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1、3. Heterogeneous Flow and Separation,3.1 Flow Past Immersed Objects,Definition of Drag Coefficient for Flow Past Immersed Objects,1. Introduction and types of drag,The flow of fluids outside immersed bodies appears in many chemical engineering applications and other processing applications. For exam
2、ple settling, drying and filtration, and so on.,2. Drag coefficient,Correlations of the geometry and flow characteristics for solid objects suspended in fluid are similar in concept and form to the friction factor-Reynolds number correlation given for flow inside conduits.,In flow through pipes, the
3、 friction factor was defined as the ratio of the drag force per unit area to the product of fluid density and velocity head.,For flow past immersed objects the drag coefficient is obtained by substituting CD for the friction factor Kf in equation (1.4-32 ),The Reynolds number for a particle in a flu
4、id is defined as,From dimensional analysis, the drag coefficient of a smooth solid in an incompressible fluid depends upon a Reynolds number and the necessary shape ratios. For a given shape,3.1-2,Drag coefficients of typical shapes,For each particular shape of object and orientation of the object w
5、ith respect to the direction of flow, a different relation of CD versus Re exists.,Correlations of drag coefficient versus Reynolds number are shown in figure.,These curves have been determined experimentally. However, in the laminar region for low Reynolds numbers, less than about 1.0, the experime
6、ntal drag force for a sphere is the same as the theoretical Stokes law equation as follows:,3.1-3,Combining Eqs. (3.1-1) and (3.1-3) and solving for CD, the drag coefficient predicted by Stokes law is,3.1-4,The variation of CD with Re is quite complicated because of the interaction of the factors th
7、at control skin drag and form drag.,For a sphere, as the Reynolds number is increased beyond the Stokes law range, separation occurs and a wake is formed.,Further increases in Re cause shifts in the separation point. At about Re = 3105 the sudden drop in CD is the result of the boundary layer becomi
8、ng completely turbulent and the point of separation moving downstream.,In the region of Re about 1103 to 2105, the drag coefficient is approximately constant for each shape and CD = 0.44 for a sphere.,3.1.2 Flow through Beds of Solids,1. Introduction,A system of considerable importance in chemical a
9、nd other process engineering fields is the packed bed, which is used for a fixed-bed catalytic reactor, adsorption of a solute, absorption, filter bed, and so on,In the theoretical approach used, the packed column is regarded as a bundle of crooked tubes of varying cross-sectional area.,The theory d
10、eveloped in Chapter 1 for single straight tubes is used to develop the results for the bundle of crooked tubes.,2. Laminar flow in packed beds,Certain geometric relations for particles in packed beds are used in the derivations for flow. The void fraction in a packed bed is defined as,The specific s
11、urface of a particle av is defined as,For a spherical particle, ,where a is the ratio of total surface area in the bed to total volume of bed (void volume plus particle volume),Since (1 - ) is the volume fraction of particles in the bed,The average interstitial velocity in the bed is u and is relate
12、d to the superficial velocity u based on the cross section of the empty container by,3.1-9,To determine the equivalent channel diameter De, the surface area for n parallel channels of length L is set equal to the surface-volume ratio times the particle volume S0L (1 - ) .,3.1-6,where S0 is the cross
13、-sectional area of the bed,The void volume in the bed is the same as the total volume of the n channels,3.1-7,Combining Eqs. (3.1-6) and (3.1-7) gives an equation for De,3.1-8,For flow at very low Reynolds numbers, the pressure drop should vary with the first power of the velocity and inversely with
14、 the square of the channel size,in accordance with the Hagen-Poiseulli equation for laminar flow in straight tubes.,The equations for u (equation 3.1-9 ) and De (equation 3.1-8 ) are used in the Hagen-Poiseuille equation,or,3.1-11,The true L is larger because of the tortuous path. Experimental data
15、give an empirical constant of 150 for 72,Equation (3.1-12 ) is called the Blake-Kozeny equation for laminar flow, void fractions less than 0.5, effective particle diameter Dp, and Rep 10:,3.1-12,3.2 Motion of Particles through Fluids,Many processing steps, especially mechanical separations, involve
16、the movement of solid particles or liquid drops through a fluid. The fluid may be gas or liquid, and it may be flowing or at rest.,the estimation of dust and fumes from air or flue gas, the removal of solids from liquid wastes, and the recovery of acid mists from the waste gas of an acid plant,Examples are：,Three forces act on a particle moving through a fluid: the external force, gravitational or centrifugal;,(2) the buoyant force, which acts parallel with the external force but in the opposite direction; and,
