Abstract
Based on the consideration of intermediate state called "Fluidized Beds", an attempt has been made to represent the fluid friction through fixed beds by the drag coefficient of a single particle. This is because the beds of solid particles change from fixed to fluidized state, with the increase in velocity of upward current, until they disppear in pneumatic transporting suspensions.
The critical velocity where the fixed beds begin to fluidize, umf, is described in the author's previous article17) as independent of the shape of particles or to the voidage of the beds as expressed by Eq. (1). Can the shape and voidage term, then, be eliminated in the same way from the equation of fluid friction through fixed beds? Empirically, the pressure drops in fluidized beds are nearly equal to the density difference of solid and fluid.9), 11), 12) To apply Eq. (1) to the ranges of velocities lower than umf where the solid beds do not fluidize, an attempt is made to substitute the density term of eq. (1) for the pressure drop term of Eq. (2). Then, the pressure drops through loosely packed beds before fluidization can be expressed as Eq. (8), in laminar region, without using the shape or voidage factor of solid particles.
Experiments are made with the apparatus as shown in Fig. 1, and it is found that the data of the friction factor fv" defined by Eq. (8) are nearly constant -about 600- in laminar region. When Eq. (8) is further applied to the transition or turbulent region, the values of fv" remain no longer constant but increase with the increase in fluid velocity or the modified Reynolds' Number, Rep, as shown by curve AB in Fig. 2. The values of fv" for irregular particles are slightly greater than those for smooth spheres in Fig. 2. Recalculated results from literature data are also shown as a comparison in Table 1 and Fig. 3.
Pressure drops in fluidized beds are independent of the fluid velocity and remain nearly constant as indicated in Eq. (2). From Eq. (8), with the help of Eq. (2), the friction factor fv" for fluidized beds is derived which is found to be inversely proportional to the fluid velocity or the modified Reynolds' Number. The results are shown by the straight lines of AC, BD, etc. in Fig. 2. When the fluid velocity is a little higher than the terminal velocity, ut, of the individual particle, all solids are transported by the fluid stream and the fluidized bed disappears. Then, instead of Eq. (8), Eq. (10) becomes applicable to the single particle. The curve CD in Fig. 2 shows the experimental data of fv" for single particles as the limits of the fluidized beds.
In the case of irregular particles generally used, the values of fv" are slightly greater than in that of smooth spheres (dotted line in Fig. 2), similar to the case of fixed beds. From Eq. (10) and Eq. (11), we can derive fv" as follows: 2fv"=(3/4)CDRep…(12) for the terminal velocity of single particles. Likewise, from Eq. (12), a constant value of 2fv"=(3/4)×24=18 can be obtained for smooth spheres, in laminar region.
The above description may be summarized as follows:
a) fv" in fixed beds, Eq. (8), varies along the curve AB with the increase in fluid velocity untill fluidization begins at umf [c.f. Eq. (1) and (2)].
b) In fluidized beds, fv" decreases inversely proportional to the increase in fluid velocity along the straight line AC or BD.
c) Fluidized bed disappears at the point where either AC or BD line intersects the curve CD of single particles.
