1、3.Heterogeneous Flow and Separation 3.1 Flow Past Immersed ObjectsDefinition 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 example
2、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 friction
3、factor was defined as the ratio of the drag force per unit area to the product of fluid density and velocity head.220 220 For flow past immersed objects the drag coefficient is obtained by substituting CD for the friction factor Kf in equation(1.4-32)2222uCuKpDff3.1-1 therefore2/2uCAFpDDfThe Reynold
4、s number for a particle in a fluid is defined as0ReuDpFrom 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(Re)DC3.1-2 Drag coefficients of typical shapes For each particular shape o
5、f object and orientation of the object with 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
6、,less than about 1.0,the experimental drag force for a sphere is the same as the theoretical Stokes law equation as follows:03uDFpD03uDFpDpDDuF033.1-3 Combining Eqs.(3.1-1)and(3.1-3)and solving for CD,the drag coefficient predicted by Stokes law isRe24242/020uDuAFCpD3.1-4 The variation of CD with Re
7、 is quite complicated because of the interaction of the factors that 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 s
8、udden drop in CD is the result of the boundary layer becoming 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.Introduct
9、ion A system of considerable importance in chemical and 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 tube
10、s of varying cross-sectional area.The theory developed 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 i
11、n a packed bed is defined as The specific surface of a particle av is defined asppvSaFor a spherical particle,pvDa62ppDS36ppDv)1(6)1(pvDaawhere 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
12、bedThe average interstitial velocity in the bed is u and is related to the superficial velocity u based on the cross section of the empty container by uu 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 rati
13、o times the particle volume S0L(1-).pveDLSaLSLDn6)1()1(003.1-6 where S0 is the cross-sectional area of the bedThe void volume in the bed is the same as the total volume of the n channelsLDnLSe2043.1-7Combining Eqs.(3.1-6)and (3.1-7)gives an equation for De132peDD3.1-8 For flow at very low Reynolds n
14、umbers,the pressure drop should vary with the first power of the velocity and inversely with the square of the channel size,in accordance with the Hagen-Poiseulli equation for laminar flow in straight tubes.232DuLpThe equations for u(equation 3.1-9)and De(equation 3.1-8)are used in the Hagen-Poiseui
15、lle equation22229413232pDuDuLpor 322172pDuLp3.1-11 The true L is larger because of the tortuous path.Experimental data give an empirical constant of 150 for 72Equation(3.1-12)is called the Blake-Kozeny equation for laminar flow,void fractions less than 0.5,effective particle diameter Dp,and Rep1000,
16、the drag coefficient is approximately constant at 0.40 to 0.45,and letspptdgu75.1(3.2-19 )so the equation isCD=0.44Equation(3.2-19)is Newton law and applies only for fairly large particles falling in gases or low-viscosity fluids.Criterion for settling regime To identify the range in which the motio
17、n of the particle lies,the velocity term is eliminated from Reynolds number by substituting ut from Eq.(3.2-16)to give,for the Stokes law range 2318Repptpgdud（3.2-21 ）Re=K3/18.Re1.0,to provide a convenient criterion K,let312ppgdK（3.2-22）Then,from Eq(3.2-21),Re=K3/18.Setting Re=1.0 and solving gives
18、K=2.6.If the size of the particle is known,K can be calculated from Eq(3.2-22).If K so calculated is less than 2.6,Stokes law applies.Substitution for ut from Eq.(3.2-19)shows that for the Newtons law range Re=1.75K1.5.In the range between Stokes law and Newtons law(2.6K68.9),the terminal velocity i
19、s calculated from Eq(3.2-13)using a value of CD found by trial from Fig.Setting this equal to 1000 and solving gives K=68.9.Thus if K is greater than 68.9,Newtons law applies.problem1lSettling of a spherical particle in a air is followed Stokes law,if the temperature changes from 25 to 50,the termin
20、al velocity will();if settling in liquid,the terminal velocity will()vThe terminal velocity is the velocity that the acceleration that a particle moves through the fluid approaches to()problem2lA single spherical particle settling freely in the fluid and it is laminar flow,when the particle diameter
21、 increases,the terminal velocity u will ;when the viscosity of fluid increases,u will ;if the fluid is a gas,what happens to u if the temperature increase?lAs shown by equations(3.2-16)and(3.2-19),the terminal velocity ut varies with the square of diameter of particle in the()range,whereas in the()r
22、ange it varies with 0.5 power of the diameter of particle l For a given packed bed,Blake-Kozeny equation indicates that the flow is()to the pressure drop and()proportional to the fluid viscosity.1、试计算直径为30m的球形石英颗粒（其密度为2650kg/m3），在20水中和20常压空气中的自由沉降速度。To calculate the terminal velocity of a spherical
23、quartz particle,30 m in diameter and 2650kg/m3 in density,settling in the water and the air at the temperature of 20,respectively.m/s1002.81001.11881.9)9982650()1030(18)(43262gdust11038.21001.19981002.810302346ttduRe Solution:d=30m、s=2650kg/m3（1）=1.0110-3Pas and=998kg/m3for water at t=20checkIt is followed stokes low ut=8.0210-4m/sm/s1018.71081.11881.9)21.12650()1030(18)(25262gdust)210(144.01081.121.11018.710304526ttduRe（2）=1.8110-5Pas =1.21kg/m3 for the air at t=20Assuming that the type of flowing is followed stokes lawcheckut=7.1810-2m/s。
