Mixing characteristics of irregular binaries in a promoted gas--solid fluidized bed: a mathematical model.
Solid mixing is a common mixing operation widely used in different industries. In fact, this operation is almost always practiced wherever particulate matter is processed. This is strongly influenced by different mobilities of the mixed components, which depend on the particle properties. However, in industrial solids mixing, it is often required to mix particles differing widely in physical properties viz. size, density, and/or shape. The role of particle size and density and the air flow rate on the segregation or demixing behaviour in a gas--solid fluidized bed has already been reported (Nienow et al., 1972). The degree of axial mixing of particles in fluidized beds is important for many continuous or batch processes, and control thereof is desirable. In fluidized beds consisting of particles with different size and/or density a concentration profile will develop over the height of the bed at moderate gas velocities (Hartholt et al., 1997). Most of the investigators who discuss the problem of solid mixing in a fluidized bed have assumed that the solid mixing stems from random movements of particles and this assumption has rarely been questioned. If it is correct it follows that solid mixing will occur by interparticle diffusion or eddy diffusion as in true fluids (Rowe et al., 1965) and bubble rise. Because of the bubble rise, some solids are seen flowing up and others flowing down the bed.
Solid exchange between a bubble wake and the emulsion phase is one of the fundamental rate processes that directly affect the direct mixing of fluidized beds (Chiba and Kobayashi, 1977 and Kunii and Levenspiel, 1969).Work relating to the mixing of segregating particles in a fluidized bed is scanty. Nicholson and Smith (1966) studied the axial mixing of particles differing in density in a fluidized bed and thereof proposed a first-order rate equation to describe the progress of mixing in the short mixing time. Gibilaro and Rowe (1974) formulated a qualitative model of particle mixing in fluidized beds based on four physical mechanisms viz. overall particle circulation, interchange between wake and bulk phases, axial dispersion and segregation. Fan and Chang (1979) studied the fluidization and solid mixing characteristics of very large particles where bubble or slug induced drift and gross solid circulation appeared to be the predominant solid mixing mechanisms. The degree of axial mixing of particles in fluidized beds is important for many continuous as well as batch processes and the control thereof is desirable.
Correlations for Mixing Index
Naimer et al. (1982) have developed the general expression for mixing index which is widely used for all systems in the form as given below:
[I.sub.M] = [X.sup.*/[X.sub.bed] (1)
Nienow et al. (1978) have proposed the correlation for the equilibrium mixing index for an equal-size, density-variant binary mixture in a three-dimensional fluidized bed as follows:
M = [(1 + [e.sup.-z]).sup.-1] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
For a size variant, equal density system of particles, Fan et al. (1990) have developed the following model for the mixing index:
[I.sub.M] = K x [([bar.d].sub.p]/d.sub.F].sup.k] [(U/U - [U.sub.F]).sup.n] (4)
Role of Bubbles on Mixing
It is a well-known fact that some solids flow up and others flow down because of bubble rise during fluidization in a gas--solid fluidized bed. This up-flow and down-flow with an interchange between the streams is the basis for various counterflow models that have been proposed to account for the vertical mixing of solids. Van Deemter (1967) divided the solids into two streams for a tall enough bed of solid particles and developed two models for up-flowing stream and for down-flowing stream. The horizontal movement of solids was first studied by Brotz (1952) in a shallow rectangular bed from where he got the information to evaluate the horizontal dispersion coefficient [D.sub.sh]. A similar approach was used by other investigators (Mori and Nakamura, 1965; Hirama et al., 1975; Borodulya et al., 1982). Heertjes et al. (1967) suggested that the wake material scattered into the freeboard by the bursting bubbles could contribute significantly to the horizontal movement of solids. Hirama et al. (1975) and Shi and Gu (1986) used partition plates in the freeboard just above the bed to study this effect. All of these investigators used rather shallow beds of height between 5 and 35 cm. In contrast, Bellgardt and Werther (1984) made measurements in a much larger bed, namely a 2 m x 0.3 m bed about 1 m deep. Quartz sand (dp = 450 [micro]m) was fluidized, and careful measurements confirmed that vertical mixing was much faster than the horizontal mixing, thus justifying the use of a one-dimensional dispersion model in the horizontal direction. Kunii and Levenspiel (1991) developed a mechanistic model based on the Davidson's bubble model and proposed the following expression for the horizontal dispersion coefficient for both fast and intermediate bubbles:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
DEVELOPMENT OF MATHEMATICAL MODEL
An attempt has been made to develop a theoretical model with the system parameters on the basis of "Counterflow Solid Circulation Models" (Kunii and Levenspiel, 1991). Considering both vertical and horizontal movement of the jetsam particles as some particles displace horizontally due to the bursting of bubbles the dispersion model in the form of the differential equation can be written as follows:
For solids upward motion, that is, in upward direction:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
For solids downward motion, that is, in downward direction:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
when the superficial velocity of the fluidizing medium is more than that of jetsam/flotsam particles, assuming that the whole solid material is divided into two streams; one stream having fraction [f.sub.u] moves up and the other stream with fraction [f.sub.d] moves down. Thus, the movement of solids is a continuous process during fluidization. It is almost impossible to determine the exact fraction of solids moving up or down. Therefore it has been assumed that always half of the whole bed material moves in upward direction while the other half moves in the downward direction during fluidization.
Again with the assumption of [f.sub.d] = [f.sub.u], [U.sub.u] = [U.sub.d], [C.sub.ju] = [C.sub.jd] and writing f, u, and [C.sub.j] for these variables respectively in the above Equations (7) and (8), then adding these two equations the following equation is obtained where (W/2 [[rho].sub.s])/[V.sub.B] is used for f:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
This is the differential equation describing the concentration of jetsam as a function of bed height. Vertical mixing rate as a function of gas velocity in rather small beds is given (Kunii and Levenspiel, 1991) as follows:
[D.sub.sv] = 0.06 - 0.1 [u.sub.o] (10)
Horizontal dispersion coefficient as mentioned in the book (Kunii and Levenspiel, 1991) is given by Equation (6). For Geldart-BD solids [alpha] has been taken as 0.77.
Equation (6) has been simplified using the expressions for the bubble diameter, bubble rise velocity, bed voidage fraction, minimum fluidization velocity, fraction of bed in bubbles etc. (Kunii and Levenspiel, 1991).
Equation (6) in simplified form is as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Now Equation (9) can be written as:
[partial derivative].sup.2] [C.sub.j]/[partial derivative][z.sup.2] + F[u.sub.o]/F[D.sub.sv] + [D.sub.sh] [partial derivative[[C.sub.j]/[partial derivative]z] = 0 (12)
where, F = W/2[rho].sub.s] [V.sub.B]
Now describing the coefficient of [partial derivative][C.sub.j]/[partial derivative]z as a function of height as:
F[u.sub.o]/F[D.sub.sv] + [D.sub.sh] FDsv + Dsh = f (z) (13)
Equation (12) can be written as:
[partial derivative].sup.2][C.sub.j]/[partial derivative][z.sup.2] + f (z) [partial derivative][C.sub.j]/[partial derivative]z = 0 (14)
Solving the above differential equation by variable separable method the concentration of jetsam particles can be written as:
[C.sub.j] = [integral] [e.sup.-] [integral] f(z)dz] dz (15)
Now substituting the [D.sub.sh] and [D.sub.sv] from Equation (11) and Equation (10), respectively, Equation (13) can be expressed as follows:
f(z) = A + [B.sub.z]/C + [D.sub.z] (16)
where, A = [Fu.sub.o] [C.sub.1] + [Fu.sub.o][D.sub.2]
B = 0.0414 x [Fu.sub.o][D.sub.2]
C = (0.06 + 0.1[u.sub.o])F[C.sub.1] + (0.06 + 0.1[u.sub.o])F[D.sub.2] + K[K.sub.1]
D = (0.06 + 0.1[u.sub.o])0.0414 x F[D.sub.2] + 0.0828K[K.sub.1]
The solution of Equation (15) in terms of A, B, C, and D can thus be written as:
[C.sub.j] = [integral] [e.sup.-] (B/D)z x (1 + D/C).sup.[BC-AD]/[D.sup.2] dz (17)
Again on simplification, Equation (17) can be written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
This gives the idea for the concentration of jetsam particles for any system at any height of the bed from the distributor. Thus, the mixing index at any height can be written as:
[I.sub.M] = [C.sub.j] x W/J (19)
Figure 1 gives a schematic diagram of the experimental setup. The binary mixtures of irregular particles are fluidized in a 15 cm x 100 cm Perspex column. The components of the mixture have been mixed in the ratio of 10:90, 25:75, 40:60, and 50:50. For a particular composition of the mixture, the initial static bed height and the superficial velocity of the fluidizing medium have been altered four times. The process has been repeated for four different size/density ratios of the homogeneous/heterogeneous binary mixtures respectively in unpromoted as well as promoted beds. The samples have been drawn for analysis for the static bed condition as well as for the fluidized bed condition. In the static bed condition the samples have been drawn layer-wise by applying vacuum after the fluidized bed is brought back to static bed condition by shutting off the air supply suddenly. In the fluidized bed condition the samples have been drawn through the side ports during fluidization process. The samples drawn at different heights have been analyzed for the distribution of jetsam particles and calculation of their concentration. The scope of the experiments is presented in Tables 1A, B, and 2A, B.
[FIGURE 1 OMITTED]
DEVELOPMENT OF EXPERIMENTAL MODELS
The model developed from dimensional approach for the unpromoted and promoted fluidized beds are as follows:
1. For homogeneous binary mixtures A. Unpromoted fluidized bed (i) Static bed condition
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
(ii) Fluidized bed condition
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
B. Promoted fluidized bed
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
2. For heterogeneous binary mixtures
A. Unpromoted fluidized bed
(i) Static bed condition
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
(ii) Fluidized bed condition
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
B. Promoted fluidized bed
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
RESULTS AND DISCUSSION
The developed model for the concentration of jetsam particles, thereby for the mixing index has been verified with a number of homogeneous and heterogeneous binary mixtures by varying the system parameters. Finally the values of the mixing index obtained through the theoretical model for unpromoted and promoted beds have been compared with both the homogeneous and heterogeneous binary mixtures. On comparing the values of the mixing index at different heights for the promoted beds with those of unpromoted ones for both the systems, it is found that the unpromoted fluidized beds are having higher jetsam concentration in almost all the cases indicating more mixing index than the promoted beds. A sample plot for the homogeneous binary mixture is shown in Figure 2. This in turn implies that better mixing is obtained with the unpromoted bed than the promoted ones, where resistance is offered in the horizontal plane. Reason for this may be that with the promoter the bubble rise is obstructed by the discs of the promoter, which in turn reduce the rise of jetsam particles upwards with the bubbles. Some particle transport might occur from the upper side of the lower disc to the bottom of the next upper disc.
[FIGURE 2 OMITTED]
The values of the mixing index calculated by the dimensional analysis approach have been compared with those obtained from the experimental observations as well as from the theoretical model for different types of beds with both the systems (homogeneous and heterogeneous binary mixtures). The average error values for mixing index obtained from the comparison of calculated mixing index values by the dimensional analysis approach and the experimental methods are listed in Table 3.
Mixing index values obtained from the theoretical model (Equation (19)) and the numerical models (developed by the dimensional analysis, Equations (20) to (25)) have been compared with the experimental ones for different types of fluidized beds for both homogeneous and heterogeneous binary mixtures in Figures 3 and 4, respectively. It was observed that the values obtained with the developed theoretical model are lower than both the experimental ones as well as the developed dimensional correlations for all types of fluidized beds in both the systems. The reason for this may be due to the "gulf-streaming effect" and the assumption of the uniform concentration in a layer of particles at any height of the bed, which may not be true in reality.
During the process of fluidization some particles move upwards and some downwards inside the fluidizer. It is difficult to know that at any instant of time how much portion of the bed materials is moving upward and how much downward. For the simplification of the modelling it was assumed that at any instant of time during the process of fluidization 50% of the bed materials is moving up and the balance 50% of the bed materials is moving in the downward direction.
Theoretical model has been developed on the assumption that 50% of the bed materials move up as the upward stream and the balance 50% move down as the downward stream during fluidization. Apparently, segregation in the axial direction might have been resulted from preferential transportation of lighter particles upwards with rising bubbles and from interparticle competition to fill the voidage created by the rising bubbles (Fan and Chang, 1979). The samples were drawn from the ports made on either side of the column alternately and were analyzed on the basis of the assumption of uniform concentration for a particular layer of particles across the cross-section of the column at any height. This may not be true in totality which in turn results in higher values of the mixing index over the theoretical values. Lower values of mixing index by the theoretical model might have been obtained due to these assumptions which may not be true in an operating fluidized bed.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Theoretical Analysis for the Model
Effect of various system parameters viz. size/density of the particles, initial static bed height, composition of the mixture, and the superficial velocity of the fluidizing medium on the jetsam concentration have been studied for both the systems with the unpromoted and promoted fluidized beds, respectively. A sample plot for the heterogeneous binary mixtures is shown through Figure 5A to D with the promoted bed. It is observed that with the increase in flotsam density or in other words decreasing the ratio of jetsam to flotsam densities the jetsam concentration decreases at any height and also the jetsam concentration decreases with the increase in bed height. Although the same tendency is observed with the homogeneous binaries but the effect of jetsam and flotsam size ratio on jetsam concentration at any height is insignificant in comparison with the heterogeneous binaries. In both the systems, the distribution of jetsam particles in any layer of the bed has been found to decrease with the increase in jetsam percentage in the overall mixture, with the increase in the initial static bed height and with the decrease in size/density ratio of the binary mixture. Also the distribution of the jetsam was found to decrease with the increase in the superficial velocity above the minimum fluidization velocity (Figure 5). It is also observed that the concentration of jetsam decreases with the height of the bed irrespective of the any system parameter involved. This implies that the segregation tendency is observed with all the system parameters for the developed model as the jetsam concentration gradually decrease with the increase in bed heights for any system. It is also noted from the Tables 1B and 2B that the ratio of minimum fluidization velocity of jetsam to that of flotsam for three mixtures in the case of homogeneous binaries and one mixture in the case of heterogeneous binaries is greater than 2.0 whereas it is less than 2.0 in the case of other mixtures indicating clear segregation tendency with the former mixtures compared to other mixtures in both the systems studied (Chen and Keairns, 1975). It was also observed that the mixing index for the homogeneous binaries is better than the heterogeneous binary mixtures indicating the better mixing operation in the former case.
[FIGURE 5 OMITTED]
The degree of mixing depends very much on gas velocity and even strongly segregating system can either be separated or well mixed by controlling this. Knowledge of the minimum fluidization velocity is crucial if the behaviour of a fluidized bed is to be properly analyzed. The [U.sub.mf] is a simple concept and easy to measure in a mono-component fluidized bed however, it is complex in both definition and measurement for any binary system. In a well-mixed bed of two solids, the void fraction depends strongly on the mean size ratio and volumetric fractions of its components and its values can be significantly lower than for a monosized bed of particles.
The developed model has been tested against the existing experimental data. The distributions of the jetsam particles are variable in the direction of the bed height. The numerical results are in satisfactory agreement with the existent experimental data.
The depth of the jetsam layer, the fluidization velocity and the particle properties, especially the minimum fluidization velocities of the two components, determines the concentration of jetsam in the upper stratum of a strongly segregating bed at steady state.
The developed experimental models can be used widely for analyzing the mixing and segregation characteristics of both the homogeneous and the heterogeneous binary mixtures of particles over a good range of the operating parameters. The developed theoretical model establishes that, the concentration of jetsam (and hence the mixing index) decreases with the height of the particle layer in the bed measured from the distributor. The presence of promoter/baffle reduces the mixing aspect for both the homogeneous and the heterogeneous binaries. This needs more work to improve upon the model so that the difference between the values of the mixing index for the experimental and the theoretical can be minimized. Further work is being carried out to fix up an optimum fraction of the bed material with respect to its distribution in the upward and the downward streams during the fluidization process, so that the theoretical model can be improved. This will ultimately reduce the difference in values of the mixing index obtained from the theoretical and the experimental models.
NOMENCLATURE [C.sub.j] concentration of jetsam particles at any height in the bed (amount of jetsam particle in the sample drawn at a height in kg/amount of that in the original mixture, kg) d diameter of particle, m [d.sub.b] bubble diameter, cm [D.sub.C] diameter of the column, m [D.sub.E] equivalent diameter of the column, m [D.sub.SH] horizontal dispersion coefficient, [m.sup.2]/s [D.sub.SV] vertical dispersion coefficient, [m.sup.2]/s F flow rate of solids moving up or down per bed volume, [m.sup.3] of the solid/[m.sup.3] of the bed volume f fraction of solids moving up or down per bed volume, [m.sup.3] of the solid/m3 of the bed volume [H.sub.b] height of particles layer in the bed from the distributor, m [H.sub.s] initial static bed height, m [I.sub.M] mixing index, dimensionless J weight of jetsam particles taken in the bed, kg K coefficient of the correlation Ks interchange coefficient K exponent of parameter M equilibrium mixing index N exponent of parameter u Velocity of the stream of particles moving up or down, m/s [u.sub.b] bubble velocity, cm/s [u.sub.br] bubble rise velocity, cm/s [U, [u.sub.o] superficial velocity of the fluidizing medium, m/s [U.sub.F] minimum fluidization velocity of the mixture, m/s [U.sub.TO] take over velocity defined as the value of U corresponding to M = 0.5 [V.sub.B] volume of the bed, [m.sup.3] W weight of the total bed material, kg [X.sup.*] percentage of jetsam particle in any layer [[bar.X].sub.bed] percentage of jetsam particle in the bed Z height of any layer of particle in the bed measured from the distributor, (varying from 0-0.2 m) Greek symbols [delta] fraction of bed in bubble [alpha] a factor, the ratio of wake diameter to bubble diameter [epsilon] bed voidage fraction [rho] density of particle, kg/[m.sup.3] Suffixes F, f flotsam fl fluidizing condition j jetsam m mixture mf minimum fluidization condition u upward component d downward component o operating condition p particle s solids w wake solids Abbreviations Dia_ratio ([d.sub.j]/[d.sub.f] x ([d.sub.m]/[d.sub.f]) Dens_factor [rho].sub.f]/[rho].sub.j] x [rho]m/[rho].sub.j] MI-cal calculated values of mixing index MI-exp experimental values of mixing index MI-th mixing index values obtained from the theoretical model
Manuscript received March 13, 2007; revised manuscript received July 18, 2007; accepted for publication August 9, 2007.
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A. Sahoo * and G. K. Roy
Chemical Engineering Department, National Institute of Technology, Rourkela 769008, Orissa, India
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Table 1A. Scope of the experiment (for homogeneous binaries) Serial Bed Size of Size of Ratio of no. material jetsam dp x flotsam dp x jetsam to [10.sup.3], m [10.sup.3], m flotsam in the mixture 1 Dolomite 1.015 0.725 25:75 2 Dolomite 1.015 0.725 25:75 3 Dolomite 1.015 0.725 25:75 4 Dolomite 1.015 0.725 25:75 5 Dolomite 1.015 0.725 10:90 6 Dolomite 1.015 0.725 40:60 7 Dolomite 1.015 0.725 50:50 8 Dolomite 1.29 0.725 25:75 9 Dolomite 1.44 0.725 25:75 10 Dolomite 1.7 0.725 25:75 Serial Average Initial static Heights of no. particle size bed height layers for of the mixture [H.sub.s], x the withdrawal dp x [10.sup.2], m of samples, [10.sup.3], m [H.sub.b] x [10.sup.2], m 1 0.798 12 2,4,6,8,10,12 2 0.798 14 2,4,6,8,10,12,14 3 0.798 16 2,4,6,8,10,12,14,16 4 0.798 20 2,4,6,8,10,12,14,16,18,20 5 0.754 20 2,4,6,8,10,12,14,16,18,20 6 0.841 20 2,4,6,8,10,12,14,16,18,20 7 0.870 20 2,4,6,8,10,12,14,16.18,20 8 1.008 20 2,4,6,8,10,12,14,16,18,20 9 1.083 20 2,4,6,8,10,12,14,16,18,20 10 1.213 20 2,4,6,8,10,12,14,16,18,20 Table 1B. Bed material properties (for homogeneous binaries) Bed material Component dp x [U.sup.mf], [10.sup.3], m m/s Mixture-1 Larger material 1.015 0.585 Smaller material 0.725 0.376 Ratio of above two 1.400 (unitless) 1.556 (unitless) Mixture-2 Larger material 1.290 0.759 Smaller material 0.725 0.376 Ratio of above two 1.780 (unitless) 2.016 (unitless) Mixture-3 Larger material 1.440 0.843 Smaller material 0.725 0.376 Ratio of above two 1.986 (unitless) 2.242 (unitless) Mixture-4 Larger material 1.700 0.976 Smaller material 0.725 0.376 Ratio of above two 2.345 (unitless) 2.590 (unitless) Table 2A. Scope of the experiment (for heterogeneous binaries) Serial Bed material Density of Density of Number flotsam jetsam particles [P.sub.p] x [P.sub.p] x [10.sub.3], [10.sub.3], kg/ kg/ [m.sup.3] [m.sup.3] 1 Coal and iron 1430 4760 2 Refractory brick and iron 2550 4760 3 Latrite and iron 3390 4760 4 Dolomite and iron 2940 4760 5 Dolomite and iron 2940 4760 6 Dolomite and iron 2940 4760 7 Dolomite and iron 2940 4760 8 Dolomite and iron 2940 4760 9 Dolomite and iron 2940 4760 10 Dolomite and iron 2940 4760 Serial Bed material Ratio of Average Number jetsam to particle density flotsam in of the mixture the [P.sub.p] x mixture [10.sub.3], kg/[m.sup.3] 1 Coal and iron 25:75 2262.5 2 Refractory brick and iron 25:75 3102.5 3 Latrite and iron 25:75 3732.5 4 Dolomite and iron 25:75 3395.0 5 Dolomite and iron 10:90 3122.0 6 Dolomite and iron 40:60 3668.0 7 Dolomite and iron 50:50 3850.0 8 Dolomite and iron 25:75 3395.0 9 Dolomite and iron 25:75 3395.0 10 Dolomite and iron 25:75 3395.0 Serial Bed material Initial Number static bed height [H.sub.s], x [10.sub.2], m 1 Coal and iron 20 2 Refractory brick and iron 20 3 Latrite and iron 20 4 Dolomite and iron 20 5 Dolomite and iron 20 6 Dolomite and iron 20 7 Dolomite and iron 20 8 Dolomite and iron 16 9 Dolomite and iron 18 10 Dolomite and iron 22 Serial Bed material Heights of Number layers for the withdrawal of samples, [H.sub.b], x [10.sub.2], m 1 Coal and iron 2,4,6,8,10,12,14,16,18,20 2 Refractory brick and iron 2,4,6,8,10,12,14,16,18,20 3 Latrite and iron 2,4,6,8,10,12,14,16,18,20 4 Dolomite and iron 2,4,6,8,10,12,14,16,18,20 5 Dolomite and iron 2,4,6,8,10,12,14,16,18,20 6 Dolomite and iron 2,4,6,8,10,12,14,16,18,20 7 Dolomite and iron 2,4,6,8,10,12,14,16,18,20 8 Dolomite and iron 2,4,6,8,10,12,14,16 9 Dolomite and iron 2,4,6,8,10,12,14,16,18 10 Dolomite and iron 2,4,6,8,10,12,14,16,18,20 Table 2B. Bed material properties (for heterogeneous binaries) Bed Component [[rho].sub.p] [U.sub.mf], material x [10.sub.3], m/s kg/[m.sub.3] Coal and Heavier material 4760.0 1.055 iron Lighter material 1430.0 0.469 mixture Ratio of above 3.329 (unitless) 2.249 (unitless) Refractory Heavier material 4760.0 1.055 brick Lighter material 2550.0 0.703 and iron Ratio of above 1.867 (unitless) 1.502 (unitless) mixture Dolomite Heavier material 4760.0 1.055 and Lighter material 2940.0 0.773 iron Ratio of above 1.619 (unitless) 1.366 (unitless) mixture Latrite Heavier material 4760.0 1.055 and Lighter material 3390.0 1.849 iron Ratio of above 1.404 (unitless) 1.244 (unitless) mixture Table 3. Averaged error values for each of the semi-empirical models presented (with reference to Equations (20) to (25)) in comparison with the experimental values for different types of beds and bed materials Material Type Homogeneous mixture UP-St. UP-Fl. Promoted Bed Type Bed Bed bed Reference Equation Equation Equation equations (20) (21) (22) Std. Dev. 6.399 5.995 6.642 Mean Dev. 0.587 -0.624 -0.619 Material Type Heterogeneous mixture UP-St. UP-Fl. Promoted Bed Type Bed Bed bed Reference Equation Equation Equation equations (23) (24) (25) Std. Dev. 13.759 10.787 11.455 Mean Dev. -1.881 -2.446 4.754
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|Author:||Sahoo, A.; Roy, G.K.|
|Publication:||Canadian Journal of Chemical Engineering|
|Date:||Feb 1, 2008|
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