger than that of char from SA char (by a factor of about 800 at 850 8C) leads to the conclusion that postcombustion of attrited fines is extensive for Robinia char fines (i.e., n ù 1 and b(1 1 n) ù 0), whereas it is limited for SA char fines. In the networks reported in Fig. 3a and 3b, pathways that, according to the experimental evidence, provide minor routes to fixed carbon conversion have been indicated as dotted lines. The typical approach to the characterization of carbon attrition phenomena followed in previous



Fig. 3. The fate of fixed carbon during fluidized-bed combustion [10]. (a) Robinia [11]; (b) South African coal [10]. The square-shaped blocks represent the two “phases”: coarse (block C) and fine (block F) particles. F0 4 feed rate; F0C 4 rate of coarse char generation after primary fragmentation; F0F 4 rate of fine char generation after primary fragmentation; FCF 4 fines generation rate by attrition; FCP 4 combustion rate of coarse char; FFP 4 fines postcombustion rate; FP 4 overall combustion rate; Ec 4 elutriation rate.

studies ([2] and references therein) has been that of neglecting fines postcombustion and to assume FCF 4 Ec (as in Fig. 3b). This approach, however, cannot be extended to the characterization of attrition of high-volatile fuels, and methods for quantitative determination of attrition rates of chars that do not rely on collection of elutriated fines must be developed. Theoretical Background The novel technique for carbon attrition rate measurement is based on the comparison of the apparent fixed carbon conversion rates of the actual test fuel (Robinia) with that of another solid fuel taken as a reference fuel. The latter must have the property of undergoing negligible carbon fines generation with respect to direct coarse char combustion (FCF K FCP). Char from Snibston coal has the lowest propensity to attrition among the coals studied in previous investigations [2], even smaller than SA coal. It has been therefore selected as the reference fuel. The following assumptions are made: 1. Combustion of coarse char particles takes place in the boundary layer diffusion controlled regime, on the basis of order of magnitude evaluation of external diffusional versus intrinsic kinetic plus intraparticle diffusional resistances for typical char particle sizes [12].

2. Particles are spheres of diameter dc. 3. Conversion of char particles takes place according to the shrinking-particle conversion model. Internal burning takes place only in a thin cortical region close to external particle surface [4,5]. 4. Fuel particle temperature is that of the emulsion phase. This simplifying assumption relies on the consideration that combustion rate barely depends on particle temperature under external diffusion controlled regime (assumption 1). 5. Carbon dioxide is the only product of coarse char combustion. This is supported by the finding that measured CO/CO2 ratio was always less than 0.05. 6. Gas flow pattern in the bed corresponds to perfect mixing: large bubble-dense phase mass transfer index was evaluated under the experimental conditions used and, in addition, U/Umf k 1. 7. Change of gas concentrations in the freeboard due to postcombustion occurring in that section of the reactor is negligible. This is partly justified by the consideration that the freeboard has been kept cold in the experiments to minimize postcombustion. The balance on fixed carbon in the reactor during batch combustion reads 1

dWc 4 FCP ` FCF 4 FCP ` FFP ` Ec dt


where Wc is the carbon mass in the coarse char, and carbon flowrates conform to notation in Fig. 3. The coarse char combustion rate, after assumptions 1 and 5, can be expressed in terms of the particle Sherwood number, Sh: FCP 4 12pShdcDO2 CO2


where DO2 is oxygen diffusivity in the nitrogen–oxygen mixture at the bed temperature (assumption 4) and CO2 oxygen concentration in the bed, equal to that at the combustor outlet (assumptions 6 and 7). On the other hand, it is FCP ` FFP 4 12Q(CCO2 ` CCO)


where Q is the volumetric gas flow rate and CCO2 and CCO carbon dioxide and monoxide concentrations in the bed. The parameter a 4 FCF/FCP is now introduced, defined as the ratio of the rate of carbon fines generation by attrition to the rate of coarse char combustion. After equation 3, equation 2 reads 1

dWc 4 12pSh(1 ` a)dcDO2CO2 dt 4 12pShappdcDO2CO2


where Shapp 4 (1 ` a)Sh is an apparent particle


Fig. 4. Sherwood number as a function of char particle diameter, for different fluidization velocities. Fuel: Snibston coal. Sand mean size 4 0.57 mm; inlet oxygen concentration, 21%.

Fig. 5. Sherwood number as a function of char particle diameter, for different sand mean sizes. Fuel: Snibston coal. Fluidization velocity 4 0.4 m/s; inlet oxygen concentration, 21%.

Sherwood number resulting from the combined action of coarse char combustion and fines generation by attrition. Test fuel Carbon elutriation is neglected in the fixed carbon balance: (Ec ù 0 ⇔ FCF ù FFP in Fig. 3a). Considering equations 2, 4, and 5: Shapp 4

Q(CCO2 ` CCO) pdcDO2CO2


The char particle diameter, according to hypothesis 3, is given by dc(t) 4 dco

1W 2 Wc




Sh 4

Q(CCO2 ` CCO) pdcDO2CO2


where the instantaneous values of dc and Wc are obtained, as for the test fuel, from equations 7 and 8, respectively. The instantaneous values of the apparent Sherwood number for the test fuel and of the true Sherwood number for the reference fuel are determined from CO and CO2 concentration profiles in the exhaust gases according to equations 6 and 9, respectively. The parameter a for the test fuel and, in turn, the rate of carbon fines generation by attrition FCF are eventually calculated once values of Shapp and Sh have been obtained in experiments with the test fuel and the reference fuel, respectively, under the same experimental conditions.


where dco and Wco are, respectively, the initial diameter and mass of the char particle. Wc is calculated, upon integration of equation 2 and considering equation 4, as Wc(t) 4 Wco 1


# 12Q(C 0


` CCO)dt8


Reference fuel In this case, fines generation by attrition of coarse char FCF is neglected in the fixed carbon balance (FCF 4 0 ⇔ a 4 0 ⇔ 1 dWc/dt ù FCP). Equation 6 reduces to

Results Figures 4 and 5 show the true Sherwood number, Sh, as a function of particle diameter for Snibston char. Data refer to different gas superficial velocities U (Fig. 4) and bed solids size ds (Fig. 5). Only data for particle diameters dc larger than 1 mm are reported. Below this threshold, combustion might not be entirely controlled by diffusion in the boundary layer. Data refer to experiments carried out using air. Within the experimental error, Sh evaluated from tests carried out using nitrogen–oxygen mixtures with different inlet oxygen concentrations (4.5% and 10%) were the same as those in Figs. 4 and 5. Theoretical curves obtained from correlations proposed



Fig. 6. Apparent Sherwood number as a function of char particle diameter, for different fluidization velocities and sand mean sizes. Fuel: Robinia. Inlet oxygen concentration, 21%.

[14]. The agreement with the correlation of Prins et al. [15] is better. Figures 6 and 7 report the apparent particle Sherwood numbers Shapp evaluated in experiments with Robinia. The influence of gas superficial velocity and bed material size (Fig. 6) and of inlet oxygen concentration (Fig. 7) is assessed. Curves in these figures are not smooth. In fact, it appeared that sudden jumps in carbon oxides concentrations occurred during combustion. These jumps could be related to the occurrence of secondary fragmentation of char particles, according to the mechanism proposed by Sundback et al. [16]. Curves used in the evaluation of Shapp were chosen among those that presented the smallest fluctuations, to minimize the influence of secondary fragmentation. Fig. 6 reports also best fit lines through data points obtained with Snibston char (Figs. 4 and 5). The apparent Sherwood number for Robinia is always much larger than the true Sherwood number obtained with the Snibston char under equal operating conditions. Shapp does not depend appreciably on gas superficial fluidization velocity and on inlet oxygen concentration. It increases as the bed solids size increases. Values of the parameter a 4 FCF/FCP calculated from Shapp and Sh under the different operating conditions tested are reported in Table 2. It is noted that values around 1 are obtained, practically independent of gas superficial velocity, bed solids size, and oxygen concentration. Discussion

Fig. 7. Apparent Sherwood number as a function of char particle diameter, for different inlet oxygen concentrations. Fuel: Robinia. Fluidization velocity 4 0.8 m/s; sand mean size 4 0.36 mm.

by La Nauze and Jung [14] and by Prins et al. [15] are reported for comparison. Sh changes linearly with char particle diameter. Sh is not influenced appreciably by gas superficial velocity (Fig. 4). It increases somewhat as the bed inert solids size is increased (Fig. 5). These trends are not consistent with predictions of La Nauze and Jung

Experimental results, and in particular the notably large value a ù 1 for Robinia, indicate that in the combustion tests and irrespectively of the operating conditions, about half of the coarse char actually burns via fines generation by attrition and subsequent postcombustion. It is interesting to compare these results with those obtained by Chirone et al. [11] with the same fuel but using a different experimental semiquantitative method. Based on the consideration that the CO/CO2 ratio as primary products of carbon oxidation depends on the size of the burning particle, these authors concluded that the pathway fines-char-generation-by-attrition → finespostcombustion contributes to overall fixed carbon conversion to an extent that is comparable, and even larger, than direct combustion of coarse char. Lin et al. [17] carried out corncob combustion experiments in a fluidized-bed reactor. They observed that solid combustion took place at a rate that was consistent with an apparent particle Sherwood number of 22. Substantial coarse char attrition and fines postcombustion (Fig. 3a) might explain also in this case the admittedly large values of the Sherwood numbers observed. It is interesting to compare the values of a obtained in the present work with previously published

FLUIDIZED-BED COMBUSTION OF A BIOMASS CHAR TABLE 2 Values of the parameter a calculated for Robinia and South African coal Robiniaa

U 4 0.4 m/s U 4 0.8 m/s

ds 4 0.3–0.4 mm

ds 4 0.425–0.6 mm

1.0 (4.5%) 0.9 (10%) 1.0 (21%) 1.0 (4.5%) 1.1 (10%) 0.9 (21%)

0.9 (21%)

1.0 (21%)

South African coal (batch experiments)b ds 4 0.3–0.4 mm U 4 0.5 m/s U 4 0.8 m/s

0.013 (4.5%) 0.032 (4.5%) 0.015 (10%) 0.009 (21%) 0.057 (4.5%)

U 4 1.1 m/s

South African coal (continuous experiments)c

U 4 0.8 m/s U 4 1.3 m/s

U 4 1.6 m/s

ds 4 0.2–0.4 mm

ds 4 0.6–0.85 mm

0.078 (e 4 1.0) 0.05 (e 4 1.1) 0.039 (e 4 1.3) 0.098 (e 4 1.0) 0.08 (e 4 1.1) 0.068 (e 4 1.2) 0.061 (e 4 1.3) 0.097 (e 4 1.0) 0.098 (e 4 1.1) 0.063 (e 4 1.2)

0.037 (e 4 1.0) 0.016 (e 4 1.1) 0.017 (e 4 1.2)

0.062 (e 4 1.0) 0.044 (e 4 1.2) 0.025 (e 4 1.4)


work (in brackets: inlet oxygen concentration). out from [4,13] (in brackets: inlet oxygen concentration). cWorked out from [18] (in brackets: excess air factor). bWorked

data for a South African bituminous coal [4,13,18]. To this end, attrition data have been worked out (by expressing them as Ec/FCP ù FCF/FCP) and reported in Table 2. Two features emerge clearly from the comparison. First, values of a for the coal are always far smaller than those for Robinia. This reflects the already noted smaller propensity to attrition of low-volatile fuels. It must be noted, however, that in spite of the smaller attrition rate observed with low-volatile fuels, its relevance to the loss of combustion efficiency can be emphasized by the correspondingly smaller intrinsic combustion reactivity of attrited fines. Second, the values of a for SA coal are extremely sensitive to the operating conditions, whereas those for Robinia are not. This observation calls for deeper


consideration of the attrition mechanisms that might be at work in the two cases. Salatino and Massimilla [5] pointed out that purely mechanical attrition and percolative fragmentation provide extreme cases of a continuous range of phenomenologies. Within this range, mechanical stresses and loss of particle connectivity by internal burning cooperate in determining the actual attrition rate, according to the combustion-assisted attrition concept. The limiting case of fines generation by percolative fragmentation is one in which the ratio of the carbon consumption rate by peripheral percolation to that by combustion is a constant determined solely by the initial particle voidage and by the voidage at the solid percolation threshold [6,7,19]. At the other extreme, the limiting case of purely mechanical attrition would imply that the rate of carbon attrition FCF is independent of oxidizing conditions but dependent on fluidized-bed parameters, like the excess superficial gas velocity and bed solids size. Accordingly, a would be dependent on parameters affecting both FCF and FCP. In particular, a } 1/CO2 would be expected in this case. When analyzing fluidized-bed combustion of a bituminous coal, Walsh [7] observed that the dependence of a on oxidizing conditions was stronger than expected on the basis of a purely percolative approach but milder than expected on the basis of a purely mechanical attrition mechanism. He postulated that attrition could be looked at as the sum of contributions from percolative fragmentation and mechanical attrition [8]. Similar conclusion was drawn by Brown et al. [20]. Salatino and Massimilla [4,5] embodied the effects of mechanical attrition and of the loss of connectivity of carbon particles associated with internal burning into a combustionassisted attrition model that was able, inter alia, to reproduce the less-than-reciprocal dependence of a on oxygen concentration observed with low-volatile fuels. Within this general framework, the relative constancy of a for Robinia suggests that its attrition behavior should closely approach the purely percolative asymptote. The large initial char porosity (0.91) and its large intrinsic combustion reactivity might provide keys to the explanation of this behavior. Conclusions The combustion-attrition behavior of a biomass char during fluidized-bed combustion has been effectively characterized by means of a novel experimental technique. The elutriation rate of char under practical combustion conditions is very limited. Conversion of fixed carbon takes place essentially along two competitive pathways: (1) direct coarse char combustion and (2) generation of elutriable carbon fines by attrition of the coarse char, followed by their postcombustion over their residence time in the reactor.



Pathway 2, usually neglected in the quantitative assessment of fixed carbon conversion with reference to low-volatile solid fuels, is extremely important with reference to the biomass char at hand. Almost half of fixed carbon fed to the reactor is actually burnt along pathway 2, the remainder following the usual pathway 1. The large contribution of pathway 2 leads to an enhanced apparent coarse char particle Sherwood number, by a factor of about two. The rate of carbon fines generation by attrition has been related to the parallel occurrence of coarse char combustion, and the influence of main operating variables of the reactor has been considered. On this ground, it is concluded that the attrition behavior of char from Robinia can be looked at in the framework of the established combustion-assisted attrition models and closely approaches the percolative fragmentation limit. This implies that the rate of carbon fines generation by attrition can be expressed to a reasonable approximation as a fixed fraction of the overall carbon consumption rate. These findings can have important implications in the operation and design of fluidized-bed combustors. In particular, the unbalance between fine and coarse char combustion should make fluidized-bed combustor operation with high-volatile fuels more sensitive to parameters that affect intrinsic combustion kinetics (like temperature) and fine elutriation rate (like gas superficial velocity) than in operation with low-volatile fuels. REFERENCES 1. Beer, J. M., Massimilla, L., and Sarofim, A. F., Inst. Energy Symp. Ser. 4:Disc.IV-5 (1980). 2. Chirone, R., Massimilla, L., and Salatino, P., Prog. Energy Combust. Sci. 17:297–326 (1991). 3. Chirone, R., D’Amore, M., and Massimilla, L., in Twentieth Symposium (International) on Combustion, The Combustion Institute, 1984, pp. 1505–1511. 4. Salatino, P. and Massimilla, L., Chem. Eng. Sci. 40:1905–1916 (1985).

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