PARTICULATE THEOREY...

Transcript

1. CHAPTER 7 An Introduction to Particle Systems PAOLA LETTIERI 7.1 INTRODUCTION Particles and processes involving particles are of enormous importance in the chemical and allied industries. It has been estimated that partic- ulate products, which include pharmaceuticals, detergents, agrochemi- cals, pigments and plastics, generate $1000M annually for the US economy and Du Pont estimates that 60% of its manufactured products rely heavily on particle science and technology. Despite this, particle technology is a relatively new subject and although the science is advancing rapidly there is still much reliance on empiricism. The behaviour of powders is often quite different from the behaviour of liquids and gases. Engineers and scientists are used to dealing with liquids and gases whose properties can be readily measured, tabulated and even calculated. With particle systems the picture is quite different. The flow properties of certain powders may depend not only on the particle size, size distribution and shape, but also on surface properties, on the humidity of the atmosphere and the state of compaction of the powder. These variables are not easy to characterise and so their influence on the flow properties is difficult to predict. In the case of particle systems it is almost always necessary to perform appropriate measurements on the actual powder in question rather than to rely on tabulated data. The measurements made are generally measurements of bulk properties, such as bulk density and shear stress. Given the wide range of industrial applications that involve particle systems, the influence of the operating conditions such as temperature, pressure, velocities, reactor design and any other special conditions, such as the presence of liquid in the reactor, can also significantly affect the 203
2. fluid–particle contacting efficiency and the interaction between particles, thus affecting the overall flow properties of the powders. In light of the complexity of such systems, research in particle tech- nology has been devoted to the development of both experimental techniques and theoretical methodologies to determine the particle properties, the fluid–particle and particle–particle interactions. Substan- tial mathematical effort is currently being directed towards the devel- opment of computational fluid dynamics (CFD) modelling as a new tool to support engineering design and research in multiphase systems. The recent development of mathematical modelling of particle systems behaviour, together with the increased computing power, will enable us to link fundamental particle properties directly to powder behaviour and predict the interaction between particles and gaseous or liquid fluids. The aim of this chapter is to provide a brief introductory guide to the: (i) Characterisation of solid materials (size analysis, shape and density); (ii) Interaction between particles and a fluid; (iii) Fluidisation, in relation to fluidised beds (as an example of a common particle process). A number of relevant worked examples are also presented, to provide the reader with the direct application of the theory presented. 7.2 CHARACTERISATION OF SOLID MATERIALS The starting point of any study of particle systems is the characterisation of particles in terms of their size, shape and density. Figure 1 shows the range of sizes of some commonly encountered particulate materials compared with other quantities, such as the wavelengths of electromag- netic radiation and the sizes of molecules. 7.2.1 Size and Size Distribution Only a minority of systems of industrial interest contain powders with a uniform particle size, i.e. monodisperse. Most systems generally show a distribution of sizes (polydisperse) and it is then necessary to define the average dimension. There are many different definitions of particle size,1,2 the most commonly used, particularly in fluidisation, is the so called volume-surface mean or the Sauter mean diameter. This is the 204 Chapter 7
3. k sulphuric mist p gas molecules colloidal silica ground talc pollens nebulizer drops sea salt nuclei H2 O2 F2 Cl2 CO2 C6 H6 CO N2 H2 OH Cl CH4 SO2 C4 H10 Equivalent sizes Electromagnetic waves Technical definitions Common Particles and gas dispersoids Atmospheric dispersoids Particle diameter, microns 0.0001 0.001 0.01 0.1 1 10 100 (1mm) 1000 (1 cm) 10000 gas dispersoids solid liquid soil dust fume 10 100 1000 Angstrom units Theotretical mesh 10000 625 Tyler Screen Mesh U.S. ScreenMesh X-ray ultraviolet visible near infrared far infrared microwaves solar mist spray smog clouds & fog rain mist drizzle Int. std. classification system clay fine sand coarse sand gravel silt viruses bacteria human hair oil smoke fly ash tobacco smoke coal dust metallurgical dust and fumes cement dust ammonium chloride fume carbon black pulverized coal hydraulic nozzle drops paint pigments Figure 1 Particle size ranges (Adapted from Ref. 5) 205 An Introduction to Particle Systems
4. diameter of a particle having the same external surface-to-volume ratio as a sphere and is given by: dVS ¼ 1 P xi =dpi ð1Þ where xi is the mass fraction of particles in each size range given by the sieve aperture dpi . A number of methods are available for measuring particle size, such as sieving, light scattering and microscopy. In sieve analysis the sample is placed on the top of a stack of sieves whose mesh size decreases with height of the stack. The stack is vibrated and as the powder falls through it is separated into fractions, which are then weighed, thereby giving the mass of particles between each sieve size. Standard sieves are in sizes such that the ratio of adjacent sieve sizes is the fourth root of two (e.g. 45, 53, 63, 75, 90 mm). Other possible geometrical diameters can be used to determine the mean particle diameter of a polydisperse system. Examples are the surface average, ds , and volume average diameters, dv ; where ds is defined as the diameter of a sphere having the same surface area as the particle and dv is the diameter of a sphere having the same volume as the particle. These are given by: ds ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X xi d2 pi q ; dv ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X xi d3 pi 3 q ð2Þ A comparison between the surface average and volume average diameters with the Sauter mean diameter shows that the latter is always skewed towards the finer end of the distribution and, therefore, is the smaller equivalent mean particle diameter in value. Table 1 provides a quantitative comparison of the mean diameters dvs , ds and dv calculated using Equations (1) and (2) for two industrial fluid catalytic cracking catalysts (FCC1 and FCC2)3 whose particle size distribution is shown in Figure 2. See worked example E1 (section 7.5.1) for the determination of the mean particle diameter and size distribution of the FCC1 catalyst sample. Table 1 Geometrical mean particle diameters for two FCC catalysts dvs ðmmÞ ds ðmmÞ dv ðmmÞ FCC1 71 91 102 FCC2 57 104 124 206 Chapter 7
5. The relative diameter spread sd /d50 can then be used to compare the width of size distribution of particles having different mean diameter. Plotting the cumulative size fraction versus particle size, the diameter spread is defined as: sd ¼ d84% À d16% 2 ð3Þ The median size diameter, d50 , is the size corresponding to the 50% value on the graph of cumulative size fraction versus particle size, see example E1. The values of sd /d50 are used to give an indication of the width of the size distribution, as summarised in Table 2.4 7.2.2 Particle Shape This is a fundamental property affecting powder packing, bulk density, porosity, permeability, flowability, attrition and the interaction with Figure 2 Particle size distributions of two industrial FCC catalysts3 Table 2 Width of size distribution based on relative spread sd /d50 Type of distribution 0 Very narrow 0.03 Narrow 0.17 Fairly narrow 0.25 Fairly wide 0.33 Fairly wide 0.41 Wide 0.48 Wide 0.6 Very wide 0.7 Very wide 40.8 Extremely wide 207 An Introduction to Particle Systems
6. fluids. It is generally considered that shape can be described in terms of two characteristics, form and proportions. Form refers to the degree to which a particle approaches some standard such as a sphere, cube or tetrahedron, while proportion distinguishes one spheroid from another of the same class. A simple form of shape factor is the sphericity, f which is defined as the ratio of surface area of a sphere having the same volume as the particle to the surface area of the particle. Thus: f ¼ surface area of equivalent volume sphere surface area of the particle ¼ dv ds  2 ð4Þ and 0 o f o1. dv and ds are as defined in the previous section. The sphericity f can be calculated exactly for such geometrical shapes as cuboids, rings and manufactured shapes, see Table 3. However, most particles are irregular and there is no simple generally accepted method for measuring their sphericity. Values for some common solids have been published (see Ref 4). However, these should be regarded as estimates only. Table 4 shows that f is between 1 and 0.64 for most materials. Other methods are available for quantifying shape factors and these are described in detail in Refs. 2 and 5. Using a Scanning Electron Microscope (SEM), for example, the shape and surface characteristics of Table 3 Sphericity for some regular solids Shape Relative proportions Sphericity f Spheroid 1:1:2 0.93 1:2:2 0.92 1:1:4 0.78 1:4:4 0.70 Ellipsoid 1:2:4 0.79 Cylinder Height ¼ diameter 0.87 Height ¼ 2 Â diameter 0.83 Height ¼ 4 Â diameter 0.73 Height ¼ 1/2 Â diameter 0.83 Height ¼ 1/4 Â diameter 0.69 Rectangular parallelepiped 1:1:1 0.81 1:1:2 0.77 1:2:2 0.77 1:1:4 0.68 1:4:4 0.64 1:2:4 0.68 Rectangular tetrahedron – 0.67 Rectangular octahedron – 0.83 208 Chapter 7
7. particles can be observed. Photographs of two fluid catalytic cracking catalysts particles are reported as typical examples,3 see Figure 3. The sphericity of the FCC2 catalyst particle is more accentuated and there- fore a higher value of the shape factor is expected compared to FCC1. Using the Ergun Equation (see section 7.2.2, equation 25), a value of the particle’s sphericity can be evaluated, which gives a sphericity fFCC1 ¼ 0.70 for FCC1 and a value fFCC2 ¼ 0.92 for FCC2. 7.2.3 Particle Density The density of a particle immersed in a fluid is the particle density, defined as the mass of the particle divided by its hydrodynamic volume, Vp .2 This is the volume ‘‘seen’’ by the fluid in its fluid dynamic inter- action with the particle and includes the volume of all open and closed pores, see Figure 4. rp ¼ mass of particle volume the particle would displace if its surface were non-porous ¼ M Vp ð5Þ Table 4 Sphericity of some common solids Sphericity f Crushed coal 0.75 Crushed sandstone 0.8–0.9 Round sand 0.92–0.98 Crushed glass 0.65 Mica flakes 0.28 Sillimanite 0.75 Common salt 0.84 Figure 3 SEM pictures of two FCC catalyst particles3 209 An Introduction to Particle Systems
8. For non-porous solids the particle density is equal to the true, skeletal, or absolute density, rABS , which can be measured using either a specific gravity bottle or air pycnometer: rp ¼ mass of particle volume of solids material making up the particle ð6Þ For porous materials rp o rABS and cannot be measured with such methods. A mercury porosimeter can be used to measure the density of coarse porous solids but is not reliable for fine materials, since the mercury cannot penetrate the voids between small particles. In this case, helium is used to obtain a more accurate value of the particle density. Methods to measure the particle density of porous solids can be found in Refs. 2 and 5. In the case of a bed of particles immersed in a fluid, the bulk density of the bed rb , which includes the voids between the particles, is defined as follows: rb ¼ ð1 À eÞrb ð7Þ where e is the bed voidage, i.e. the volume fraction occupied by the fluid. 7.3 INTERACTION BETWEEN PARTICLES AND FLUIDS The fluid–particle and particle–particle interactions dominate the fluid dynamics of particle systems and are of major importance in pre- dicting the behaviour of complex operations, such as fluidisation, pneumatic transport and flow of slurries. In this section, the simple case study of a single particle immersed in a fluid is the starting point for the subsequent examination of the behaviour of fluidisation processes, which involve many particles in up-flowing fluids. This extends what is presented in Chapter 3, in which the concepts of Fluid Flow are Figure 4 Hydrodynamic volume of a particle (Adapted from Ref. 2) 210 Chapter 7
9. introduced and parameters such as drag coefficient and Reynolds Number are defined. 7.3.1 Single Particles We consider the motion of a single spherical particle immersed in a stationary fluid and falling at a constant velocity and we examine the forces exerted on the particle to estimate the steady velocity of the particle relative to the fluid, the particle terminal fall velocity, ut . It is important to emphasise at this point that the equilibrium conditions experienced by the particle falling at ut are equivalent to that of a motionless particle suspended in an upwardly flowing fluid with velocity ut , as represented in Figure 5. A stationary particle suspended in a fluid experiences a buoyancy force Fb , evaluated from Archimedes’ principle as the weight of fluid dis- placed, rf gv, where rf is the fluid density, g is the acceleration due to gravity and v is the volume of the particle. If the particle begins to move the fluid will exert an additional force, the drag force, made up of two components; the skin friction drag, which is a direct result of the shear stress at the surface due to fluid viscosity, and the form drag due to differences of pressure over the surface of the particle. The drag force on a single particle falling through a fluid is a function of a dimensionless drag coefficient, CD , the projected area of the particle and the inertia of the fluid.6 For a sphere: Fd ¼ CD pd2 p 4 rf u2 2 ð8Þ From a balance of forces on a stationary particle, as represented in Figure 5: Gravitation À Buoyancy À Drag ¼ acceleration ¼ 0 Figure 5 Fluid–particle interaction 211 An Introduction to Particle Systems
10. Hence: CD p 8 d2 p rf u2 ¼ pd2 p 6 ðrp À rf Þg ð9Þ or CD Re2 t ¼ 4 3 Ar ð10Þ where Ret ¼ rf ut dp m ; Ar ¼ d3 p rf ðrp À rf Þg m2 ð11Þ From Equation (10), ut can be written as follows: ut ¼ 4dp ðrp À rf Þg 3rf CD !1=2 ð12Þ Equation (12) is implicit, as ut is a function of CD . At low Reynolds numbers (Rep o 1) Stokes’ law applies and: Fd ¼ CD pd2 p 4 rf u2 2 ¼ 3pdp mu ð13Þ with CD ¼ 24 Rep and Rep ¼ rf udp m ð14Þ In the region where Newton’s law applies (750 o Rep o 3.4 Â 105), CD is almost constant at 0.445. For the intermediate flow regime, a correlation for CD has to be chosen. A correlation that covers much of the range of interest for fluidisation is due to Dallavalle:7 CD ¼ 0:63 þ 4:8 ffiffiffiffiffiffi Re p p !2 ð15Þ The relationship between drag coefficient and Reynolds number is shown in Figure 6. Changes in drag coefficient correspond to changes in 212 Chapter 7
11. the flow pattern around a sphere, as also represented in Figure 6. At low Reynolds numbers, up to Rep ¼ 1, the flow pattern past the particle is symmetrical (i.e. the flow does not separate); in the intermediate regime, with values of 1 o Rep o 500, the flow will tend to separate, giving rise to the formation of the so-called Karman vortices. A further increase in the Reynolds number will result in fully separated flow. The higher the Reynolds number, the lower the drag coefficient, i.e. the resistance applied to the particle to flow. When the flow regime is not known, the analytical calculation of ut from Equation (12) requires choosing an equation for CD , such as, for example, Equation (15), and solving a ‘‘non-trivial’’ system of two non- linear equations. To avoid the analytical solution, an iterative procedure can be applied by guessing an initial value of CD or Rep :  Step 1: Guess an initial value of CD (or Rep ).  Step 2: Calculate value of ut from Equation (12).  Step 3: Calculate Reynolds number from Equation (11).  Step 4: Check initial guess for CD by computing it from knowledge of Rep and using, for example, Equation (15). If CD is equal to the initial guess, then the ut value calculated at step 2 is correct, otherwise go to step 5.  Step 5: Choose a new value of CD and go back to step 2. Alternatively, a graphical solution can be applied to determine ut from an empirical plot based on the two dimensionless groups reported in Figure 6 Drag coefficient of a sphere and flow pattern development 213 An Introduction to Particle Systems
12. Equations (16) and (17), which are depicted in Figure 7.8 Worked example E2 (section 7.5.2) shows the application of this procedure to the calculation of ut . dà p ¼ dp rf ðrp À rf Þg m2 !1=3 ¼ Ar1=3 ð16Þ uà t ¼ ut r2 f mðrp À rf Þg " #1=3 ¼ Ret Ar1=3 ð17Þ 7.3.2 Flow Through Packed Beds We now examine the case in which the particles are stationary and in direct contact with their neighbours, which is known as a packed bed. In a packed bed, the fluid–particle interaction force is insufficient to support the weight of the particles. Hence, the fluid that percolates through the particles loses energy due to frictional dissipation. This results in a loss of pressure that is greater than can be accounted for by 0.01 0.1 1 10 100 1 10 100 1000 10000 dp * ut * spheres only: top line φ φ =1 bottom line φ=0.5 Figure 7 Empirical diagram for the calculation of the particle terminal fall velocity (Adapted from Ref. 8) 214 Chapter 7
13. the progressive increase in gravitational potential energy. We now determine this additional energy loss, which is also the energy required to overcome the weight of the particles in a fluidised bed. The total drop in fluid pressure across a length L of bed is given by: Dp ¼ DP þ rf gL ð18Þ where rf gL represents the hydrostatic contribution and DP is the portion lost due to the frictional interaction between the fluid and the particles. It represents the energy lost by the fluid and dissipated as heat. In laminar flow, the flow through a packed bed of solid particles has been analysed in terms of the fluid-flow through parallel straight tubes, in light of the similarity in the expression of the pressure drop through a porous media as given by Darcy9 and that in cylindrical tubes proposed by Hagen–Poiseuille.6 When a fluid passes through a bed of porous material the pressure drop per unit length of bed is given by Darcy’s law as: DP L ¼ k m U ð19Þ where U is the volumetric flux (volumetric flow rate per unit cross section of the empty tube), m is the fluid viscosity and k is the perme- ability of the medium. The Hagen–Poiseuille equation for laminar flow through cylindrical tubes is: DP L ¼ 32 m U D2 ð20Þ The application of the analogy involves simply replacing the fluid flux U and the bed diameter D in the Hagen–Poiseuille equation with equivalent terms relating to flow through porous media in order to take into account that a fraction of the cross-sectional area of the bed is occupied by particles. Following this reasoning, the expression for the pressure drop through a bed of spherical particles for laminar flow can be obtained: DP L ¼ 72ð1 À eÞ2 e3 mU d2 p ð21Þ where the flux U in the Hagen-Poiseuille equation has been taken as indicating the superficial velocity of fluid relative to the tube wall (volume flow of gas/cross section of the bed). Using some empirical evidence, the constant in Equation (21) has been subsequently increased from 72 to 150 215 An Introduction to Particle Systems
14. and Equation (21) with the new constant value has become known as the Blake and Kozeny equation (see Ref. 10 for the full theoretical develop- ment). Equation (21) applies to the flow region where the pressure drop is due solely to viscous losses. For beds of particles larger than about 150 mm in diameter, inertial forces become important. For the turbulent flow regime the pressure drop in a straight tube can be written as: DP ¼ 4f 1 2D rf LU2 ð22Þ where f is the friction factor. Substituting the fluid flux U and the diameter D in Equation (22), we obtain: DP ¼ 3f ð1 À eÞ e3 rf LU2 dp ð23Þ It was found experimentally that 3f ¼ 1.75. With this value, Equation (23) becomes known as the Burke–Plummer equation: DP ¼ 1:75ð1 À eÞ e3 rf LU2 dp ð24Þ Experiments have shown that equations (21) and (24) can be com- bined to cover for any flow condition, giving what is known as the Ergun equation:11 DP L ¼ 150ð1 À eÞ2 e3 mU d2 p þ 1:75ð1 À eÞrf U2 e3dp ð25Þ where the first term is the viscous term and the second is the inertial term. In general, the Ergun equation for flow through a packed bed of non- spherical and polydisperse-sized particles becomes: DP L ¼ 150ð1 À eÞ2 e3 mU ðfdsv Þ2 þ 1:75ð1 À eÞ e3 rf U2 fdsv ð26Þ The Ergun equation may be rearranged to form dimensionless groups, as graphically represented in Figure 8. Using Equation (25) we obtain: DP L   dp rf U2   e3 1 À e   ¼ 150 mð1 À eÞ dp rf U þ 1:75 ð27Þ 216 Chapter 7
15. It is worth noting the similarity between Figure 8 and the friction factor for tube flow represented in Figure 6. This highlights the physical meaning of pressure drop across the bed as energy dissipation due to friction between the particles. 7.4 FLUIDISED BEDS As the velocity of the fluidising fluid (gas or liquid) is increased through a bed of solid particles, there comes a point where the drag force exerted by the fluid on the particles, which is proportional to the global pressure drop across the bed, is balanced by the buoyant weight of the suspension. At this point, the particles are lifted by the fluid, the separation between them increases and the bed becomes fluidised. Thus: DP ¼ weight of particles and fluid À upthrust bed cross-sectional area ¼ Mg A ð28Þ where M is the mass of particles in the bed and A is the cross sectional area of the bed. For a bed of particles of density rp , fluidised by a fluid of density rf to form a bed of depth L and voidage e in a vessel of cross-sectional area A, Equation (28) becomes: DP ¼ ALð1 À eÞrp g þ ALerf g  à À ALrf g A ð29Þ 0.1 1 10 100 1000 0.1 1 10 100 1000 10000 Re cD Er B B Ergun equation Blake-Kozeny equation Burke-Plummer equation Figure 8 The Ergun equation (Adapted from Ref. 11) 217 An Introduction to Particle Systems
16. from which: DP L ¼ ðrp À rf Þð1 À eÞg ð30Þ An important property of fluidised beds follows immediately from this simple relation; as the bed expands, the product (1Àe) L, which repre- sents the total volume of particles per unit cross section, remains unchanged. As the fluid flux is increased, L increases and (1Àe) decreases so as to maintain their product at a constant value. As a consequence, the pressure drop through the fluidised bed remains constant for further increases in fluid velocity, as shown in Figure 9. 7.4.1 Minimum Fluidisation Velocity The fluid velocity at which the particles become suspended is the minimum fluidisation velocity, umf . For spherical particles, the velocity at minimum fluidisation can be predicted by combining Equations (25) and (30), given that the values of the bed voidage at minimum fluidi- sation, emf , are known a priori: ðrp À rf Þg ¼ 150 ð1 À emf Þ e3 mf m umf d2 p þ 1:75 e3 mf rf u 2 mf dp ð31Þ which can also be written as: Ar ¼ 150ð1 À emf Þ e3 mf Remf þ 1:75 e3 mf Re2 mf ð32Þ From this, the minimum fluidisation velocity can be obtained: umf ¼ e3 mf ðrp À rf Þd2 p g 150ð1 À emf Þm ð33Þ Equation (33) is strictly applicable only to conditions of viscous flows where the pressure drop is solely due to viscous energy losses. If Equation (31) is expressed as a function of the sphericity factor, f, as shown in Equation (26), then a value of f can be back-calculated by knowing umf and emf , as done for the FCC catalysts shown in Figure 3. If emf is not known, than we can use the equation proposed by Wen-Yu12, which expresses umf solely as a function of the physical 218 Chapter 7
17. characteristics of the fluid and the particle. Wen-Yu used experimental evidence to reformulate Equation (32) as follows: Ar ¼ 1650 Remf þ 24:5 Re2 mf ð34Þ This formulation assumes values of the voidage at minimum fluidisation of around 0.38. Superficial gas velocity (cm/s) H (m) Umf Umf Hmf Umb Hmb (b) ∆P (N/m2) Increasing velocity Decreasing velocity Mg/A (a) Ergun Eq.(30) Figure 9 Pressure drop and bed expansion profiles for gas–solid systems of fine particles 219 An Introduction to Particle Systems
18. For fine spherical particles (below about 100 mm) umf is obtained from the viscous term of Equation (34) umf ¼ d2 p ðrp À rf Þg 1650m ð35Þ For larger particles, it becomes: u2 mf ¼ dp ðrp À rf Þg 24:5rf ð36Þ The sensitivity of umf on the bed voidage is shown in the worked example E3 (section 7.5.3), in which the calculation of umf using the Ergun and Wen-Yu equations is compared. Most fluidisation processes are operated at high temperatures and pressures. It is important, therefore, to be able to predict changes in fluidisation with the operating conditions. Using Equations (35) and (36), the effect of temperature and pressure can be determined. With increasing temperature, gas viscosity increases while gas density de- creases. For small particles, the fluid–particle interaction is dominated by the viscous effects. Equation (35) shows that umf varies with 1/m, and umf should therefore decrease with temperature. For large particles, the inertial effects dominate; Equation (36) predicts that umf will vary with (1/rf )0.5, umf should therefore increase with temperature. With increasing pressure, gas viscosity is essentially independent of pressure, while gas density increases. For small particles, umf remains almost constant. For large particles, umf decreases with pressure. 7.4.2 Types of Fluidisation Regimes At fluid velocities higher than at minimum fluidisation, different types of fluidisation behaviour are possible: (i) uniform expansion; (ii) bubbling or slugging; (iii) turbulent; (iv) lean fluidisation. In liquid–solid systems, and for gas–solid systems of small particles, increasing the fluidising velocity beyond umf gives rise to a uniform expansion of the bed. The homogeneous expansion of a gas–solid system is generally described using the Richardson–Zaki equation:13 u ¼ ut en ð37Þ 220 Chapter 7
19. Equation (37) describes the relationship between fluidising velocity and voidage as a function of the particle terminal fall velocity and of a parameter n which depends on the particle Reynolds number according to the following relations:10 n ¼ 4:8 Reto0:2 n ¼ 4:6 ReÀ0:03 t 0:2oReto1 n ¼ 4:6 ReÀ0:1 t 1oReto500 n ¼ 2:4 Ret 4500 ð38Þ In gas–solid systems, as the gas velocity is further increased, gas bubbles form within the bed to give a bubbling fluidised bed. The velocity at which the first bubble forms is called the minimum bubbling velocity, umb . An empirical correlation typically used to predict umb at ambient conditions has been given by Abrahamsen and Geldart:14 umb ¼ 2:07 expð0:716 F45 Þ dp r0:06 g m0:347 ð39Þ where F45 is the mass fraction of particles smaller than 45 mm. The pressure drop and bed expansion profiles shown in Figure 9 represent the different stages from fixed to bubbling fluidisation for a typical gas–solid system of fine particles. Knowing the bed height, the average bed voidage can be obtained: e ¼ 1 À M rp HA ð40Þ An increase in gas velocity beyond minimum bubbling leads to more vigorous bubbling with larger bubbles that may eventually become as large as the vessel diameter, giving a slugging bed. This may occur typically with high aspect ratio (H/D) beds for Geldart Group B particles. Turbulent fluidisation occurs for even higher velocities. Here the two-phase structure disappears and bed takes on a foam-like appearance. Still higher gas velocities result in particles being trans- ported out of the bed, until only a low concentration of rapidly moving particles remains in contact with the gas, as in pneumatic conveying systems.2 These types of systems are used for transporting particulate solids in a conveying fluid, where the solids are typically at least three times denser than the transporting fluid. A typical example is the transport of particles in a pipeline using gas. The distinction between 221 An Introduction to Particle Systems
20. pneumatic conveying systems and hydraulic conveying systems lies in the ratio between the solid and fluid density, with the solid density being generally of the order of magnitude of the fluid density in the latter. In pneumatic conveying systems, two types of flows are generally encoun- tered. ‘‘Dilute phase’’ is characterised by low particle concentrations (less than 1% by volume) and relatively high velocities (greater than 20 m sÀ1), which results in high attrition of the particles. Dilute phase conveying is carried out in systems in which the solids are fed to the pipe from hoppers at a controlled flow rate and cyclone separators are used to separate the solids from the gas stream at the end of the line. ‘‘Dense phase’’ conveying systems are characterised instead by high particle concentrations (greater than 30% by volume) and low velocities (1–5 m sÀ1). In this case the solids are not entirely suspended in the gas stream; particle attrition is reduced due to the low solid velocities, reducing in turn pipeline erosion and product degradation. Both types of flows are used in horizontal and vertical transport systems. Figure 10 shows various types of fluid–particle contacting, from fixed bed, to bubbling fluidisation, slugging and pneumatic transport. The minimum fluidisation velocity represents the lower limit of the range of operative conditions at which a fluid-bed process can be operated, while the particle terminal fall velocity represents the upper limit beyond which the particles will start leaving or elutriating from the bed. To avoid or reduce carryover of particles from a fluidised bed, gas velocity has to be kept between umf and ut . For polydisperse systems, in calculating umf , the mean diameter dvs is used for the particle size distribution present in the bed. In calculating ut , the smallest size of solids present in appreciable quantity in the bed is used. gas or liquid low velocity gas or liquid gas liquid or gas at high pressure gas gas or liquid high velocity Fixed bed Slugging Bubbling Lean Minimum fluidization Particulate Figure 10 Types of fluid–particle contacting 222 Chapter 7
21. Using the dimensionless parameters introduced with Equations (16) and (17) (substituting u* t and ut with u* and u), for a given particle system and a given fluidising velocity, the fluidisation regime can be determined using the diagram reported in Figure 11.15 Similarly, the operating velocity needed to attain a certain fluidisation regime can also be determined from Figure 12, see worked example E4 (section 7.5.4). 7.4.3 Classification of Powders In an attempt to classify the different fluidisation behaviours, Geldart16 proposed an empirical classification, which divides fluidisation behaviour Figure 11 Empirical diagram for the prediction of the fluidisation regime (Adapted from Ref. 15) 223 An Introduction to Particle Systems
22. according to the mean particle size and gas and particle densities. Boundaries between these groups were proposed in the form of a dimensional plot of (rp À rg ) versus dp , as shown in Figure 12. Geldart proposed four typical fluidisation regimes for materials fluidised with air at ambient conditions, which he indicated as A, B, C and D. Group C materials are usually cohesive and very difficult to fluidise. The behaviour of this group of powders is strongly influenced by the interparticle forces, which are greater than those exerted by the fluidising gas. As a result, the particles are unable to achieve the separation they require to be totally supported by drag and buoyancy forces and true fluidisation does not occur, with the pressure drop across the bed being lower than the theoretical value given by Equation (30). Bubbles, as such, do not appear; instead the gas flow forms channels through the powder. An attempt to fluidise a Group C powder is shown in Figure 13. Fluidisation of Group C materials can sometimes be improved using mechanical stirring or vibrations, which break up the stable channels. In the case of plastic materials, fluidisation may be promoted by adding some sub-micron particles, which modify the contact geometry, thereby reducing the interparticle forces. Group A powders are those which exhibit a stable region of non- bubbling expansion between umf and umb , as shown in Figure 10. C A B D 10 50 100 500 1000 Mean particle diameter (µm) Density difference (ρp - ρg ) (kg/m3) 102 103 104 Figure 12 Classification of fluidisation behaviour for air at ambient conditions (Adapted from Ref. 16) 224 Chapter 7
23. Geldart4 distinguished these powders as those for which umb /umf 4 1. At gas velocities above umb bubbles begin to appear, which constantly split and coalesce, and a maximum stable bubble size is achieved. The flow of bubbles produces high solids and gas back-mixing, which makes the powders circulate easily, giving good bed-to-surface heat transfer. Group B powders are characterised by having umb ¼ umf . Bubbles rise faster than the interstitial gas velocity, coalescence is the dominant phenomenon and there is no evidence of a maximum bubble size, as defined for Group A materials. Bubble size increases with increasing fluidising gas velocity, see Figure 14. The interparticle forces are con- sidered to be negligible for these powders. Group D powders are large particles that are distinguished by their ability to produce deep spouting beds. The distinction between this and the previous groups concerns also the rise velocity of the bubbles, which is, in general, less than the interstitial gas velocity. The gas velocity in the dense phase is high and solids mixing is relatively poor, consequently, back-mixing of the dense phase is small. Segregation by size is likely when the size distribution is broad, even at high gas velocities. The flow regime around these particles may be turbulent, causing particle attri- tion and, therefore, elutriation of the fines produced. 7.4.4 Interparticle Forces, Measurement of Cohesiveness and Flowability of Solid Particles at Ambient Conditions When trying to describe the fluidisation of different materials, the nature of the forces acting between adjacent particles becomes of major Figure 13 Fluidisation behaviour of a group C powder: an attempt to fluidise a group C powder produces channels or a discrete plug 225 An Introduction to Particle Systems
24. Figure 14 Fluidisation behaviour of a Group B powder: effect of gas flow on bubbles in a two-dimensional fluidised bed, (i) lower gas velocity, (ii) higher gas velocity. A max. stable bubble size is never achieved 226 Chapter 7
25. importance in order to predict their contribution to the fluid-bed behaviour. As described in the previous section, fluidisation of Geldart Group C materials is dominated by interparticle forces, whose effects become negligible in the fluidisation behaviour of Group B powders. The role of the interparticle forces on the fluidisation behaviour of Group A powders is, on the other hand, far from being unequivocally understood, mainly due to the difficulty in recognising the nature of the forces involved and, therefore, of quantifying their effect on the fluid- isation behaviour. Particle–particle contacting can be the result of different mechanisms of adhesion, which in turn can influence, in different ways, the fluidisa- tion behaviour of solid materials, see Ref. 17. An extensive review on the subject is also reported in Israelachvili.18 Table 5 lists various mecha- nisms of adhesion with and without material bridges. It would be desirable to have simple tests capable of characterising the fluidisation behaviour or ‘‘flowability’’ of particulate materials on the basis of their bulk properties. To this end, Carr19 developed a system to characterise bulk solids with respect to flowability. Table 6 summarises the properties which are determined. In Carr’s method a numerical value is assigned to the results of each of these tests, and is summed to produce a relative flowability index for that particular bulk material. Given the extensive use of these empirical techniques in academia and industry, a brief review on the subject is reported here. Nevertheless, it should be emphasised that these techniques allow measurements of the flow- ability or cohesion of materials solely in their stationary or compressed status and at ambient conditions. A direct relationship between these Table 5 Mechanisms of adhesion Without material bridges With material bridges Van der Waals forces Capillary forces Electrostatic forces Solid bridges Magnetic forces Sintering Hydrogen bonding Table 6 Properties of bulk solids to determine flowability with the Carr method Bulk density loose, rBDL Angle of repose Bulk density packed, rBDP Angle of fall Hausner ratio Angle of difference Compressibility Angle of spatula Cohesion Angle of internal friction 227 An Introduction to Particle Systems
26. measurements and the fluidisation behaviour can therefore prove very difficult, in particular when operating conditions cause the properties of the materials to change. 7.4.4.1 Loose and Packed Bulk Density. The loose bulk density is determined by gently pouring a sample of powder into a container through a screen, and is measured before settlement takes place. The packed or tapped bulk density is determined after settling and deaera- tion of the powder has occurred due to tapping of the sample. Details on the techniques are reported by Geldart and Wong.20 7.4.4.2 Hausner Ratio and Powder Compressibility. The ratio between the loose and packed bulk density, rBDL and rBDP respectively, is known as the Hausner Ratio (HR) and is used as an indication of the cohe- siveness of the materials, see Ref. 20. In addition to the HR, the powder compressibility is also used to define cohesiveness. This is expressed as 100(rBDP À rBDL )/rBDP . 7.4.4.3 Cohesion. This test is used for very fine powders (below 70 mm). Material is passed through three vibrating sieves in series. The material left on each sieve is weighed and a cohesion index is determined from the relative amounts retained. Carr19 defined cohesion as the apparent surface force acting on the surface of powders, which are composed of millions of atoms. The number of points of contact within the powder mass determines the effect of this force. Thus, cohesiveness increases with decreasing particle size, since the number of contact points increases as the particle size decreases. 7.4.4.4 Angle of Repose, Angle of Fall and Angle of Difference. The angle of repose is defined as the angle between a line of repose of loose material and a horizontal plane. This is determined by pouring the powder into a conical pile from a funnel and screen assembly. Materials with good flowability are characterised by low angles of repose. The angle of fall is determined by dropping a small weight on the platform on which a loose or poured angle of repose has been formed. The fall causes a decrease of the angle of repose forming a new angle identified as the angle of fall. The more free-flowing the powder, the lower the angle of fall. The angle of difference is determined by the difference between the angle of repose and the angle of fall. The greater this angle the better the flow. 7.4.4.5 Angle of Spatula and Angle of Internal Friction. The angle of spatula is a quick measurement of the angle of internal friction. This is the angle, measured from the horizontal, that a material assumes on a 228 Chapter 7
27. flat spatula that has been inserted into a pile and then withdrawn vertically. A free-flowing material will have formed one angle of repose on the spatula’s blade. A cohesive material will have formed several angles of repose on the blade, the average of these is then taken. The higher the angle of spatula of a material, the less its flowability. The angle of internal friction has been defined as the angle at which the dynamic equilibrium between the moving particles of a material and its bulk solid is achieved. This is of particular interest for flows in hoppers and bins. The measurement of this angle is described below. Measurements of Angles of Friction. Shear cells are generally used to determine the properties which influence powder flow and handling.21 Some of these properties are called ‘‘failure properties’’. These include the effective angle of internal friction, the angle of wall friction, the failure (or flow) function, the cohesion and ultimate tensile strength. The require- ment for getting powders to flow is that their strength is less than the load put on them, i.e. they must fail, hence the name ‘‘failure properties’’. From soil mechanics, a solid is characterised by a yield locus that defines the limiting shear strength under any normal stress. Plotting shear stress t and normal stress sn , the yield locus for a Coulomb material intersects the t axis at a value of t which is defined as cohesion C at an angle fi , which is defined as the angle of internal friction of that material. In a shear cell, such as the Jenike shear cell,22 the test powder is consolidated in a shallow cylindrical chamber which is split horizontally. The lower half of the cell is fixed and a shear force is applied to the upper moveable part at a constant low rate (see example E5, section 7.5.5, where experiments were obtained at a constant rate of 0.03 rpm, corresponding to a linear velocity of 1 mm minÀ1). Shearing can be carried out for each of a series of normal loads on pre-consolidated samples, so that at the end of the test the relationships between the shear stress and normal stress at various bulk densities are obtained. Construction of the Static Internal Yield Locus. Once the shear cell is loaded with the powder specimen, the pre-consolidation load Nc is applied on the cell lid and the pre-shearing phase starts. When a steady-state value is attained for the measured shear stress, the pre- shearing is stopped. Then the shearing phase starts under a normal load N1 lower than Nc and a peak value of the shear stress is reached corresponding to the material failure. The sequence of pre-shearing and shearing phases is then repeated, keeping the same value of the normal load for each pre-shearing phase, Nc , and using decreasing normal loads for each shearing phase, i.e. N1 4N2 4N3 4. . .4NN . 229 An Introduction to Particle Systems
28. Following the theory of Jenike,23 the solid normal stress acting in the shear plane can, in general, be calculated by simply dividing the external normal load by the shear surface area, which is assumed to correspond to the cross-sectional area of the cell. si ¼ Ni Acell ð41Þ Figure 15 reports a typical shear stress chart and the derived yield locus corresponding to the applied consolidation stress. The yield locus is the interpolation of the experimental points (s, t) corresponding to failure. It is worth pointing out that different yield loci are obtained when applying different consolidation stresses sc . The choice of the normal stresses used in the shearing phases and the number of phases used in each experiment, on the other hand, are arbitrary, as all the (s, t) points obtained in the shearing phases will belong to the same yield locus, provided that the same consolidation stress is applied in the pre- shearing phases. The values of the static angle of internal friction j and the cohesion C were worked out as the slope and the intercept with the t axis of the yield locus, respectively, as shown in Figure 15. The circle passing through sc and tangent to the yield locus is called the Mohr Circle. The two intercepts of this Mohr circle and the s axis are the major principal stresses s1 and s2 corresponding to the consolidation applied. By repeating the whole experiment with different consolidation loads Nc , a family of incipient yield loci can be obtained. As a result, different angles of internal friction and values of the cohesion can be obtained as a function of the major principal stress s1 . Jenike carried out many experimental measurements on free-flowing and cohesive materials. He found that the yield locus of a dry mate- rial would be a straight line passing through the origin, as shown in steady state failure time Shear stress chart Static yield locus C pre-shearing ii > > 2 σ1 ϕ τ σc σc σc σc σ σi σi σii σii σi σ σ shearing Figure 15 Construction of the static yield locus 230 Chapter 7
29. Figure 16(a). The term ‘‘Cohesionless’’ was therefore used to describe materials which have a negligible shear strength under zero normal load (sn ¼ 0). On the other hand, Jenike found that the yield loci of cohesive materials differ significantly from a straight line and have a non- zero intercept, indicated by C. Moreover, the position of the locus for a cohesive powder is a strong function of the interstitial voidage of the material. Fig 16(b) shows the typical yield locus for cohesive materials. Construction of the Dynamic Internal Yield Locus. The dynamic yield locus represents the steady state deformation, as opposed to the static yield locus which represents the incipient failure. The dynamic yield locus is constructed by plotting on a (s, t) plane the principal Mohr circles obtained for various consolidation stresses. The dynamic yield locus will be the curve or straight line tangent to all circles, as shown in Figure 17. The dynamic angle of internal friction d and cohesion Cd are independent of the consolidation stress. d and Cd are obtained as the slope and the intercept at s¼0 of the dynamic yield locus of the powder. An approximation of the dynamic yield locus is given by the effective yield locus, provided that Cd ¼ 0. The effective yield locus, shown in Figure 17, is the straight line drawn through the origin and tangentially to the Mohr’s circle represented in Figure 15. Since it passes through the origin, it represents the angle of internal friction that the material would have if it were cohesionless. This angle is denoted by the effective angle of internal friction of the material, je (Figure 18). The difference between the effective and the internal angle of friction can be an indication of the degree of cohesiveness of the powder. An application of the above procedure is presented in the worked example E5, which Shear stress, τ τ (N m-2) Normal stress, σ σ( N m-2 ) (a) (b) φ i yield locus angle of internal friction C Figure 16 Yield loci of (a) free-flowing materials; (b) cohesive materials 231 An Introduction to Particle Systems
30. illustrates the results obtained from experiments carried out on two different alumina materials and ballotini.24 7.4.5 Some Industrial Fluid-Bed Applications The fluidised bed is only one of the many reactors employed in industry for gas–solid reactions, as reported by Kunii and Levenspiel.25 When- ever a chemical reaction employing a particulate solid as a reactant or as a catalyst requires reliable temperature control, a fluidised bed reactor is often the choice for ensuring nearly isothermal conditions by a suitable selection of the operating conditions. The use of gas-solid fluidised beds dynamic yield locus c δ ' ' 2 δ σ ' 2 σ '' ' 2 σ ' ' 1 σ ' 1 σ '' ' 1 σ σ τ Figure 17 Dynamic yield locus ϕ σ τ ϕe internal angle effective angle Figure 18 Effective angle of internal friction 232 Chapter 7
31. spans across many industries, ranging from the pharmaceutical, refining and petroleum sectors to power-generation and can be classified de- pending on whether the physical mechanisms of heat and mass transfer are the governing factors in the process or whether chemical reactions are taking place in the fluid-bed. Examples are given in Table 7. Examples of chemical reaction processe that employ a fluid-bed include catalytic cracking of petroleum, metallurgical processes, tita- nium refining, where the fluidised bed reactor is used for extracting titanium from naturally occurring ore, coal gasification, combustion and incineration, where the recent emphasis is on using fluidised bed reactors for the treatment of clean biomass and urban waste in order to produce a clean syngas from which electric energy can subsequently be obtained. Applications where mass and heat transfer are the key mechanisms include processes such as granulation, drying and mixing. It has been shown earlier that the fluidisation properties of a powder in air at ambient conditions may be predicted by establishing in which Geldart group it lies. Table 8 shows that most fluidised bed industrial processes are operated at temperatures and pressures well above ambi- ent. It is important to note that at operating temperatures and pressures Table 7 Some industrial applications of fluidised beds Physical mechanism Application Heat and/or mass transfer between gas/ particle Solids drying Absorption of solvents Food freezing Heat and mass transfer between particle/ particle or particle/surface Coating of pharmaceutical tablets Granulation Mixing of solids Plastic coating of surfaces Heat transfer between bed/surface Heat treatment of glasses, rubber, textile fibres Chemical mechanism Application Gas/gas reaction in which solid acts as catalyst Oil cracking Manufacture of: Acrylonitrile Polyethylene Gas/gas reaction in which solids are transformed Coal combustion Gasification Incineration of waste Catalyst regeneration 233 An Introduction to Particle Systems
32. above ambient a powder may appear in a different group from that which it occupies at ambient conditions.26 This is due to the effect of gas properties on the grouping and may have serious implications as far as the operation of the particle system is concerned. Reliable design of commercial scale particle systems requires a good understanding of the highly complex flow phenomena and also requires detailed knowledge of how the hydrodynamics are affected by both reactor design and plant scale-up. Factors such as powder characteris- tics (i.e. density, surface properties, fines content), operating conditions (i.e. temperature, pressure, velocity) and reactor design (i.e. cooling coils, dip-legs, grid), plus any special conditions, such as the presence of Table 8 Operating conditions employed in fluidised bed processes Process Example/Products Process conditions Drying of solids Inorganic materials 60–1101C; 1 atm Pharmaceuticals 601C; 1 atm Calcination Limestone 7701C; 1 atm Alumina 550–6001C, 1 atm Granulation Soap powders 51C; 1 atm Food; Pharmaceuticals 20–401C; 2–3 atm Coating Food; Pharmaceuticals 20–801C; 2–3 atm Roasting Food industry products 2001C; 1 atm Sulfide ores (FeS2 ) 650–11001C; 1 atm Synthesis Reactions Phthalic anhydride 340–3801C; 2.7 atm Acrylonitrile 400–5001C; 1.5–3 atm Ethylene dichloride 260–3101C; 1–10 atm Maleic anhydride 400–5001C; 4 atm Polyethylene (low/high den- sity) 75–1201C; 15–30 atm Thermal cracking Ethylene; propylene 7501C; 1 atm Some other applications Combustion and incineration of waste solid 800–9001C; 1–10 atm Gasification of coal and coke/ solid waste 8001C; 1–10 atm Biofluidisation, cultivation of micro-organism 301C; 1 atm Semiconductor industries 300–11001C; 1 atm 234 Chapter 7
33. liquids and other additives in the reactor, can significantly affect the gas–solid contacting efficiency. In light of this and of its wide range of industrial applications, particle science and technology remains a com- plex and challenging area for scientific research. 7.5 WORKED EXAMPLES 7.5.1 Example E1 Table 9 shows the results of the sieve analysis obtained for a sample of a FCC1 powder. The values reported in the first two columns of the table are the standard diameters of the sieve apertures. From these the mean diameter dp is obtained for each two adjacent sieve sizes and the values are reported in the third column. From the mass fraction of powder in each sieve (values in the fourth column) the weight percentage is obtained and reported in the fifth column. Thus, the sum of the mass fraction over the mean diameter allows the calculation of the volume- surface mean particle diameter of the distribution using Equation (1): dVS ¼ 1 P xi =dp ð1Þ Plotting the weight percentage versus the mean particle diameter dp allows the size distribution of the powder to be obtained, as shown in Figure 2. The relative diameter spread s/d50 is then calculated to obtain an indication of the width of the size distribution of the sample under analysis. The diameter spread, sd , is given by: sd ¼ d84% À d16% 2 ð3Þ and the median size diameter, d50 , are determined by plotting the cumulative size fraction, reported in Table 9, versus particle size, see also Figure 19. This gives a value for sd /d50 ¼ 0.37, indicating a wide particle size distribution according to Table 2. 7.5.2 Example E2 A non-spherical FCC catalyst particle falls in a column of nitrogen at ambient conditions and attains a terminal fall velocity ut . The density and viscosity of the nitrogen are rf ¼ 1.2 Â 10À3 g cmÀ3 and m ¼ 1.8 Â 10À4 g cmÀ1 sÀ1 respectively. Calculate the terminal velocity of the particle given its physical characteristics, dp ¼70 mm, sphericity f ¼ 0.7 235 An Introduction to Particle Systems
34. and rp ¼ 1.40 g cmÀ3, using a graphical solution method based on Equations (16) and (17). Solution: First, calculate d* p from Equation (16): dà p ¼ dp rf ðrp À rf Þg m2 !1=3 ¼ 0:0070 0:0012ð1:4 À 0:0012Þ980 ð0:00018Þ2 " #1=3 ¼ 2:6 Table 9 Worked example E1: calculation of mean particle diameter Min particle diameter Max particle diameter Mean particle diameter Mass fraction % Mass fraction in sieve Mass fraction/ mean diameter Cumulative mass fraction 0 38 19 0.015 1.5 7.89E-04 1.5 38 45 41.5 0.027 2.7 6.51E-04 4.2 45 53 49 0.086 8.6 1.76E-03 12.8 53 63 58 0.157 15.7 2.71E-03 28.5 63 75 69 0.204 20.4 2.96E-03 48.9 75 90 82.5 0.201 19.8 2.44E-03 69.0 90 106 98 0.132 13.2 1.35E-03 82.2 106 125 115.5 0.085 8.5 7.36E-04 90.7 125 150 137.5 0.050 5.0 3.64E-04 95.7 150 180 165 0.025 2.5 1.52E-04 98.2 180 212 196 0.011 1.1 5.61E-05 99.3 212 250 231 0.004 0.4 1.73E-05 99.7 250 300 275 0.001 0.1 3.64E-06 99.8 300 355 327.5 0.001 0.1 3.05E-06 99.9 355 425 390 0.001 0.1 2.56E-06 100 425 500 462.5 0.001 0.1 2.16E-06 100 500 600 550 0.000 0.0 0.00E-00 100 Total 100.0 1.40E-02 Mean particle diameter (mm) 71.54 Mean particle diameter at 16% 84% of cumulative mass fraction 51 102 Diameter spread (mm) 26 Median particle diameter (mm) 69 (at 50% of the cumulative mass fraction) Particle size distribution 0.37 236 Chapter 7
35. Next, we find uà t from the empirical diagram reported in Figure 7, shown again here as Figure 20, choosing the curve that corresponds to f ¼ 0.7. This gives: uà t ¼ 0:28 Finally, from Equation (17), we obtain ut : ut ¼ uà t mðrp À rf Þg r2 f !1=3 ¼ 0:28 0:00018ð1:4 À 0:0012Þ980 ð0:0012Þ2 " #1=3 ¼ 15:5 cm sÀ1 7.5.3 Example E3 A fine spherical silica-based catalyst with the physical characteristics reported below is being investigated for the development of a new fluidised bed process. Fundamental parameters such as the bed voidage and minimum fluidisation velocity, umf , have to be determined. dp ¼ particle diameter ¼ 75 mm rp ¼ particle density ¼ 1770 kg mÀ3 rf ¼ fluid density ¼ 1.22 kg mÀ3 m ¼ fluid viscosity ¼ 1.8  10À5 kg mÀ1 sÀ1 (i) Calculate the voidage of the bed, e, (volume fraction occupied by the voids) when the packed bed of solid particles occupies a depth L ¼ 0.5 m in a vessel of diameter D ¼ 0.14 m. The mass of 0 20 40 60 80 100 19 49 69 98 138 196 275 390 Particle Size (µm) Cumulative % in sieve d50% d16% d84% Figure 19 Worked example E1: cumulative percentage under size versus particle size 237 An Introduction to Particle Systems
36. particle is 8.5 kg. Knowing the voidage, determine the number of particles making up the bed, assuming that they are perfectly spherical. (ii) Determine umf using the Ergun equation for pressure drop through packed beds of spherical particles. Different values of the voidage at minimum fluidisation (emf ¼ 0.38, 0.42, 0.45) are given in order to estimate the sensitivity of the calculation of umf on emf when using the Ergun equation. Determine umf also using the Wen-Yu equation. Solution: (i) e ¼ volume occupied by the gas=total volume of bed e ¼ ðVbed À Vparticle Þ=Vbed ¼ 1 À Vparticle =Vbed e ¼ 1 À M=ðrp LAÞ ¼ 1 À M=ðrp LpD2=4Þ Substituting the values we obtain: e ¼ 0:38 0.01 0.1 1 10 100 1 10 100 1000 10000 dp * ut * spheres only: top line φ =1 bottom line φ =0.5 Figure 20 Worked example E2: empirical diagram for the calculation of the particle terminal fall velocity 238 Chapter 7
37. Knowing the voidage, the number of particles can be deter- mined. The total volume occupied by the particles in the bed is equal to: V ¼ ALð1 À eÞ which is also equal to the number of particles multiplied by the volume of a single particle, thus: ALð1 À eÞ ¼ Np p d3 p 6 from which: Np ¼ 2:2 E þ 10 (ii) The Ergun equation for pressure drop through packed beds of spherical particles (Equation 32) may be applied to give umf as follows: Ar ¼ 150ð1 À emf Þ e3 mf Remf þ 1:75 e3 mf Re2 mf ð32Þ where: Remf ¼ rf umf dp m ; Ar ¼ d3 p rf ðrp À rf Þ g m2 The viscous effects dominate the fluid–particle interaction of small particles (below 100 mm), thus the inertial term of the Ergun equation can be neglected. Hence, the minimum fluidisation velocity can be obtained from: umf ¼ e3 mf ðr p À r f Þ d2 p g 150 ð1 À emf Þ m ð33Þ By substituting the given numerical values, umf can be obtained for the different values of the voidage: emf umf (cm sÀ1) 0.38 0.32 0.42 0.46 0.45 0.6 239 An Introduction to Particle Systems
38. For fine spherical particles, when using the equation proposed by Wen–Yu, umf is obtained from the viscous term of Equation (35): umf ¼ d2 p ðrp À rf Þ g 1650m ¼ 0:33 cm sÀ1 The results show that the values calculated using the Ergun equation are very sensitive to the value used for the voidage at minimum fluidisation. The value obtained from the Wen–Yu equation corresponds to that obtained from the Ergun Equation (25) with a voidage of approximately 0.38, as reported in Section 7.3.2. 7.5.4 Example E4 As part of the fluid-bed process development described in E3, the operative conditions and the corresponding fluidisation regime have to be determined. Calculate the superficial gas velocity which is needed to expand the bed and obtain an average fluid-bed voidage e¼0.62. Consider viscous flow regime. Use Figure 11 to check what fluidisation regime corre- sponds to the calculated superficial gas velocity. Find the flow regime at which the fluid-bed would be operated if fluidised at 0.25 m sÀ1 and 1.5 m sÀ1. Solution: The Richardson–Zaki equation (Equation 37) is used in fluidisation to describe the homogeneous expansion. It relates the superficial gas velocity to the fluid bed voidage and particle terminal fall velocity: u ¼ ut en ð37Þ where for viscous flow regime, n¼4.8. Knowing the particle terminal fall velocity, ut , the superficial gas velocity which is needed to expand the bed and obtain an average fluid- bed voidage e ¼ 0.62 can be determined. Using a similar procedure to that adopted in E2, we can use Figure 11 (repeated again here as Figure 21) to obtain the value for ut , as follows: From the data given, we can calculate d* p dp ¼ particle diameter ¼ 75 mm rp ¼ particle density ¼ 1770 kg mÀ3 240 Chapter 7
39. rf ¼ fluid density ¼ 1.22 kg mÀ3 m ¼ fluid viscosity ¼ 1.8  10À5 kg mÀ1 sÀ1 dà p ¼ Ar1=3 ¼ dp rf ðrp À rf Þg m2 !1=3 ¼ 3 Using the diagram in Figure 22, the intersect of the ut curve with the coordinate corresponding to the calculated d* p gives a value for uà t ¼ 0.43. Figure 21 Worked example E4: empirical diagram for the prediction of the fluidisation regime 241 An Introduction to Particle Systems
40. From uà t we can therefore back-calculate a value for the terminal fall velocity, ut : ut ¼ uà t r2 f mðrp À rf Þg " #À1=3 ¼ 0:26 m sÀ1 Thus, the superficial gas velocity needed to expand the bed and obtain a voidage equal to 0.62 is: u ¼ ut e4:8 ¼ 0:026 m sÀ1 We can also check that the regime that corresponds to the calculated superficial gas velocity is homogeneous fluidisation, for which the Richardson–Zaki equation is valid. To do so, we calculate the value of u* corresponding to u¼0.026 m sÀ1, giving u* ¼ 0.015, and using Figure 21 we can verify that the corresponding fluidisation regime is homogeneous fluidisation. Using Figure 21, the fluidisation regime corresponding to 0.30 m sÀ1 and 1.5 m sÀ1 can also be determined: u ¼ 0.15 m sÀ1 u* ¼ 0.25 i.e. Bubbling fluidisation u ¼ 1.5 m sÀ1 u* ¼ 2.5 i.e. Fast-turbulent fluidisation 7.5.5 Example E5 The following example reports the experimental results obtained for the effective angle of internal friction and cohesion for three samples of powders. 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 time (s) τ τ (Pa) σ c σ c σ c σ c σ c σ 1 σ 2 σ 3 σ 4 σ 5 1106.2 Pa 904.8 Pa 1106.2 Pa 703.4 Pa 1106.2 Pa 501.9 Pa 300.5 Pa 1106.2 Pa 1106.2 Pa 199.8 Pa Figure 22 Worked example E5: experimental shear stress chart (sc ¼ 1106.2 Pa) 242 Chapter 7
41. Two alumina materials, A1 and A2 belonging to Geldart Group A, and ballotini belonging to Geldart group B (see section 7.4.3) are examined. The physical characteristics of the three materials are sum- marised in Table 10. A1 is virtually free of fines (particles having size below 45 mm), while A2 contains 30%wt of fine particles. In light of this, the alumina sample A1 is expected to be more cohesive than A2. As reported in section 7.4.3, the ballotini are instead expected to be free from any dominant effect of the interparticle forces. For each powder, four static yield loci are obtained by applying the normal loads reported in Table 11, to which the corresponding consol- idation stresses, sc , are calculated from Equation (41). Following the procedure described in section 7.4.4.7, the values of the normal stresses si applied for each consolidation stress sc , are reported in Table 12 and also shown in Figure 22. Figure 22 shows a typical experimental shear stress chart, obtained for A2, using a compaction with normal stress equal to 1106.2 Pa. Figure 23 shows the corresponding yield locus from which the static angle of internal friction, j, was worked out from the slope of the yield locus. The cohesion, C, is obtained from the intercept with the shear stress axis. As outlined in section 7.4.4.5, the static angle of internal friction and the cohesion of a granular material are a function of the consolidation stress. Therefore, they can be expressed also as a function of the major principal stress s1 . Table 13 reports the values obtained for the effective angle of internal friction and the cohesion for all the powders Table 10 Worked example E5: material physical characteristics dp (mm) F45 (%) rp (kg mÀ3) A1 75 3.2 1730 A2 49 30 1730 Ballotini 300 — 2500 Table 11 Worked example E5: Consolidation stresses in internal yield loci experiments Normal load (N:) Normal stress sc (Pa) 300 1106.2 210 804.1 150 602.6 90 401.2 243 An Introduction to Particle Systems
42. investigated. Sample A2, which contains a large amount of fines exhibits the highest values for cohesion and effective angle of internal friction. The ballotini in contrast, are less cohesive, as reflected in the lowest values for both C and je À j. 7.6 CONCLUDING REMARKS In this chapter we have introduced some of the fundamental parameters which characterise solid materials, including particle size analysis, shape, Table 12 Worked example E5: Shearing stresses in internal yield loci experi- ments sc ¼ 1106.2 Pa sc ¼ 804.1 Pa sc ¼ 602.6 Pa sc ¼ 401.2 Pa si (Pa) si (Pa) si (Pa) si (Pa) 904.8 602.6 401.2 300.5 703.4 401.2 300.5 250.1 501.9 300.5 199.8 199.8 300.5 199.8 149.4 149.4 199.8 149.4 – – 0 200 400 600 800 1000 1200 0 500 1000 1500 2000 2500 σ (Pa) τ (Pa) Figure 23 Worked example E5: experimental static yield locus – Alumina A2, sc ¼ 1106.2 Pa Table 13 Worked example E5: failure properties of the powders Alumina A1 Alumina A2 Ballotini C (Pa) (s1 ¼ 700 Pa) 98 160 24 je Àj (o) (s1 ¼ 700 Pa) 22 25 6 244 Chapter 7
43. density and shear stress, and how they influence the flow properties of powders. The description of the fluid–particle interaction, for both the case of a single particle suspended in a fluid and when flow passes through a packed bed, has also been presented. We have introduced the concept of fluidisation and the principle forces acting on a particle under equilibrium conditions. We have also shown how particle char- acteristics influence the fluidisation behaviour and presented some industrial applications. A number of relevant worked examples have also been presented to provide the reader with the direct application of the theory. NOMENCLATURE A Cross-sectional area of the bed m2 Ar Archimedes number — C Cohesion Pa Cd Cohesion from dynamic yield locus Pa CD Drag coefficient — dp Mean particle diameter m dpi Sieve aperture m ds Surface diameter m dv Volume diameter m dsv Surface/volume diameter m d* p Dimensionless particle diameter Equation (16) — D Bed diameter m F Friction factor — Fb Buoyancy force N Fd Drag force on a single particle N F45 Fraction of fine particles below 45 mm g Acceleration due to gravity m sÀ2 K Permeability Pa sÀ1 H Bed height m L Length of bed m m Mass of a single particle kg n Richardson–Zaki parameter Equation (38) — Nc Pre-consolidation load kg Ni Normal load in shearing phase i kg Np Number of particles in the bed — 245 An Introduction to Particle Systems
44. (continued ) Re Reynolds number — Remf Reynolds number at minimum fluidisation — Rep Particle Reynolds number — Ret Particle terminal Reynolds number — U volumetric flux m3 sÀ1 mÀ2 u Superficial gas velocity m sÀ1 uf Fluid phase velocity m sÀ1 umf Minimum fluidisation velocity m sÀ1 umb Minimum bubbling velocity m sÀ1 ut Particle terminal fall velocity m sÀ1 uà t Dimensionless particle terminal fall velocity Equation (17) — Vp Volume of particle m3 xi Mass fraction of particles in each size range — Greek symbols d Dynamic angle of internal friction Deg e Bed voidage — emf Minimum fluidisation voidage — f Particle sphericity — fi Angle of internal friction Deg je Effective angle of internal friction Deg m Gas viscosity Ns mÀ3 rBDL Loose bulk density kg mÀ3 rBDP Packed bulk density kg mÀ3 rABS Absolute density kg mÀ3 rb Bulk density kg mÀ3 rg Gas density kg mÀ3 rf Fluid density kg mÀ3 rp Particle solid density kg mÀ3 DP Bed pressure drop Pa s Normal stress N mÀ2 sc Consolidation stress N mÀ2 sd Mean deviation m sd /d50 Relative diameter spread — s1 , s2 Major stress N mÀ2 si , sii Shearing stress N mÀ2 t Shear stress N mÀ2 246 Chapter 7
45. REFERENCES 1. T. Allen, Particle Size Measurements, Chapman and Hall, London, 1996. 2. M. Rhodes, Introduction to Particle Technology, Wiley, London, 1998. 3. P. Lettieri, J.G. Yates and D. Newton, Homogeneous bed expan- sion of FCC catalysts, influence of temperature on the Richardson– Zaki index n and on the particle terminal fall velocity, Powder Technol., 2002, 123, 221–231. 4. D. Geldart, Gas Fluidization Technology, Wiley, London, 1986. 5. J.P.K. Seville, U. Tuzun and R. Clift, Processing of Particulate Solids, Chapman & Hall, London, 1997. 6. B.S. Massey, Mechanics of Fluids, 6th edn, Van Nostrand Reinhold, London, 1989. 7. J.M. Dallavalle, Micromeritics, Pitman, New York, 1948. 8. A. Haider and O. Levenspiel, Drag coefficient and terminal velocity of spherical and nonspherical particles, Powder Technol., 1989, 58, 63–70. 9. H.P.G. D’Arcy, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris, 1856. 10. L.G. Gibilaro, Fluidization Dynamics, Butterworth Heinemann, Oxford, 2001. 11. S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog., 1952, 48(2), 89–94. 12. C.Y. Wen and Y.H. Yu, Mechanics of fluidization, Chem. Eng. Prog. Symp. Ser., 1966, 62, 100–111. 13. J.F. Richardson and W.N. Zaki, Sedimentation and Fluidization: Part I, Trans. Instn. Chem. Engrs., 1954, 32, 35–53. 14. A.R. Abrahamsen and D. Geldart, Behaviour of gas-fluidized beds of fine powders Part I. Homogeneous expansion, Powder Technol., 1980, 26, 35–46. 15. J.R. Grace, Contacting modes and behaviour classification of gas– solid and other two-phase suspensions, Can. J. Chem. Eng., 1986, 64, 353–363. 16. D. Geldart, Types of fluidization, Powder Technol., 1973, 7, 285. 17. J.P.K. Seville, C.D. Willett and P.C. Knight, Interparticle forces in fluidization: a review, Powder Technol., 2000, 113, 261–268. 18. J. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1991. 19. R.L. Carr, Particle behaviour storage and flow, Br. Chem. Eng., 1970, 15(12), 1541–1549. 247 An Introduction to Particle Systems
46. 20. D. Geldart and A.C.Y. Wong, Fluidization of powders showing degrees of cohesiveness – I. Bed expansion, Chem. Eng. Sci., 1984, 39(10), 1481–1488. 21. J. Schwedes, Consolidation and flow of cohesive bulk solids, Chem. Eng. Sci., 2002, 57(2), 287–294. 22. A.E. Jenike, Gravity Flow of Bulk Solids, Utah Engineering Exper- imental Station, University of Utah, Bulletin 108, 1961. 23. A.E. Jenike, Storage and Flow of Solids, Utah Engineering Exper- imental Station, University of Utah, Bulletin 123, 1964. 24. G. Bruni, D. Barletta, M. Poletto, P. Lettieri, A rheological model for the flowability of fine aerated powders, Chem. Eng. Sci., 2006, in press. 25. D. Kunii and O. Levenspiel, Fluidization Engineering, Butterworths, London, 1991. 26. J.G. Yates, Effects of temperature and pressure on gas–solid fluid- ization, Chem. Eng. Sci., 1996, 51(2), 167–205. 248 Chapter 7