Drag function

Partially Compatible Both of the drags function, but at half their normal effective strength. 

One of the most important functions in the description of fluidization is the drag function which measures the ratio of pressure gradient to gas volume flux. The definition of this drag function is discussed in (14) through recourse to the correlation of Richardson (22). 

In the solid phase momentum balance there are physical models needed to describe the solid pressure p, the shear viscosity /i, the bulk viipcosity A, and the drag function (3. These models are briefly described in the following chapters. A more comprehensive discussion and a comparison with experiments can be found in Boemer et al. (1995b). 

The dimensionless drag function is composed of a viscous (laminar) and inertial (turbulent) term and can be specified according to bed material properties. The Ergun correlation [127] found wide application for coal bed characterization, which is... 

The same relation is expressed in Equations (3.91) using the dimensionless Reynolds (Re) and Archimedes (Ar) numbers, see Equations (3.87) and (3.81), where is the drag function of a particle or bed depending on the Reynolds number Re and the bed voidage e. For fluid-particle interaction the following states can be distinguished ... 

Reh [138] suggested the coordinates (3/4) (R fAr) = f(Re) because a single spherical particle has the drag function (Re) = (3/4)Co. Consequently, the ordinate expresses the reciprocal value of the drag coefficient Co, hence... 

Figure 3.7 shows several orientation lines. The f-line of minimum fluidization is given in Equation (3.94). In this case, the drag function described by Ergun [127] (Equation (3.95)) is integrated into the expression. Hence, e f refers to the bed voidage at the state of minimum fluidization. 

The momentum transfer between the phases occurs on the basis of the drag function. negbusi et al. [8] employed the following formulation for momentum exchange between the two phases ... 

The term essentially a drag coefficient for the dust cake particles, should be a function of the median particle size and particle size distribution, the particle shape, and the packing density. Experimental data are the only reflable source for predicting cake resistance to flow. Bag filters are often selected for some desired maximum pressure drop (500—1750 Pa = 3.75-13 mm Hg) and the cleaning interval is then set to limit pressure drop to a chosen maximum value. 

Equation 36 must be corrected for changes in the drop shape and for the effects of the inertia of Hquid flowing through the orifice, viscous drag, etc (64). As the orifice or aperture diameter is increased, d has less effect on the drop diameter and the mean drop si2e then tends to become a function only of the system properties ... 

The drag coefficient has different functionalities with particle Reynolds number Ri in three different regimes (Fig. 14), which results in the following expressions (1). 

Pxampk 2. A smooth spherical body of projected area Al moves through a fluid of density p and viscosity p with speed O. The total drag 8 encountered by the sphere is to be determined. Clearly, the total drag 8 is a function of O, Al, p, and p. As before, mass length /, and time t are chosen as the reference dimensions. From Table 1 the dimensional matrix is (eq. 23) ... 

The drag coefficient Cg can be plotted as a function of the dimensionless product p 5/ R. Thus, equations 40 and 41 are in proper form for direct determination of the speed once the drag is given. 

Suppose that an experiment were set up to determine the values of drag for various combinations of O, p, and ]1. If each variable is to be tested at ten values, then it would require lO" = 10, 000 tests for all combinations of these values. On the other hand, as a result of dimensional analysis the drag can be calculated by means of the drag coefficient, which, being a function of the Reynolds number Ke, can be uniquely determined by the values of Ke. Thus, for data of equal accuracy, it now requires only 10 tests at ten different values of Ke instead of 10,000, a remarkable saving in experiments. 

The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Re = d pii/ where [L = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes Law gives 24... 

Nonsplierical Rigid Particles The drag on a nonspherical particle depends upon its shape and orientation with respect to the direction of motion. The orientation in free fall as a function of Reynolds number is given in Table 6-8. 

The drag coefficients for disks (flat side perpendicular to the direction of motion) and for cylinders (infinite length with axis perpendicular to the direclion of motion) are given in Fig. 6-57 as a Function of Reynolds number. The effect of length-to-diameter ratio for cylinders in the Newton s law region is reported by Knudsen and Katz Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1958). 

Since u and 6 are functions of time, they are written u t) and 6 t). The constant a could be calculated if the following vehicle data for engine torque T, wheel traction force F, aerodynamic drag D were available ... 

CR 3nd tp are the contributions from chain recoiling and interfacial dynamics (i.e. drag forces and disentanglement), respectively, and / ve is the viscoelastic loss function which has interfacial and bulk parts. / is a characteristic length of the viscoelastic medium, t is the contact time and n is the chain architecture factor. Fig. 21 illustrates the proposed rate dependency of adhesion energy. 

The resistance to liquid flow aroimd particles may be presented by an equation similar to the viscosity equation but with considering the void fraction. Recall that the shear stress is expressed by the ratio of the drag force, R, to the active surface, K27td. The total sphere surface is Ttd and Kj is the coefficient accoimting for that part of the surface responsible for resistance. Considering the influence of void fraction as a function 2( ). we obtain ... 

When we consider many particles settling, the density of the fluid phase effectively becomes the bulk density of the slurry, i.e., the ratio of the total mass of fluid plus solids divided by the total volume. The viscosity of the slurry is considerably higher than that of the fluid alone because of the interference of boundary layers around interacting solid particles and the increase of form drag caused by particles. The viscosity of a slurry is often a function of the rate of shear of its previous history as it affects clustering of particles, and of the shape and roughness of the particles. Each of these factors contributes to a thicker boundary layer. 

Figure 2.2 Drag coejficients for the sphere as a function of particle Reynolds number...

The drag coefficient also depends on shape and 0(, but in addition, because drag is partially due to friction, and frictional effects in a flow arc governed by a powerful dimensionless quantity called Reynolds number, then Cu is also a function of the Reynolds number. Re ... 

FIGURE 1.2 Schematic diagram of potential drag targets. Molecules can affect the function of numerous cellular components both in the cytosol and on the membrane surface. There are many families of receptors that traverse the cellular membrane and allow chemicals to communicate with the interior of the cell. 

FIGURE 9.2 Histograms showing the number of new drag entities entering phase I clinical development (blue bars), and concomitantly the number entering phase III development as a function of year. Adapted from [2]. 

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