193 - 206
Particle Attrition Phenomena in a Fluidized Bed YIA-CHING RAY, TSUNG-SHANN JIANG Department
of Chemical Engineering,
of Illinois at Chicago, Chicago, IL 60680
and C. Y. WEN Chemical Engineering
West Virginia University,
WV 26506 (U.S.A.)
(Received June 7, 1985 ; in revised form September 3,1986)
In many fluidization processes, the attrition of particles is a serious problem. To simulate the attrition, it is necessary to consider the attrition characteristics of materials, the breaking energy supplied by the system, and the distribution of attrition rates among the particles. The size distribution of the attrited fines is an essential element of the attrition characteristics. For some materials, a unique fine distribution suggests that the quantity of attrition may be directly proportional to the breaking energy supplied by the system. Based on the concepts of energy transformation and constraint, this study analyzes the results of attrition in fluidized beds with porous distributors in order to understand the relationship between the rate of breaking energy and the mechanical conditions. To simulate the distribution of attrition rates for a multicomponent mixture, a burface-reaction’ model is proposed. The model states that the attrition rate for each component is proportional to its surface area, and is also a function of the interactions between different materials. Therefore, for particles of the same material but of different sizes, the fractional loss due to attrition is larger for the smaller particles. If one material has the dominant fraction of the surface area in a mixture, then, under the same magnitude of breaking energy, the attrition rate of this material is almost independent of its particle size. However, for a material with only a minute fraction of the total surface area, its attrition rate is inversely proportional to its average particle size. Experimental results from various studies are used to verify our 0032-5910/87/$3.50
distribution model. The model suggests that larger coal and smaller limestone are more desirable to reduce attrition in a coal corn bustor.
Recently, more and more research has been undertaken on particle attrition in fluidized beds [l - 41. In a fluidized-bed combustor, particles of both fuel (coal, char, etc.) and sulfur absorbent (limestone, dolomite, etc.) are disintegrated by the mechanical force, in parallel with the progress of chemical reactions. Attrition increases the number of particles and reduces the particle size. Knowing the particle population is essential in the assessment of the bed performance and particle entrainment or elutriation [ 5, 61. Therefore, a fluidized bed with attritable materials cannot be designed and simulated appropriately until the attrition activity is quantified. The mechanisms to reduce the size of an object can be classified as scraping, tearing, impact, compression (crushing), pulling (tension), cutting, bending, and twisting . Different mechanisms create products of different patterns. Two theories  have been put forward to correlate the degree of size reduction with the energy consumed. According to Rittinger’s surface theory, the energy consumed is proportional to the area of new surfaces formed. Kick’s volume theory states that the energy is proportional to the volume or weight of the cornminuted product. Therefore, the so-called ‘specific surface energy’ (energy per unit surface) or ‘specific volume energy’ (energy per unit 0 Elsevier Sequoia/Printed in The Netherlands
volume) should be a size-reduction property. Since no reliable measurement has yet been developed, no consistent data on these properties are available in the literature so far. One reason is that it is very difficult to distinguish the energy absorbed by the material from that absorbed by the equipment . Besides, a seemingly simple process can involve many breaking mechanisms. It becomes extremely difficult to trace the history of each portion of the product and to determine the breaking mechanisms and energy involved [ 7,9 - 121. Many studies have suggested methods for quantifying the sizereduction processes. Indices such as hardness, abrasion, ductility, grindability, etc., have been standardized for many materials or processes [ 7,8]. Paramanathan and Bridgwater [ 13, 141 recently used an annular attrition cell to study the attrition characteristics of different materials. Zenz  also suggested several tests for fluid-particle systems. These single-valued indices are the combinations of certain material properties and process variables. The role of each variable on the indices is not discerned. Therefore, their applications to other conditions are questionable. A commercial fluidized bed usually has more than one particle material and each material has its own particle size distribution. To simulate the attrition of these particle mixtures, it is necessary to know the attrition rate of the individual component and the size distribution of the attrited product. Many studies [9 - 121 have tried to simulate the particle population in the size-reduction processes. Although they are successful in describing the results of a given experiment by using fitting parameters [ 12, 161, without more fundamental analysis, the extrapolation to different conditions remains difficult [ 17, 181. Compared with other size-reduction processes, particle attrition in a fluidized bed is perhaps simpler to study. In a fluidized bed, the breaking mechanism depends on the particle strength and the breaking force. The breaking of material can change from abrasion to splitting with increasing breaking force. With the same breaking force, splitting may occur for a weak but not for a strong material. Many studies [2,4,15,19 - 211 show that for several materials, the breaking mechanism in the fluidized bed is primarily abra-
sion. Particles of intermediate size resulting from splitting are rare [ 2,4]. Upon abrasive attrition, a mother particle is disintegrated into two parts: the fines and a single son particle, which is only slightly smaller than the mother -particle. Therefore, the amount of fines produced is a direct indication of the degree of attrition. This study concerns only the abrasive attrition. A fundamental and complete modelling theory for attrition should address the following three important aspects. A ‘property study’ quantifies the attrition characteristics of vastly different materials. A ‘mechanicalmodel study’ relates the material characteristics to such mechanical variables as energy, force, velocity, etc., in order to predict the attrition rate. A ‘distribution study’ predicts the distribution of attrition rates in a particle mixture. Vaux et al.  summarized the perceived factors causing limestone attrition in a fluidized-bed combustor. These factors are refined into two groups and listed in Table 1. One group of factors creates some mechanical forces and directly results in particle attrition. The mechanical model describes how these factors affect the attrition rate. The other group of factors alters the attrition property.
Experiments were carried out in a fluidized bed of 0.10 m diameter and 1.20 m height. The column was made of fiber glass. The distributor, a porous plate, was used to create bubbles and avoid grid jets. The elutriated fines were collected from the top by a cyclone and a fabric bag. Limestone from Greer Co., Morgantown, WV, was ground to small particles of radii ranging from 300 to 1000 pm. Next, these ground particles were put into the fluidized bed and attrited until the steady-state attrition was reached, which means that the influence of grinding on the particle surface is erased. Then, they were sieved into narrowly sized groups. Either a single size or a combination of two sizes was used as the bed particles. During the attrition, the fine loss was measured to calculate the attrition rate. In each measurement, the bed particles were screened to determine whether the particle size had changed significantly.
195 TABLE 1 Factors affecting limestone attrition in a fluidized-bed combustor Group
Region of occurrence
Screw and feeding tube
Pneumatic conveyer; Transfer line
Inertial impaction; High-velocity impact
High-velocity impact, abrasion
Low-velocity impact and abrasion from stirring by bubbles
Volume of bed between grid and freeboard
One time (after feeding)
Inside fluidized bed
Stress caused by change in crystal
One time (after feeding)
Inside fluidized bed
Stress caused by change in crystal lattice
One time or continuous
Inside fluidized bed
Internal gas pressure
One time (after feeding)
Inside fluidized bed
*Group I is characterized by the process mechanical force and Group II by the material property change.
Since the attrition of this limestone is a very slow process in a cold bed, significant change in the size of mother particles was not detected. After the measurement, the fine loss was made up by adding new particles so that the particle weight was kept constant. The minimum fluidization velocity was determined by use of the pressure drop curve. To measure the distribution of attrition rates for different size components, an experimental scheme were developed. Attrition of this limestone generated fines and son particles slightly smaller than the mother particles. Particles of intermediate size between fine and son particles were rarely detected. Therefore, after a slight fraction of the mother particles were attrited, the son particles could be completely collected by using a sieve with a mesh size slightly smaller than that of the mesh collecting the mother particles. In some experiments, particle mixtures of two single-sized components were used. The size difference between the two components was at least a mesh interval apart.
After a very slight degree of attrition, the lower size limit of the son particles originated from the bigger mother particles was still larger than the upper size limit of the smaller mother particles. Therefore, son particles of the two components could be completely separated by screening. The rate of weight loss in each component indicated its mass attrition rate. How the attrition rates were distributed among different size components could thus be determined.
Previous studies [ 2, 31 indicate that freshly ground particles usually have an initial period of rapid attrition before the attrition pace slows down to a certain steady level. Studies on other size-reduction processes  also recognize that particles possess memory which reflects the effects of their treatment before feeding. After this memory fades out, the attrition becomes a normal Markov
0” 0 t= 2
Fig. 1. Characteristics of initial attrition. Data reported for three different steady-state attrition rates: 0, 7.80 lo+ kg/h;o, 1.22 x 1O-4 kg/h;n, 2.70 x 1O-4 kg/h.
process  in which the attrition characteristics of the material are uniquely defined and free of the memory other than that of attrition. These particles are said to have reached a steady-state condition . The attrition rate of material in the steady-state attrition condition is referred to as the steady-state attrition rate, as opposed to the initial attrition rate of material with preattrition memory. Our experiments indicate that freshly ground particles approach the steady-state attrition condition after approximately 3 to 5 wt.% of the original mass is stripped away. We measured both the initial and steady-state attrition rates for particle samples of the same mass and size under various fluidizing velocities. The ratio of the initial to steady-state attrition rates is shown
in Fig. 1 as a function of the degree of attrition. Here the degree of attrition is defined as the mass fraction of the original ground particles lost due to attrition. We define R, (kg/s) as the total mass rate of attrition and R (l/s) as the fractional rate of attrition, which is the ratio of the total mass rate of attrition R, to the total mass M of the mother material. There appears to be a characteristic relationship between the ratio of the initial to steady-state attrition rates and the degree of attrition, regardless of the fluidizing velocity. This suggests that a mechanical model may concern only the effects of various mechanical conditions upon the steady-state attrition rate. For freshly ground particles, the attrition rate may be simulated by combining the mechanical model with the
increase in the fluidizing velocity or sand size. Considering the scope of their study (twofold in the velocity increase and fourfold in the sand size change), we think that the effects of particle size and velocity on the fine size distribution are not significant. This conforms to our observation in Fig. 2. Many studies [2,3, 13,18,22] report the attrition rates without mentioning the fine distributions. It is not known whether the materials used have basic fine size distributions. A study  with coal and limestone in a laboratory Hardgrove machine suggests that particles of a certain size ratio in the same machine would break into products of the same size ratio, which implies that larger particles create larger products. Therefore, the existence of a basic fine distribution is perhaps unique to the abrasive attrition and can hardly apply to
characteristic relationship between the initial and steady-state attrition rates. It must be pointed out, however, that although particle samples of the same size and material reach the same steady-state attrition rates under the same mechanical conditions, their initial attrition rates may differ, depending upon the preparation history. Figure 2 shows the size distribution of the attrited fines taken from cases of different particle sizes and fluidizing velocities. Merrick and Highley [l] observe in their fluidized-bed combustor that increasing the fluidizing velocity has little effect on the size distribution. Arena et al.  measured the size distribution of carbon fines in a fluidized-bed combustor using sand as the inert bed material. Their results indicate that there is only a slight increase in the size of coal fines with
Fig. 2. Size distribution of attrited limestone fines. Data are for different mother particle sizes d and excess velocities U,; 0, d = 1540 pm, U, = 0.76 m/s; 0, d = 770 pm, U, = 0.63 m/s;A, d = 770 pm, U, = 0.63 m/s.
other size-reduction processes. We think that the abrasion mechanism tends to produce fines of size characterizing the structure of the material. Unlike crystalline materials, which distribute the stress more evenly and break more uniformly, amorphous materials such as coal and limestone may consist of grains, pore structure, and agglomerates or particles of one material embedded in a matrix of another . Thus, this kind of amorphous material attrites into fines of a certain characteristic size distribution, referred to as having “natural grain size” [ 71. Materials inside the fluidized bed do not necessarily have natural grain size. To investigate the attrition characteristics of a given material, it is necessary to measure and compare the fine distributions in different situations. If the material does not have natural grain size, attrition rates under different conditions become less comparable and meaningful interpretation is more difficult. If the attrited fines are much smaller than the mother particles, the newly created surface area can be attributed mostly to the fines. For materials having natural grain size, the total surface area of fines is directly proportional to their total volume or mass. This implies that both Rittinger’s surface theory and Kick’s volume theory [7,8] lead to the same conclusion that the mass rate of fines produced is proportional to the rate of effective breaking energy. Thus, these two theories are not necessarily irreconcilable
Although porous distributors are not widely used in the commercial process, the study of the bubbling bed is a prerequisite for the development of the mechanical model for any fluidized bed. Whatever the distributor, the upper section of a fluidized bed is usually a bubbling zone . This bubbling zone and the distributor zone together cpntribute to the attrition inside the bed . The effects of bed height, wall, baffle and gas velocity can be assessed in the bubbling bed. Then, the effects of distributor can be investigated by subtracting the contribution of the bubbling zone. Some reports
Fig. 3. Bubbling
[2, l&20,24 - 281 are available for the effect of the distributors other than the porous plate. Although several studies [l - 4,291 have investigated the attrition in the bubbling bed with porous distributors, a satisfactory model has not been achieved yet. In this section, we restrict our attention only to develop a mechanical model for a bubbling bed with a porous distributor by use of the concept of energy transformation. Considering the bubbling bed as shown in Fig. 3, we can calculate the rate of gas energy E, supplied to the bed in terms of the difference of the rate of pressure energy between inlet and outlet gases: E, = UA,AP
where AP (EPi -P,) is the pressure drop across the fluidized bed, U the superficial gas velocity, and At the cross-sectional area of the bed. When the superficial gas velocity U exceeds the minimum fluidization velocity Umf, the pressure drop across the bed remains constant, i.e., AP = AP,,. Therefore, eqn. (1) becomes E, = UAt Al’,,,, = MgU
where M is the total mass of bed particles and g the magnitude of the gravitational acceleration. In the second line of eqn. (2), a force balance across the bed after minimum fluidization is employed. Attrition occurs only after the minimum fluidization velocity is exceeded [ 1,2, 41. Therefore, the rate of kinetic energy received by particles Ek can be attributed to that portion of the gas energy exceeding minimum fluidization [ 21: Ek = MgU,
here U, (’ U - V,,) is the excess gas velocity exceeding minimum fluidization. As discussed in the section on Property study, for material having natural grain size, the total mass rate of attrition Rt is proportional to the rate of effective breaking energy Eb : R, = aE,
in which a is an attrition coefficient, defined as the mass of fines generated by a unit of effective breaking energy. Since E,, is related to Ek, we can write E,, = rl%
in which 77represents the efficiency of energy transformation. We can express eqn. (4) as R, = aqE, = aqMgU,
Equation (6) expresses the total mass rate of attrition as a generalized function of the excess gas velocity and total bed mass. If 7) is constant, we can then conclude that R, is proportional to both U, and M. Ulerich et al. [ 21, after reviewing several previous studies, suggested that R, is proportional to U,. For a given mixture of coal and sand in a hot fluidized combustor, Arena et al.  also reached the same conclusion. Figure 4 shows our experimental results with different values of TJ,and particle sizes. The bed material is 100% limestone and the bed load is fixed. Our data indicate that R, increases linearly with U, at relatively high values of U,. However, this linear relationship, when extrapolated, does not pass through the origin. The intercept at Rt = 0 is a positive value, U,. This suggests that, at relatively high values of U,, Rt = ak(Ue =ak[u-(&f
uk) + uk)l
in which ok is a constant. Kono [ 291, studying attrition with aluminum-silica particles, reported data indicating that at high U,, a linear relationship like eqn. (7) applied, while at small U,, R, decreased more slowly than a linear function and approached the origin asymptotically. Material strength usually has a yield point before breakage. Since at small U, the kinetic force is not sufficiently large to exceed the yield strength of the particle, no
breakage occurs. In other words, q in eqn. (5) is small at small U, and increases with U, until it reaches a constant value. With constant efficiency, the increase in R, is proportional to the increase in U,, as shown in Fig. 4 and eqn. (7). For brittle materials such as coal, the yield strength is small. The efficiency of energy conversion reaches a constant almost immediately after minimum fluidization is exceeded. Therefore, uk is very small. This explains why, in some studies [ 1, 2,4, 301, the attrition rate is found to be directly proportional to U,. Figure 4 also shows that in a bed with 100% limestone particles, the total attrition rate is independent of the particle size at the same U,. It is worth noting that data in Fig. 4 are obtained from cases of single size as well as particle mixtures of two-size components. The particle size affects the attrition rate only through its effects on U,,. This agrees with the results of Ulerich et al. [ 21. In light of eqn. (6), the above result supports the characteristics of natural grain size and confirms the postulate that the attrition coefficient a is a material property. The effect of the static bed height h,, (or the bed load M) on the total attrition rate R, needs clarification. Ulerich et al.  proposed a mechanical model based on the kinetic force between particles stirred by the bubble motion. Since the bubble velocity increases with bed height due to bubble coalescence, the collision force between particles increases with bed height as well. With this argument, they concluded that the total mass rate of attrition Rt is proportional to the square of the bed height or, equivalently, the fractional attrition rate R is proportional to the bed height. Experiments of Kono  indicated that R is proportional to (h,f)0.78. On the other hand, both Merrick and Highley [l] and Donsi et al.  discovered that R, is proportional to hmf or that R is independent of hnf in their experiments with coal. We reason that the rate of effective breaking energy Eb across the fluidized bed cannot exceed the rate of kinetic energy Ek provided by the gas. According to eqn. (3), the rate of kinetic energy is proportional to the bed load or hmf. If the efficiency of transforming Ek to Eb is constant with increasing bed load, the total mass rate of attrition R, is proportional to the static bed height hmf. Perhaps, in a
f $0.30 a X a? 0.20
U, (m/s) Fig. 4. Effect of excess gas velocity on attrition rate in a bed of single material limestone. The solid line is the best fit to the data of single-sized particles. Symbol
710 545 385
1.00 1.00 1.00
0.84 0.59 0.43
certain range of bed height, the efficiency of energy transformation in eqn. (5) increases with bed height because of the bubble activity, so that Rt may have a stronger dependence on hmf. However, this should not be extrapolated to the higher value of bed height because of the energy constraint mentioned above. In fact, as the bed height
reaches the limit of slugging or maximumly attainable bubbles, the extra bed load does not vary the bubbling condition. In this situation, the reasoning for increasing particle interaction or collision efficiency  is no longer valid. The model based on the consideration of impact or mechanical force  can be helpful to envision qualitatively the
efficiency of energy transformation, but it may not be suitable to evaluate the overall attrition rate. The abrasion mechanism suggests that the breaking force is dispersive. Collision of high-speed particles is probably not a dominant phenomenon. Instead, we suggest that the emulsion of particles is like an easily deformed body. The mechanical force exerted upon this body will be quickly dispersed and shared by the neighboring particles. In this situation, we think it will be easier to analyze the problem in terms of the energy transformation than the mechanical force involved.
Up to now, opinions from various attrition studies are contradictory about the effect of particles on the attrition rate [l, 2, 4,30, 321. Although the total attrition rate can be easily measured for a material, no research has yet demonstrated experimentally what fraction of the attrited fines is generated by a given size of the mother particles. Gardner and Austin  used a radioactive tracer technique to observe the breakage of narrowly sieved particles in a grinding study. In principle, their method can be utilized in the study of attrition, but the experiment is difficult to perform. In this section, we will refer to ‘component’ as particles of the same material and size. A mixture consists of many components, which are either of different materials or of different particle sizes. For component i with particle radius ri and mass Mi, we will speak of Rtt as the mass attrition rate (or attrition rate) and Ri (= RtI/MI) as the fractional attrition rate. For a multi-sized mixture of a single material, Merrick and Highley [l] postulated that the mass attrition rate of a component is proportional to the excess velocity, the mass of the component, and the mass fraction wi of components smaller than rl: R,i = klU,MiWi = klU,MXiwi
Here Xi (rMi/M) is the mass fraction of component i, M the total mass of the particle mixture, and k 1 a constant. Equation (8), as
indicated by wt, implies that the larger particles are attrited faster. In contrast, Fan and Srivastava  proposed that the mass attrition rate is proportional to the volume or mass of the component, independent of particle size, that is, R,i = k,Mi = k2Mxi
in which k2 is also a constant. Ulerich et al.  analyzed experimental results from various sources and concluded that the total mass attrition rate Rt (=ZR,i) for a single material is independent of particle size when U, is fixed in a bubbling bed. This agrees with our previous discussion (see the section on Mechanical-model study). In a hot combustor with coal and sand, Arena et al.  and Chirone et al.  found that the total coal attrition rate is proportional to U, and the total surface area of coal particles:
(10) Where k3 is a constant, F the surface-mean radius, and superscript c represents coal particles. Equation (10) suggests that the larger coal particles are attrited more slowly. In addition, Arena et al.  found that the coal attrition rate increased with the particle size of sand. Here, we develop an attrition rate distribution model by analogy to the endothermic surface reaction. The total ‘reaction’ rate is constrained by the rate of input energy. The ‘reaction’ rate for an individual component is proportional to its surface area and ‘selectivity’. The basic assumptions for our model are as follows: (1) Abrasion is the dominant breaking mechanism. (2) The materials involved have ‘natural grain size’. (3) The particle mixture is uniformly mixed without any segregation. There is a representative value for the minimum fluidization velocity U,, of the mixture . (4) Particles are in the steady-state attrition condition. The attrition property of each material can be uniquely defined. (5) The rate of effective breaking energy E,i associated with component i is propor-
tional to the total surface area Ai of this component. We define
as an energy-sharing coefficient for component i. This is equivalent to the reaction selectivity for the surface reaction. (6) Processes under the same mechanical conditions provide the same total rate of effective breaking energy Eb for particle systems of either only one component or mixtures. This means that the mechanical model constructed from the case of single component can be applied to that of mixtures. The effects of distributor, bed load, gas velocity, etc., can be assessed in the same way. For a mixture of N components, assumption (6) states that
Rti = ai% = athAi%
where ai is the attrition coefficient for component i. In the second line of eqn. (13), we have utilized both eqns. (11) and (12). The total surface area of component i can be written as
in which ci and pi are the shape factor and mass density for component i, respectively. With eqns. (13) and (14), we can have a general equation for the attrition rate of component i as Rti
(A) Single material, different
In this case, material properties, such as the attrition coefficient, energy-sharing coefficient, and density, are all the same. We will assume that the shape factors are identical as well. Equation (15) can thus be reduced to
The total mass rate of attrition becomes
Rt = f Rtj
which is independent of particle size distribution as discussed above. However, the ratio of attrition rates between components of sizes 1 and 2 is Rtl r12 - 42
To verify the validity of the model, the following two special cases are discussed.
The mass rate of attrition for component i can be expressed as
C (bjxjlejrjpj) I=1
with the understanding that
=- xl/r1 x2/r2
In contrast to eqns. (8) and (9) [l, 321, eqn. (19) suggests that in a mixture, the larger particles are attrited more slowly. There are no direct experimental verifications for eqns. (8) and (9). Table 2 compares our experimental results with eqn. (19). The reasonably good agreement confirms the validity of our surface-reaction model for the distribution of attrition rates. This is the first time that the relationship between the ‘size distribution’ and the ‘attrition rate distribution’ is demonstrated experimentally. (B) Two materials c and s:
The total mass attrition rate of material c can be calculated from eqn. (15) as
rate for mixtures rzX
0.50 0.50 0.25 0.50 0.66 0.50 0.66
385 545 770 770 770 770 770
RF = a’ AC =a
545 770 385 385 385 385 385
0.50 0.50 0.75 0.50 0.33 0.50 0.33
0.19 0.225 0.046 0.19 0.19 0.172 0.26
0.14 0.165 0.204 0.33 0.15 0.331 0.219
0.33 0.39 0.25 0.52 0.34 0.503 0.479
1.36 1.36 0.23 0.58 1.27 0.52 1.19
1.42 1.41 0.17 0.50 1.00 0.50 1.00
xc/F’ + Bsc(xsecpc/Fsesps)
is a relative energy-sharing coefficient of material s with respect to material c, A is the surface area, F is the surface-mean radius and x is the mass fraction of each material. Superscripts c and s refer to materials c and s, respectively. If xc is very small or AC is much smaller than AS, then eqn. (20) reduces to RF = CIACEb
Rt2 x lo3
with cl and c2 as constants. This agrees with the result of Arena et al. [ 41, as shown in eqn. (10). In their fluidized-bed coal combustor, a large amount of small sand was used as inert material. Therefore, the coal attrition rate becomes proportional to the exposed surface of coal. However, if the surface area of material c becomes dominant, eqn. (20) approaches eqn. (4), which shows that the size distribution of material c has no effect on its attrition rate. In eqn. (2O), Eb and a’ can be obtained from comparative experiments with a single material. But B,, is probably determined by some property differences between materials or even by the composition of the mixture.
We performed two comparative experiments, one with 100% limestone having two size components, the other in which the smaller limestone particles were replaced by sand of the same size. The sand has about the same density as the limestone and is unattritable. The minimum fluidization velocity is the same for both samples, so, at the same gas velocity, E, can be assumed to be the same too. Therefore, the attrition rates of limestone particles in both cases can be compared. The relative energy-sharing coefficient Bs. of sand with respect to limestone can be calculated according to eqn. (20) and is shown in Table 3 for our experimental result with two size compositions. The relative energy-sharing coefficient varies as the composition changes. Arena et al.  reported that in their combustor experiment,, the coal attrition rate increased with the sand particle size. This kind of interaction between different materials is anticipated in our distribution model. Since the surface area of sand decreases with increase in its particle size, according to eqn. (20), the attrition rate of coal will thus increase. Unfortunately, Arena et al.  did not report sufficient information for us to determine a’ and B,, from their data. In particular, factors affecting B,, are not well understood. Therefore, further study on this respect of material interaction is needed. In some size-reduction processes, the sizereduction rate decreases with the presence of a significant. amount of fines. This is referred to as the cushioning effect of the fines [ 361. The fines share the breaking energy proportional to their surface areas, as described in assumption (5). However, this portion of breaking energy is wasted because the fines do
204 TABLE Effect Case
3 of unattritable Material
sand on attrition
ri x lo6
rate of limestone Xi
not break further. Thus, the breaking energy actually causing the size reduction is reduced, and so is the size-reduction rate. This suggests that in any attrition process, the presence of fines is not only the result of previous attrition but also a factor for determining the future attrition.
In general, coal is burned in a fluidized-bed combustor with a dominant amount of limestone. The loss of attrited coal reduces the combustion efficiency. The elutriated limestone causes particulate pollution. Through our distribution model, the strategy to deal with both problems can be qualitatively discussed here. The following discussion is based on the use of the same material composition and comparable mechanical conditions. By comparable conditions, we mean that the systems provide the same total rates of effective breaking energy. If the mass fraction of coal is small, eqn. (22) suggests that its mass attrition rate is inversely proportional to the coal particle size. The smaller the coal particle size, the higher the coal attrition rate. Besides, the size of limestone particle also affects the coal attrition rate according to eqn. (20) or eqn. (22). Reducing the size of limestone increases the overall surface area of bed particles and thus decreases the fraction of the coal surface and, in turn, its attrition rate. However, because of its dominant composi-
tion, the limestone attrition rate is not significantly influenced by the sizes of both coal and limestone according to eqn. (4). Therefore, in order to reduce attrition, particle feed of smaller limestone and larger coal seems more desirable. But reducing attrition is just one aspect of the overall optimal strategy. The performance of the solid-gas reactions usually depends upon particle size as well as gas concentration. Gas concentration is also indirectly influenced by the particle size because of U,,. Furthermore, such factors as feeding, elutriation, overflow, etc., also affect the process. Therefore, the process can be accurately simulated only by using a population model which accommodates all these relevant factors.
CONCLUSIONS It is hoped that this work can enhance our understanding about attrition in a systematic and quantitative way. The ways to quantitatively describe the attrition characteristics of a given material are discussed by use of concepts such as attrition coefficient, natural grain size, ratio of initial to steady-state attrition rate, and energy-sharing coefficient. A mechanical model to assess the effects of various mechanical factors on the attrition rate for a bubbling bed with a porous distributor is also proposed. The concepts of energy transformation and constraint are introduced. The ‘surface-reaction’ model is proposed to predict the distribution of component attrition rates for a mixture. The
model takes into account the effects of mass fraction, density, size distribution, and attrition properties of the mixture. Since the attrition rate for each component is proportional to its surface area, the fractional loss due to attrition is larger for the smaller particles in a multi-sized mixture. The total attrition rate is determined by the rate of breaking energy supplied by the system. Different materials interact and share the breaking energy according to their surface areas, densities and other properties. For material with dominant composition or surface area, the attrition rate is almost independent of its average particle size; but for material with minute surface area, the attrition rate is inversely proportional to its average particle size, provided that the rate of breaking energy is the same for all. In a mixture, the ratio of attrition rates between components is independent of the mechanical conditions. The model suggests that larger coal and smaller limestone are more desirable for reducing attrition in a coal combustor.
W(J m2) c2
4 Ek ei
Lf kl kz k3 M
This work was partially sponsored by a grant from the United States Department of Energy, Morgantown Energy Technology Center (Grant No. 8610106640, UAC 24003281). The authors would especially like to thank Dr. Jer Yu Shang of Morgantown Energy Technology Center for his encouragement to initiate this project. One of the authors, Dr. C. Y. Wen, passed away during the progress of this research project. Two of the authors (YCR and TSJ) are grateful to the National Science Foundation (Grant No. CPE-8:07873) for financial support. T. S. Jiang would also like to thank the Union Oil Company for Fellowship support.
Mi N P @nlf
Rt Rti r F
r12 LIST OF SYMBOLS
particle surface area of component i, m2 cross-sectional area of fluidized bed, m2
attrition coefficient, defined as the mass of fines generated by a unit of effective breaking energy, kg/J constant defined in eqn. (7), kg/m relative energy-sharing coefficient of material s with respect to material c, defined in eqn. (21) energy-sharing coefficient of component i, defined in eqn. (11) constant defined in eqn. (22), constant defined in eqn. (22), m/J particle diameter, m total rate of effective breaking energy, J/s rate of effective breaking energy portioned to component i, J/s rate of gas energy across the bubbling bed at velocity U, J/s rate of energy for particle movement across the fluidized bed, J/s particle shape factor of component i gravitational acceleration, m/s2 static bed height, m constant defined in eqn. (8), l/m constant defined in eqn. (9), l/s constant defined in eqn. (10) total mass of bed particles, kg mass of component i, kg number of components pressure, Pa pressure drop across bed when fluidized, Pa total fractional attrition rate for the whole bed of single material, = R,/M, l/s or l/h fractional attrition rate of component i, = Rti/Mi, l/s or l/h total mass attrition rate for the whole bed of single material, kg/s or kg/h mass attrition rate of component i, kg/s particle radius, m surface-mean particle radius, m ratio of attrition rates, size 1 to size 2 particle radius of component i, m superficial gas velocity based on total cross-sectional area of the fluidized bed, m/s excess gas velocity exceeding minimum fluidization, m/s
u mf Wi
constant defined in eqn. (7), m/s minimum fluidization velocity, m/s mass fraction of particles smaller than ri mass fraction of component i
Greek symbols energy transformation rl defined in eqn. (5) particle density, kg/m3 P
Superscripts C material c, such as coal S material s, such as sand L limestone Subscripts component i component j
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