CORRESPONDENCE Heat Transfer in Fluidized Beds Sir: The authors assume that Nu > 2 is necessarily correct. They then question all that has gone before as being inconsistent because it does not support their conclusions. Why should Nu > 2? The authors do not examine this aspect of the problem. If they had, they might be inclined to reconsider their conclusions. Consider conduction in a n infinite medium around a spherical, steady source of heat, q: q = -4 m 2 k ( d t / d r )
integrating, q = 4 rrk(t1 - t 2 ) / ( 1 / r 1 - l / r 2 ) ; as
r2 + m ,
.:
> rl
q --t 4 akrl(t1 - tz)
q = 4 rrr12h(tl
now,
r2
- tt)
hrI2 = krl, or hrl/k = 1, or hd.k = 2 = Nu
This seems to be the source of the rule, Nu > 2. Is it reasonable to conclude that particle-to-fluid heat transfer in a fluidized bed is governed solely by conduction in a stagnant medium? O r is it more reasonable, in the light of much research which indicates Nu < 2 , either to seek some rational explanation for the discrepancy or to present the results of a n experiment which shows Nu > 2? Let it be supposed, for the moment, that Nu > 2 . I t is not difficult to conceive of a plausible mechanism for the observed low coefficients. All it takes is agglomeration. T h a t is not to say that agglomeration is the mechanism for the observed behavior. Agglomeration is just a handy postulate here, for the sake of argument. If the particles agglomerate to size D = nd, the necessity for Nu > 2 requires that the effective heat transfer coefficient for the agglomeration be less than that for a single particle (by the factor l / n ) . Also, the effective surface area presented by the agglomeration is less than the sum of the areas of the individual particles making up the agglomerates. Therefore, the product of effective heat transfer coefficient and effective heat transfer area is less for the agglomeration than for the individual particles. Further, for the same temperature difference, agglomeration transfers less heat and gives the appearance of low heat transfer coefficients for the individual particles. The requirement that Nu > 2 does not take into account motions by the particles or the fluid. Isn't it possible that there may be transient effects, brought about by the motions influencing the heat transfer? Only 17 of the references in the article by Barker, to which the authors refer, mention fluid-to-particle heat transfer coefficients. Those are the ones in Table I and in Figure 1 of that article. The text of that paper points out that 12 of those 17 should be considered as representative of relevant data.
While it is true that the great majority of fluid bed applications can dismiss the fluid-to-particle temperature difference as completely negligible, there are exceptions, particularly in the nuclear field. JAMES J. BARKER Consulting Engineer 70 Walden A m . Jericho, L . I., N . Y.
Professor Harriott's Refily Sir; For heat transfer from a single sphere to the surrounding fluid, the minimum Nusselt number is 2.0, corresponding to conductions in a n infinite stagnant medium. The heat transfer coefficient is based on the heat transferred per unit surface of the sphere and the temperature difference between the surface and the fluid a t infinity. If the particle is confined in an enclosure or if the fluid flows past the particle, the coefficient and the Nusselt number are increased. The question considered is: Are there any conditions under which heat transfer from a particle to the surrounding fluid would give a Nusselt number less than 2.0? Transient Effects. Introducing a particle into stagnant fluid at a different temperature gives rise to temperature transients. For short times, the instantaneous coefficient is large and varies with t-"', approaching the limiting value (NNu= 2 ) for long times. For a particle in a flowing stream, the corresponding solutions are not available, but exposing the particle to temperature cycles would probably always increase the average coefficient, since the solutions for very rapid cycling would approach that for a particle in stagnant fluid a t short contact times. Fluctuations in velocity, u, might lead to a lower coefficient, h, than a constant velocity, since h varies with a fractional power of u. However, the Nusselt number would never be less than 2.0, the limit for u = 0. Shape Effects. A single particle of approximately spherical shape but having a rough surface can have a Nusselt number less than 2.0 based on the total exterior surface. However, the Nusselt number based on the area of a sphere of the same nominal size would be 2.0 or greater. Irregular particles such as grains of sand may have 1.1 to 2 times as much surface as a sphere of the same nominal size and might therefore have minimum Nusselt numbers of 1.8 to 1 . This small effect is not a n explanation for the very low Nusselt numbers reported for fluidized beds. Agglomeration Effects. Two spheres touching each other will transfer less heat to a flowing or stagnant fluid than two widely separated spheres, and the minimum Nusselt number based on total area will be about 1.5 (the exact solution is given in a recent issue of Trans. VOL. 5 9
NO. 4
APRIL 1967
59
Inst. Chem. Engrs.). Agglomerates of many particles
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will have a minimum apparent hTusselt number roughly equal to two times the area of the envelope surrounding the particles divided by the area of the particles. However, if agglomeration occurs, there are really three heat transfer processes to consider : transfer from particles to gas within the agglomerate, transfer across the agglomerate, and transfer from the outer layers of particles to the outside fluid. The coefficients for the first and third stages of the transfer will corrPspond to Nusselt numbers greater than 2 based on the particle diameter and agglomerate diameter, respectively. The coefficient for the second stage would decrease with increasing agglomerate size and would generally be the limiting factor in heat transfer. If experimental data for heat transfer from a fluid stream to agglomerates are treated by assuming plug flow of gas and a single coefficient based on total particle area, apparent Nusselt numbers much smaller than 2.0 may result. However, these apparent values should not be used to predict local particle temperatures for a chemical or a nuclear reactor where the pattern of heat generation differs from that of the test, and where the temperature gradients within the agglomerates may be important. A better model is needed. Although agglomeration can lead to low apparent Nusselt numbers, it is probably not an important factor in the experiments on fluidized bed haat transfer. With appreciable agglomeration, the minimum fluidization velocity, Vm.f., would be much higher than predicted, since for small size, Vm.f. varies with the square of the particlc size. I n general, the minimum velocities reported in published studies fit the same correlation. Furthermore, to explain a Nusselt number as low as 0.02 would require agglomerates about 200 times as large as the particles. D e p a r t u r e from Plug Flow. If the fluid velocity in some parts of a bed is much less than the average, the fluid will come almost to equilibrium with these particles, but not so close to equilibrium where the velocity is high. The average apparent coefficient will be lower than the correct value for the average velocity. Similarly if backmixing of fluid occurs because of axial conduction, eddying of fluid, or motion of the solid, the actual driving force will be reduced, and coefficients calculated using the log-mean driving force (the plug flow equation) will be too low. In summary, either departure from plug flow or agglomeration can lead to low apparent Nusselt numbers. However, these apparent coefficients should not be used to predict temperature differences for conditions that differ from those of the tests, even if the same incorrect model is used. Such usage would correspond to analyzing kinetic data for one reaction using plug flow equations, when complete backmixing really existed, and then applying the incorrect kinetic constants to other flow rates or conversions. PETER HARRIOTT Cornell University Ithaca, N . Y.
60
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