This article focuses on processes that do not require a high flow rate of sweep gas; the complex-mode vibration-fluidized bed offers lower power needs, attrition rates, and elutriation rates than gas-fluidized beds or rotary kilns. The fluidized solids are induced to flow horizontally by inclining the trough and/or axis of vibration downward in the direction of solids flow. When viewing the operating region through a window in the side wall, the particles in the bed move in unison, like a column of marching soldiers. In light of the complexities and uncertainties in the interaction of adjacent particles in the fluidized bed, the dynamics of a single particle falling on a horizontal plate vibrating along a vertical axis should be considered first. Because the complex-mode vibration-fluidized bed can be tailored for certain applications, a number of projects are currently in the early stages of development. Promising uses include coal pyrolysis to produce fuel for gas turbines in combined-cycle power plants, the manufacture of char for superior activated carbon, recycled synthetic fiber in carpeting, and counterflow heat exchangers.
Most vibrating machines that transport granular solids and powders use a simple linear vertical motion to fluidize the material. The fluidized solids are induced to flow horizontally by inclining the trough and/ or axis of vibratiot1 downward in the direction of solids flow. When viewing the operating region through a window in the side wall, the particles in the bed move in unison, like a column of marching soldiers . With little or no fine-grain turbulence, there is very little mixing-just a plug flow down the vibrated trough. Because of the lack of mixing, these conventional vibration- fluidized beds are not used for chemical processes other than transport or drying.
To find a better system for processes requiring high rates of mixing and/ or heat transfer coupled with low rates of particle attrition and elutriation, combinations of vibratory motions have been researched, including linear, whirling, oscillatory, pitching, and rocking modes. A series of analyses and experiments have demonstrated high rates of mixing and heat transfer with a wide range of flow circulation patterns. Some of these patterns have shown major advantages over gas-fluidized beds and rotary kilns.
For processes that do not require a high flow rate of sweep gas, the complex-mode vibration-fluidized bed reduces the power requirement by a factor of 10, the attrition rate by a factor of 100, and the elutriation rate (the fraction of the finer particles carried off by the gas leaving the bed) by a factor of 10,000. Development is enor-mously complex, however, because there are more than 20 independent design variables with millions of combinations, including vibration frequency and amplitude; different vibration modes; particle size and size distribution; particle shape; particle physical properties such as density and elastic modulus; bed depth, width, and length; bed-floor detail geometry and inclination; sweep-gas flow rate; inflow rates of solid streams; and the shape, character, and location of feed spouts for solids.
Analyses of vibrating systems can be performed using rather complex mathematical models, but such models generally are rough approximations or become so intricate that a definitive solution is unobtainable. The interrelationships between the principal variables are rendered so complex by facto rs such as friction between particles that purely analytical solutions are out of the question. On the other hand, a purely cut-and-try experimental approach is bewildering.
For an investigation to be successful, mathematically tractable analytical approximations must be developed and used to outline promising experiments and indicate probable trends of the effects of major variables. Experiments must then be conducted with good instrumentation to determine the effects of modulating design parameters.
A Single Particle's Motion
In light of the complexities and uncertainties in the interaction of adjacent particles in the fluidized bed, the dynamics of a single particle falling on a horizontal plate vibrating along a vertical axis should be considered first. The motion is similar to that of a tennis ball kept bouncing by small rapid vertical motions of a tennis racket, but at a much higher frequency and lower amplitude. While falling back onto the plate from the top of its trajectory, only gravity acts on the particle; its motion in this portion of its path is easily defined. Acceleration is g, velocity is v = gt, and the distance it falls is 5 = gt2/2, where time t is measured from the beginning of the fall. The elapsed time for the fall will be half that for the cycle, or %0 second for the 30-hertz forced vibration that can be taken as typical. Thus the distance that the particle falls in that time interval is 1.36 millimeters, and the particle's downward velocity at the point of impact is 16.3 centimeters per second.
Ideally, for the particle to bounce back and provide continuity for steady-state conditions, the initial upward velocity of the rebounding particle must be the same as that approaching the impact, and the deceleration rate after it bounces up from the plate must be 1 g. Therefore, ideal plots of the distance-flight time relation for the ascent and the descent will be mirror images.
Contact between a single particle and the plate during impact will last only approximately Y2 millisecond. This amount of time is vital in not only particle motion but also heat transfer, and thus merits a quantitative estimate. This estimate can be made using Timoshenko's theory of elasticity, which gives the relations for calculating the contact time of two spheres. Following this theory, a good approximation can be obtained by taking the radius of the smaller sphere to be one-tenth that of the larger sphere. Accordingly, the duration of the impact for a pellet of silicon dioxide striking an aluminum plate is 0.00042 second, or only about 1 percent of the time for one cycle.
The motion of a vibrating-bed floor is usually sinusoidal because the driving force is commonly from either a solenoid ac magnet or from off-center weights on a pair of synchronous motors, running in opposite directions so their respective horizontal shaking forces cancel but their vertical forces are additive. With the 3-hertz, 3-millimeter double-amplitude condition chosen as typical, the vertical displacement of the vibrating table from the mean position is a sin θ, where a is the amplitude of the motion and θ is the angular displacement of the rotating weight from the initial horizontal position (in radians) . The velocity is thus a cosθ and acceleration is a(60n)2 sin θ.
Diagrams showing the displacement, velocity, and acceleration of the particle and the vibrating plate as a function of time are helpful in appraising the effects of the principal design parameters on the motion of a bouncing particle. For ideal elastic impacts, the particle would impact when the plate is at the top of its stroke, but hysteresis losses in the stressed zones cause energy losses. To compensate for these losses, the impact on the plate must occur when the upward velocity of the plate is enough to make the upward velocity of the particle equal to the downward velocity at impact. Simple experiments with small steel balls from ball bearings, glass beads, and the like indicate that rebound h eight from a static aluminum plate is only about 75 percent of the height from which the sphere is dropped. For purposes of this example, phasing the impact 40 degrees before the plate reached its peak upward displacement was assumed to illustrate the particle motion for a typical case. In this case, the acceleration of the particle during impact (which is less than 1 millisecond) is approximately 100 g.
A substantial bed depth does not have as much effect as might be expected because momentum is transferred through stacks of particles in elastic compression waves with virtually no motion of the intermediate particles. Frictional losses between columns of particles in a vibrated bed increase with the depth of the bed. Such losses vary with the particle size, size distribution, and the shape and physical properties of the particle and casing materials.
For bed depths up to 150 particle diameters, these hysteresis losses can be accommodated by changes in the phase relation between the motions of the bouncing particle and of the floor of the bed. In other words, with greater frictional losses, the impact point must occur earlier in the plate-vibration cycle for the relative velocity at impact and the momentum -input to the particles-to be increased. The particle hits the floor at a point in the cycle when the floor 's vertical velocity is dropping rapidly to zero, which makes a small change in the phase angle of the impact. This has a large effect on the relative velocity and thus on the energy input to the particle. The system is therefore relatively insensitive to variations in the frictional losses over a very wide range.
If a fluidized bed is needed in a process that requires the granular material to be mixed or circulated, complex modes of vibration may work well enough to justify the extra effort required to design the appropriate machinery. A variety of bed-flow patterns in simple- and complex- mode vibration-fluidized beds are possible, and each can be selected for a specific application.
For example, a complex-mode pattern with combined whirl and oscillatory motions, a level floor, and a weir at one end has been designed to ingest shredded nylon for depolymerization in the recycling of used carpeting. A bubbling gas-fluidized bed had been tried, but the light nylon fluff was simply blown out of the bed by the fluidizing gas flow before it could react with the depolymerizing catalyst. In tests with a favorable flow pattern, the flow into the trough between the two vortices swallowed the nylon fluff, and there was no sweep gas to blow it out of the bed.
Circulation in a bed with the floor horizontal and the outlet partially blocked to produce a nearly constant bed depth can be induced by a variety of complexmode vibratory motions, including combinations of linear, whirl and oscillatory motion. The key factor is the horizontal velocity component of the bed floor near the top of its stroke. The circulation rate is at local velocities of approximately 2 centimeters per second unless the bed is inclined steeply upward to produce strong bunkering. If that occurs, a relatively thin layer at the top of the bed courses down the slope at a much higher velocity, up to 1 meter per second. In any case, a substantially higher circulation velocity can be obtained with complex-mode vibration than with inclined linear vibratory motion.
A set of shaft and eccentric-weight configurations have been used to generate a wide variety of vibratory motions. A single- shaft arrangement will generate a pure whirl if the center of gravity of the mass being vibrated is coincident with the axis of rotation. If the center of gravity is shifted away, the rotating force will still generate a whirl at the axis, but the moment arm between the axis of rotation and the center of gravity will produce a superimposed oscillatory motion. The amplitude of this oscillation will be directly proportional to the radial distance between the shaft centerline and the center of gravity.
Complex three- and five- shaft machines allow the generation of any motion likely to be needed. These machines hold a crucial advantage for long vibrating beds: They are insensitive to shifts in the position of the center of gravity of the vibrating mass for a given set of positions of the eccentric shafts. This is the key factor in the magnitude of the pitching moment that determines the amplitude of the oscillatory motion for simpler one- and two- shaft configurations. For the two-shaft configuration, for example, the degree of bunkering of the bed might easily increase the distance between the center of gravity of the assembly and the line of action of the vibration driving force from zero to perhaps 50 millimeters. This condition might increase the amplitude of the oscillation of a point at the end of the casing from zero to even more than the basic vertical component of the total amplitude on the line of action.
Experiments have shown that even a small oscillation can have large effects on the bed-flow pattern. However, if just the end pairs of the five- shaft configuration were used to give the same average vertical amplitude for the entire mass, the shift in the center of gravity would cause only a small percentage increase in the effective mass at one end compared with the other. This would lead to a small percentage difference in the amplitude at the two ends, because each end would be driven by its own eccentric weights. Furthermore, this difference would be in phase rather than 180 degrees out of phase, as it is for one- and two-shaft configurations. Therefore, the bed-flow pattern with three- and five- shaft configurations is essentially unaffected by a small shift in the position of the center of gravity due to the bed bunkering.
The central shaft in the more-complex arrangements makes it possible to halve the effective free span of the beam represented by the bed casing, increasing its natural frequency by a factor of eight. Substantial amplitudes of motion in vibratory bending can develop if the casing is operating near its first mode frequency, which may grossly distort the bed- flow pattern, but the central shaft eases the structural difficulties in getting the natural frequency of the casing in bending up to at least three times the driving frequency. The mounting points commonly act as nodes, or points of minimum amplitude.
When a beam-bending vibration mode developed in a trough 200 millimeters deep and 1,500 millimeters long, for example, one end of the beam bowed upward while the other bent downward with a node at the middle and nodes at support points approximately 300 millimeters in from each end. Because particles tend to move away from the region of high amplitude toward a node, a bed that was supposed to be uniformly 50 millimeters deep built up such a large mass in the center that the bed material flowed over the 200-millimeter-high sides of the trough.
Higher orders of beam-bending modes may develop; this can also occur in diaphragms and shaft torsional vibrations. Thus, the engineer should ensure the natural frequency of the first mode for any vibratory motions that can be anticipated is at least three times the design frequency.
The stresses associated with the vibratory motions of interest are not high in well-designed machines because the accelerations are low-rarely more than 6 g.
These beds show promise for combined-cycle power plants, counterflow heat exchangers, and other uses.
Power Requirements and Mounting
The centrifugal-force vectors acting along the centerline of the eccentric arm clearly must equal the product of the radial acceleration and the mass concentrated at the centers of the shaft and the eccentric weight. The torque forces required to drive the vibration are less easily visualized, however. This motion is driven every bit of the way. A force normal to the axis of the eccentric arm is required throughout the circular path of the eccentric weight to force its change in direction. An equal and opposite force acts at the end of the lever arm that extends from the center of gravity of the assembly to the shaft center. This force thus acts continuously to change the direction of motion of the main mass of the machine.
The lever arm is equal to the amplitude of the vibratory motion, and the power required is the product of the mass of the machine, the acceleration, and the velocity. The acceleration is the same as the radial acceleration but 90 degrees out of phase. Therefore, the power for a pure whirl is 2n times the power for the lever arm. For simple linear motion genera ted by two shafts running in opposite directions, the horizontal forces are balanced so that the power is reduced to four times the product of the amplitude, frequency, mass, and acceleration.
The actual measurements of power consumption have correlated closely with calculated values that included allowances for the efficiency of the motor and frictional losses in the bearings, which are rather heavily loaded.
The product of the vibration amplitude and the mass of the machine exactly equals the product of the eccentric mass and its radius of motion. This radius is the length of the eccentric arm minus the amplitude of vibration of the machine.
A major requirement in installing a vibrating machine is isolating the vibratory motion from the building, which can be accomplished with an elastic mounting at each of the four corners of a rectangular machine. To provide well-defined control of the vibration amplitude, these mountings should provide flexibility such that the natural frequency of the vibrating assembly will be one third of the operating frequency of the machine or less. This condition, however, gives a large amplitude of motion in going through the natural frequency during start-up. The large amplitude is a brief transient, and the peak amplitude can be limited by using rubber as the elastic element.
Both hysteresis losses in the rubber and its nonlinear spring constant make important contributions. For a machine that is designed to operate at 30 hertz, the natural frequency for the machine on its rubber suspension should be no more than 10 hertz-and preferably less-so the elastic static deflection should be at least 0.1 inch (the best deflection is 0.5 inch). If the amplitude of the motion in the start-up transient is excessive, it can be limited to an acceptable value by using a rubber snubber or bumper.
The design of the rubber mounting partly depends on the vibratory motion chosen. For simple linear motion, commercial units based on a rubber toroid inflated with air to the desired pressure provide a variable spring constant that can be set to give the static deflection and mounting natural frequency desired. They lend themselves nicely to mounting under the base of the vibrating machine. These toroidal pneumatic rubber mountings are flexible in shear and axially, so they can be used with many vibratory motions. The machine can also be suspended from rubber straps inclined at angles chosen to vary the ratio of the spring constants in the vertical and horizontal directions for certain complex vibratory modes.
To avoid driveshaft whipping, the motor mounting must ensure reasonably good alignment of the motor with the driven shaft on the vibrating machine. With a static deflection of 1/2 inch in the rubber mountings and a vibration amplitude of)/16 inch, the mounting structure should serve to position the motors rigidly with respect to the supports for the rubber mountings. The mass of the quill shaft and the universal joints joining the motor to the vibrating machine should be kept to a small fraction of the eccentric weight to avoid problems with whipping and an undesirable input to the vibratory motion of the fluidized bed. For small machines, short lengths of rubber hose that fit tightly on tubular aluminum shafts make light universal joints that have given excellent service with motors up to at least 1.5 horsepower.
Synchronous three-phase motors with electronic controls can be used to synchronize multiple shafts. A lighter, less expensive, and more easily maintained system uses ordinary 1,750-rpm induction motors coupled with a lay shaft and bevel gears. If the total power required is transmitted through gears of any sort, the gears introduce so much noise in oscillograph images from accelerometers that the oscillographs are difficult to interpret. If double-ended motors are used with one end coupled to the vibrating machine and the other. to the synchronizing shaft, the gear loads are reduced to a few percent of the power input to the vibrating machine, and the noise in the oscillograph images can be reduced to a tolerable level.
Because the complex-mode vibration-fluidized bed can be tailored for certain applications, a number of projects are currently in the earl y stages of development. Promising uses include coal pyrolysis to produce fuel for gas turbines in combined-cycle power plants, the manufacture of char for superior activated carbon, recycled synthetic fiber in carpeting, and counterflow heat exchangers.
This article is adapted from a paper presented at the 1996 ASME International Mechanical Engineering Congress and Exhibition held in Atlanta.