Terminal entrainment velocity

The calculation of the terminal entrainment velocity is necessary for design of all gasification systems. In the case of moving-bed gasifiers, the load-limiting particle carry-over is determined by particle entrainment. In fluid-bed systems— probably the most critical—the terminal velocity is directly related to the gasifier discharging and carbon conversion rate. Finally, in entrained-flow systems, all particles must fulfill the entrainment conditions. 

To determine the terminal entrainment velocity of a particle, the dynamic force balance on the particle must be solved, where the summation of drag force and buoyancy force equals the gravity. 

The entrainment velocity Ue is calculated fi-om the particle diameter according to Equation (3.70), the apparent coal density p, the gas density p, the gravitational acceleration g, and a drag coefficient Cd- The drag coefficient can be estimated by the correlation from Morsi et al. [135] 

The shown relationships are intended for a very broad range of conditions. Particularly for particle entrainment from fluid-bed gasification, the empirical correlations in Equation (3.88) according to Kunii and Levenspiel [134] and Equation (3.89) according to Martin [137] may give better results than the more general formulations [17]. 

It must be noted that the Archimedes number Ar T is based on the Sauter diameter referring to Equations (3.81) and (3.69) and y/ denotes the dimensionless sphericity of the particles. 

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The control parameters for the experiment described in Figure 15.6 are the mass of particles per unit surface, Mp / S, the area-averaged empty-bed velocity of the fluid in the column, U = Q/S, the density ps of the particles, and the density pf of the fluid. The diameter/) of the particles, as will be seen in section 15.6.3, chiefly influences the fluid flow in a porous medium, jointly determining with the kinematic viscosity v of the fluid the minimum fluidization velocity U f and the terminal entrainment velocity Ut, between which a steady-state fluidization regime is obtained. Furthermore, for a given velocity U of the fluid, the thickness of the fluidized bed increases if the diameter of the particles decreases, for the same mass of particles in the colunrn. 

It is readily understood that the terminal entrainment velocity Ut is equal to the fall velocity Wc of the particles constituting the bed. The fall velocity is the maximum relative velocity that a particle can have in a fluid at rest, whereas the terminal entrainment velocity is the maximum relative velocity that the fluid can have with respect to a stationary particle without carrying it away. The classical laws for the fall velocity of a spherical particle in a fluid at rest (Stokes , Van Allen s, or Newton s law - see section 15.1 and Table 15.1) are therefore used to estimate the terminal entrainment velocity. 

T able 15.2. Correlations characterizing the fluidflow in a fluidized bed U is the mean streamwise velocity, D the diameter of the particles, U t the terminal entrainment velocity (equal to the fall velocity of the particle — see Table 15.1), and 0 the diameter of the column. Ret = UtD I v. 

Dilute This is a fully expanded condition in which the solids particles are so widely separated that they exert essentially no influence upon each other. Specifically, the solids phase is so fuUy dispersed in the gas that the den sity of the suspension is essentially that of the gas phase alone (Fig. 12-29). Commonly, this situation exists when the gas velocity at all points in the system exceeds the terminal setthng velocity of the solids and the particles can be lifted and continuously conveyed by the gas however, this is not always true. Gravity settling chambers such as prilling towers and countercurrent-flow spray diy-ers are two exceptions in which gas velocity is insufficient to entrain the sohds completely. 

The top surface of the suspension should always be sufficiently far below the level of the overflow weir to provide a clear zone deep enough to allow any entrained particles to settle out. As they will be present only in very low concentrations, they will settle approximately at their terminal falling velocities. Provided that this requirement is met, the depth of the thickener does not have any appreciable effect on its clarifying capacity. 

Whereas for bubbling fluidized beds the solids holdup in the upper part of the reactor and the entrainment of catalyst are often negligible, these features become most important in the case of circulating fluidized beds These systems are operated at gas velocities above the terminal settling velocity ux of a major fraction or even all of the catalyst particles used (% 1 m s 1 < umass flow rales to be externally recirculated are high, up to figures of more than 1000 kg m 2s-1... 

The eruption of these bubbles at the bed surface is responsible for the ejection of particles of all size classes into the freeboard. Very fine particles may even be entrained without the assistance of a bubble, if their terminal falling velocity is below the superficial gas velocity in both the bed and the freeboard. 

Fluidized This is an expanded condition in which the solids particles are supported by drag forces caused by the gas phase passing through the interstices among the particles at some critical velocity. The superficial gas velocity upward is less than the terminal setting velocity of the solids particles the gas velocity is not sufficient to entrain and convey continuously all the solids. Specifically, the sohds phase and the gas phase are intermixed and together behave as a boU-ingjluid (Fig. 12-28). The gas forms the continuous phase, but the bulk density is not much lower than a continuous packed bed of solids. 

This approach is usually not practical for drops less than 100 pm, since for most situations large cross-sectional areas are needed to reduce the bulk velocity below the terminal settling velocities of such drops. An air stream with a 10 ft/s vertical velocity will entrain water drops less than about 700 pm, while a bulk velocity of 1.0 ft/s will entrain drops of about 100 pm or less. Gravity techniques can be used to remove large quantities of die latger-size dispersed-phase materia prior to some other segregation technique. 

The flow rate in a fluidized bed is limited on one hand by the minimum and on the other by entrainment of solids from the bed proper. This maximum allowable velocity is approximated as the terminal settling velocity v[ of the particles. (See Section 13.3 for methods to calculate this settling velocity.) Approximate equations to calculate the operating range are as follows (P2). For fine solids and AfR.y < 0.4,... 

For the coarse particles with a terminal settling velocity Ut above the superficial gas velocity in the freeboard U, it is often assumed that they caimot be entrained if the freeboard is sufficiently high, but measurements by Geldart and Pope (1983) and Geldart et al. (1979) have shown that coarse particles may be entrained, too, if there is a sufficient entrainment flux... 

It is clear from an examination of these empirical correlations that the terminal fall velocity is an important factor in determining the value of the elutriation rate coefficient for any given size of particle. It is also apparent that the rate coefficient will increase as decreases, so that increasing pressure would be expected to increase the rate of elutriation. Chan and Knowlton (1984) studied elutriation from a bed of Ottawa sand with a wide size distribution at pressures of up to 31 bar. They found the solids entrainment rate to increase significantly with increasing pressure and fluidizing gas velocity (Fig. 6). 

The ratio has been shown to vary from about 90 for small particles to about 10 for large particles, which represents the theoretical limits of operation. In practice, these limits may be even narrower if elutriation is to be avoided because most practical fluidized beds contain a range of particle sizes. Equations (7.5.5)-(7.5.7) provide the theoretical limits of the superficial gas velocity at which elutriation can take place. Practical experience has shown, however, that in fluidized beds, the particles whose terminal falling velocity is smaller than the superficial gas velocity are not entrained (or elutriated) immediately but rather the elutriation occurs at a finite rate. 

Flue particles ia a fluidized bed are analogous to volatile molecules ia a Foiling solution. Therefore, the concentration of particles ia the gas above a fluidized bed is a function of the saturation capacity of the gas. To calculate the entrainment rate, it is first necessary to determine what particle sizes ia the bed can be entrained. These particles are the ones which have a terminal velocity less than the superficial gas velocity, assuming that iaterparticle forces ia a dilute zone of the freeboard are negligible. An average particle size of the entrainable particles is then calculated. If all particles ia the bed are entrainable, the entrained material has the same size distribution as the bed material. 

In practice, the entrained material is enriched ia fines even when the entire bed is entrainable. However, as the gas velocity is iacreased to many multiples of the terminal velocity, the composition of the entrainable material approaches the bed composition. 

Commonly, the most important feature of a nozzle is the size of droplet it produces. Since the heat or mass transfer that a given dispersion can produce is often proportional to (1/D ) , fine drops are usually favored. On the other extreme, drops that are too fine will not settle, and a concern is the amount of liquid that will be entrained from a given spray operation. For example, if sprays are used to contact atmospheric air flowing at 1.5 m/s, drops smaller than 350 [Lm [terminal velocity = 1.5 m/s (4.92 ft/s)] will be entrained. Even for the relative coarse spray of the hoUow-cone nozzle shown in Fig. 14-88, 7.5 percent of the total hquid mass will be entrained. 

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