Drag expression

The ratio between the bed and particle diameters and the Reynolds number based on bed diameter, superficial velocity, and solid density appear only in the modified drag expression, in which they are combined, see Eq. (40). These parameters form a single parameter, as discussed by Glicksman (1988) and other investigators. The set of independent parameters controlling viscous dominated flow are then... 

Introducing the above drag expression Eq. (146) in the force balance particle equation of unified model (K14), we obtain... 

H(p, q, t) Hamiltonian function in Hamiltonian mechanics h Rep) dimensionless function in particle drag expression (—) interfacial heat transfer due to phase change (J/m s) h specific enthalpy of ideal gas mixture J/kg)... 

Bioavailability is (1) the fraction of an administered dose of a drag that reaches the systemic circulation as intact drag (expressed as F) and (2) the rate at which this occurs. As an i.v. dose is injected directly into the systemic circulation, the bioavailability of an i.v. dose is by definition 100% (F = 1). Eor aU other routes of admiiustration, bioavailability is determined by the extent of drag absorption (being the result of both drag uptake from the administration site and possible first-pass effects see Section ELD.), and varies between... 

A similar set of equations proposed in the context of atomization was given by Liu et al. (44). The correlations derived for spherical particles are a useful basis on which to develop drag expressions for nonspherical particles. Available methods are presented and compared by Chhabra et al. (45). 

When the Stokes drag expression is used the sinq>le force balance Equation (3.2) becomes ... 

The basic concepts of a gas-fluidized bed are illustrated in Figure 1. Gas velocity in fluidized beds is normally expressed as a superficial velocity, U, the gas velocity through the vessel assuming that the vessel is empty. At a low gas velocity, the soHds do not move. This constitutes a packed bed. As the gas velocity is increased, the pressure drop increases until the drag plus the buoyancy forces on the particle overcome its weight and any interparticle forces. At this point, the bed is said to be minimally fluidized, and this gas velocity is termed the minimum fluidization velocity, The bed expands slightly at this condition, and the particles are free to move about (Fig. lb). As the velocity is increased further, bubbles can form. The soHds movement is more turbulent, and the bed expands to accommodate the volume of the bubbles. 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

Flow Along Smooth Surfaces. When the flow is entirely parallel to a smooth surface, eg, in a pipe far from the entrance, only the shear stresses contribute to the drag the normal stresses are directed perpendicular to the flow (see Piping systems). The shear stress is usually expressed in terms of a dimensionless friction factor ... 

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

The concentration boundary layer forms because of the convective transport of solutes toward the membrane due to the viscous drag exerted by the flux. A diffusive back-transport is produced by the concentration gradient between the membranes surface and the bulk. At equiUbrium the two transport mechanisms are equal to each other. Solving the equations leads to an expression of the flux ... 

We now have all the information necessary to develop some working expressions for particle settling. Look back at equation 3 (the resistance force exerted by the water), and the expressions for the drag coefficient (sidebar discussion on page 261). The important factor for us to realize is that the settling velocity of a particle is that velocity when accelerating and resisting forces are equal ... 

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 ... 

Therefore, the inertia forces have an insignificant influence on the sedimentation process in this regime. Theoretically, their influence is equal to zero. In contrast, the forces of viscous friction are at a maximum. Evaluating the coefficient B in equation 55 for a = 1 results in a value of 24. Hence, we have derived the expression for the drag coefficient of a sphere, = 24/Re. 

By definition, the drag foree per unit area on a single partiele at infinite dilution is related to the kinetie energy of the fluid by the expression... 

Equation 2.15 is thus an expression for the terminal veloeity ly in terms of partiele and fluid properties and the drag eoeffieient. 

Equation 2.32 thus reduees to the veloeity of a single partiele for e => 1. For the general ease, however, an expression for the hindered drag eoeffieient, cde, is needed. This ean be obtained as follows. 

Friction factor, dimensionless Flow rate of one phase, GPM Aqueous phase flow rate, GPM Cy clone friction loss, expressed as number of cy clone inlet velocity heads, based on Drag or resistance to motion of body in fluid, poundals... 

The minimum fluidisation velocity of the particles is achieved when the adsorbent becomes suspended in the liquid. This occurs when the drag forces exerted by the upward flow of the liquid phase ate equal to the weight of particles in the liquid. Therefore, at minimum fluidising conditions, it can be described by the following expression ... 

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