CORRESPONDENCE VOID FRACTION VARIATION I N T H E SPOUTED BED ANNULUS
SIR: T h e experimental observations upon which Madonna (1966) bases his argument for a large and continuous longitudinal variation of void fraction in a spouted bed annulus are the linear increases of particle velocity-at-wall with bed level measured by Thorley et al. (1956) for wheat spouted in a 6-inch diameter column with a 1-inch diameter inlet, and reproduced as Figure 5 in Madonna’s paper. Without reference to any other observations or considerations, these longitudinal particle velocity-at-wall gradients could be caused by three possible factors, which are not necessarily mutually exclusive : A. Increase of annulus void fraction with bed level. B. Cross flow of annular solids into spout along bed level. C. Decrease of annulus cross section with bed level. Mathur and Gishler (1955), the original workers in the field of spouting, maintained that the main cause of the gradients was B, with some contribution by C for a 6-inch column, but that A could be discounted. Madonna, on the other hand, attributes the entire effect to A. T h e purpose of the present communication is to argue in favor of the Mathur-Gishler explanation and to criticize the contrary argument of Madonna. Why were Mathur and Gishler justified in giving no consideration to A and in concluding that B and C could completely explain the particle velocity-at-wall gradients? 1. They observed visually both in their full columns and in their sectional columns-see, for example, their Figure 1that the particles in the annulus touch each other and behave like a moving packed bed. They could observe no significant variation in void fraction of the annulus with bed level, except for the fountain a t the very top which is looser, and the cone a t the very bottom which is possibly denser (Thorley et al.. 1956), than the main annulus. T h e fountain and core are, however, outside the region of particle velocity-at-wall measurements. 2. T h e void fraction of a moving packed bed corresponds to that of a loosely packed fixed bed (Happel, 1949). 3. Mathur and Gishler measured the void fraction of a loosely packed bed of their wheat particles as 0.432. They therefore assigned this constant value to the annulus of their spouted wheat beds. 4. Quantitative justification for this procedure, which is independent of any possibility of optical illusion, is the fact that when the 6-inch diameter, 25-inch deep wheat bed, initially in its loosely packed static condition a t a measured void fraction of 0.432, was spouted, “the expansion in the bed was observed to be about 6%, which is roughly equivalent to the volume of solids displaced from the spout, and indicates that the annulus is substantially a packed bed” (quoted from Mathur and Gishler 1955). T h e over-all void fraction of the spouted wheat bed, including both the annulus and the bed, was then approximately 1 - (0.568/1.06) = 0.464. 5 . Mathur and Gishler, and subsequently also Thorley et al. (1956), observed that in a 6-inch column the spout diameter grows appreciably with bed level, giving rise to effect C . T h e same effect is, however, much less in evidence in a 24-inch column (Thorley et al., 1956).
6. By means of high-speed movies of their sectional column, Mathur and Gishler were able to observe directly that “ a considerable cross flow of solids from the annulus into the spout takes place all along the bed height,” and they were also able to measure the upward velocity of the particles in the spout as a function of bed level. 7. Despite this cross flow of solids from annulus to spout, Mathur and Gishler assumed that, for purposes of calculating the downward flow of solids in the annulus a t any bed level, the particle velocity-at-wall measurements were representative of the entire solids in the annulus from wall to spout. This procedure was subsequently vindicated by Thorley et al. (1956, 1959) in their detailed study of solids flow in a n annulus. 8. Thorley et al. also showed that, except near the wall, horizontal components of particle velocity were quantitatively discernible in their 24-inch diameter column throughout the entire annulus below the fountain. .4 similar observation can be made qualitatively from Madonna’s streak photographs (1966) on a 6-inch column. 9. Knowing the downward solids flow in the annulus, the upward particle velocity in the spout, and the cross-sectional area of the spout, each as functions of bed level, Mathur and Gishler were able to compute by incremental solid balances the void fraction in the spout as a function of bed level. T h e values thus obtained were in excellent agreement with those calculated by an independent method, according to which gas flow distribution between spout and annulus was determined by assuming pressure drop-gas velocity behavior in the annulus equivalent to that in a loosely packed bed. This agreement is further strong support for considering factors B and C, rather than A, as the relevant causes of the observed particle velocityat-wall gradients. Wherein is the contrary argument of Madonna in favor of factor A vulnerable? T h e derivation of Madonna completely neglects horizontal components of particle velocity in the annulus, thus ignoring the evidence cited in items 6 and 8 above. I t is simply not possible in a 6-inch column for an annular “element. . . to be isolated from the internal spout and the wall in order to eliminate any possible chance of cross flow” (quoted from Madonna, 1966). T h e subsequent simplification of the threedimensional continuity equation, A . p v = 0, to the onedimensional b ( p ) / b j = 0, is therefore invalid, the more so that the integration of the latter equation over the bed depth serves to cumulate the original error in the differential equation (since the cumulative cross flow increases with y for a given downflow). Madonna’s derivation also takes no cognizance of the growth of spout diameter with bed level in the 6-inch column, as cited in item 5 above. T h e numerical results of this derivation are consequently in sharp disagreement with experimental observations. For example, Madonna’s Figure 6, which is the result of applying his derivation to the particle velocity-at-wall data of Thorley et al. (1956) on a 6-inch diameter spouted bed of wheat, shows annulus void fractions for the top three quarters of a 2-foot deep bed ranging from 0.80 to 0.95! VOL. 7
NO. 1
JANUARY 1968
157
Madonna understandably “doubts whether bed depths over 0.3 foot in Figure 6 have validity-the mathematical model assumed is imperfect” (quoted from Madonna, 1966). This understatement is tantamount to discarding 80% of his key graph. Unfortunately, even the remaining 20% of this graph shows results in serious disagreement with experiment-for example, annulus void fractions in the range 0.50 to 0.75 near the bottom of the 2-foot deep bed. As was indicated in item 4 above, even the over-all void fraction of a similar bed, including the high porosity spout along with the annulus, is only 0.46. Madonna’s Figure 7 is essentially a plot of the slopes of the four curves in Figure 5 us. bed depth. T h e fact that the particle velocity-at-wall gradient increases with bed depth for the 6-inch diameter column has been explained by Thorley et al. by the fact that “an increase of 27% was observed in the spout diameter corresponding to an increase in bed depth from 12 to 24 inches,” thus causing an “increase in cross flow, since it provides a larger interfacial area between the annulus and the spout” (quoted from Thorley et al., 1956), and also by the increase in cross flow caused by the higher gas rates required to spout deeper beds. This explanation thus invokes factor B and is consistent with experimental observation. Madonna, on the other hand, explains the same increase of particle velocity-at-wall gradients with bed depth by stating that the higher solids turnover rates associated with higher bed depths serve “to increase the void fraction a t the top of the bed” (Madonna, 1966). This statement has no experimental justification-the acthor is perhaps led to it by the necessity to explain magnitudes of void fractions and void fraction variations in the annulus which simply do not exist. Most of the particle velocity-at-wall data of Thorley et al. for their 24-inch diameter beds show no gradients with bed level. Madonna’s analysis applied to these data would therefore yield no variation of void fraction with bed level. Thus factor A, which Madonna maintains is an inherent property
SIR: T h e comments of Epstein concerning the paper titled “Void Fraction Variation in the Spouted Bed Annulus” are interesting. I t is stated that void fraction variations do exist (even if only of small magnitude). These effects may be important with regard to heat and mass transport characteristics of the bed. I n order to model the bed mathematically for all the transport processes, these variations must be evaluated-where d o we start? Starting with a one-dimensional continuity expression this writer does not expect the data (when they are found experimentally) to match the model. Here an attempt has been made to initiate modeling on the spouted bed; the results are not good, but does Epstein have a better model? No one a t this juncture can even guess at the void fractions near the top of the spouted bed when the internal spout does not oscillate; these void fractions seem very high. If oscillations of the internal spout d o occur (as they do many times) the void fractions must be even higher than for the nonoscillatory mode. T h e experimental and mathematical work has yet to be done in this general area; statements of the real validity of models should be withheld until some data have been collected. 158
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of spouted beds, disappears completely in a 24-inch diameter column. I n fact, the 24-inch column behaves little differently than the 6-inch column as regards void fraction in the annulus. The disappearance of solids velocity gradients can be rationally explained by the decreased potency of factors B and C in the larger column. Thus, as opposed to the 6-inch column, a larger proportion of the 24-inch column cross section is occupied by annulus and hence the fraction of total particles which cross flow into the spout is considerably smaller, and there is also much less variation in spout diameter and hence annulus cross section with bed level in the case of the 24-inch column (Thorley et al., 1956). I n fact, such variation as exists is mainly in the direction to oppose (spout diameter decreases slightly with bed level) the effect of cross flow on downward solids velocity. T h e above criticism of the argument for a large void fraction variation in the annulus does not preclude the possibility, suggested by Thorley et al. (1956), that a slight variation of void fraction may occur in the annulus. This variation would, however, have to be one or two orders of magnitude smaller than that proposed by Madonna, imperceptible to the naked eye, and would not be such as to influence the mechanics of a spouted bed significantly. Literature Cited Happel, J., Ind. Eng. Chem. 41, 1161 (1949). Madonna, L. .4., IND.END.CHEM.PROCESS DESIGNDEVELOP. 5, 47 . . (1966). \ - _ - -
Mathur,K:B., Gishler, P. E., A.Z.Ch.E.J. 1, 157 (1955). Thorley, B., Mathur, K. B., Klassen, J., Gisher, P. E., “Effect of Design Variables on Flow Characteristics in a Spouted Bed,” Natsnal Research Council, Ottawa, Canada, 1956: Thorley, B., Saunby, J. B., Matur, K. B., Osberg, G. L., Can. J . Chem. Eng. 37, 184 (1959). hrorman Epstein Department of Chemical Engineering The University of British Columbia Vancouver, B.C., Canada
Since this writer has only claimed approximations (even if the commenter disagrees as to the type), any statement to the effect that the mathematical result does not come close to the experimental is surprising. Do all engineering mathematical model results check experimental data very closely? I n fact, one is fortunate to see the model results compared to experimental data. One is led to believe, by some of the statements, that experimental void fraction profiles of the annulus do exist-they d o not exist. T h e observations alluded to are calculations made to determine average bed void fraction-not axial profiles. T h e excellent work of Mathur and Gishler in order to obtain flow distributions in the bed is interesting, but several points should be discussed. These experimenters selected a void fraction of 0.432 for the entire bed because they did not have void fraction profile data. Their selection of this value should not be used as arguments against the writer’s data. Even if Thorley did dismiss void fraction variations, it is believed that these same variations are of importance in transport modeling and experimentation in the spouted bed.
