Module 2: Flow Of Fluids Through Packed And Fluidized Columns

2. MODULE 2: FLOW OF FLUIDS THROUGH PACKED AND FLUIDISED COLUMNS 2.1 Flow of Fluids Through packed beds and granular columns Topics: 2.1.1 1. Introduction 2. Flow of a single liquid through a granular bed 3. Specific surface area and voidage 4. General expression for flow through packed beds: The CarmanKozeny equation 5. Streamline and turbulent flow 6. Effect of bed structure K” and bed structure 7. Other useful equations (and regions of validity) INTRODUCTION The flow of fluids through beds of stationary particles commonly occurs in chemical process systems. Expressions are therefore needed to predict the pressure drop, ΔP, across beds due to the resistance to flow. Alternatively, through measuring pressure drop, the available surface area for reaction or adsorption in the bed may be calculated. Examples of such systems are: ▪ catalytic fixed bed reactors ▪ drying columns ▪ adsorption columns ▪ gas absorption into a liquid through counter current flow in packed columns ▪ cake filtration ▪ deep bed filtration. In conventional filtration, the bed depth increases the filtration progresses, and hence either the pressure drop increases or the flow rate decreases. The flow of either a single phase through a bed of particles or the more complex flow of two phases is approached by using the concepts of laminar flow of incompressible fluids through pipes. In general, however, a complete analytic solution is not possible (especially for turbulent conditions) and empirical correlations are required. 2.1.2 FLOW OF A SINGLYE LIQUID THROUGH A GRANULAR BED The first work reported on flow through packed beds is that of Darcy in 1830. He observed a sand – water system. He found that the average velocity measured over the area of the bed, uc, was proportional to the driving pressure and inversely to the bed thickness. This relationship is commonly known as Darcy's Law: uc = K - P l Eqn 2.1 where -ΔP = pressure drop across the bed, l = bed thickness, uc = average flow velocity across the bed area = (1/A)(dV/dt), A = total cross-sectional are of the bed, V = the volume in time t and K = a constant  bed and fluid properties. The linear relationship indicates laminar flow conditions i.e. streamline flow. This is reasonable where there is a low flow through the pores since velocities and pore diameters are small. In this case, resistance to flow is due primarily to viscous drag. As streamline flow is considered, it is useful to recall the Hagen-Pouseulle equation for laminar flow in pipes: − dP d 2 u = max dl 32 2 Eqn 2.2 To describe flow through the packed bed, the effect of the fluid property (viscosity) and particle properties, combined in K, can be separated in terms of the fluid viscosity  and a permeability coefficient, B, defined that depends only on the bed properties, i.e. uc = B - P l Eqn 2.3 The value of B will indicate the ease with which the fluid will pass through the bed, and is commonly used in filtration. Values are of the order of 10-8 m2 or more broadly in the range 10-7 to 10-10 m2. Table 2.1: 2.1.3 Properties of beds of some regular shaped particles (Sourced from: Coulson and Richardson, Vol 2) Specific surface area and voidage The general structure of a packed bed can often be defined by the specific surface area presented to the fluid per unit volume of the bed, SB and the fractional voidage, e. SB has units 1/m while e is dimensionless. The fractional voidage is that volume fraction of the bed not occupied by solids (e), also called the voidage or porosity. Thus, the volume of solids is (1-e). For a single particle, the specific surface area is the surface area divided by the volume, and also has units (1/m). For a sphere: S = [d2]/[d3/6] = 6/d Eqn 2.4 Note that S and SB are not equal, due to the voidage occurring when the particles are packed into a bed. If only point contact occurs, so that almost no surface area is lost by touching, then SB = S(1-e) Eqn 2.5 Note that e can be very high, values of 0.95 are possible for fibres, for example, but values of 0.4 are more common, and would be good estimate if no other information is available. See Table 8.1 for examples. Also note that: ▪ as the particle diameter decreases, so S increases ▪ as the fractional voidage increases, so B increases and flow through the bed is easier. 2.1.4 GENERAL EXPRESSION FOR FLOW THROUGH PACKED BEDS: THE CARMAN- KOZENY EQUATION The analogy between streamline flow through a tube and through the pores of a packed bed is a useful starting point for deriving a general expression. We know (or should know) that for streamline flow through a tube: u= where u dt lt  2 d t - P 32  l t = = = = Eqn 2.6 mean velocity of the fluid diameter of the tube length of tube viscosity of fluid If we now assume that the free space in the bed consists of a series of tortuous channels (of diameter d’m and length l’), this can be rewritten as: 2 m (-P) d u1 = K   l where u1 d’m l’ K = = = = Eqn 2.7 average velocity through pore channels equivalent diameter of pore channels length of channel dimensionless constant dependent on bed structure Note that u1 and l' represent conditions in the pores and are not the same as uc and l in Darcy’s Law, which describe the total flow velocity. However, it can reasonably be assumed that l' is proportional to l. The relationship between uc and u1 can be derived by considering that a cubic packed bed (of side X) has a free volume of eX3 and a mean area for flow given by the free volume divided by the height, or eX2. The volumetric flow rate through the cube is ucX2, and hence u1, the average linear velocity through the pores, is given by: 2 u1 = uc X u = c 2 eX e Eqn 2.8 This is reasonably accurate for random packings, but is not true for all regular packings. Imagine a bed of regularly packed spheres as an example, in which e changes at each point in the packing (e = 0.476 overall, but varies from 0.215 through to 1 depending on whether across particles or between successive layers). An expression for d'm is required. Kozeny proposed that d m = e = SB e S(1 - e) Eqn 2.9 This is justified as e volume.of .voids. filled .with.liquid = SB wetted .surface.area.of .bed or e cross − sec tional.area. perpendicular.to. flow = SB wetted . perimeter For non-cylindrical pipes under turbulent flow conditions, the diameter may be approximated by the hydraulic diameter through empirical correlation, where the hydraulic diameter is given as [(cross-sectional area/wetted perimeter)x4]. Hence: e/SB = 0.25 hydraulic diameter Using: u1 = uc/e l' α l d m = e = SB e S(1 - e) and 2 d m (-P) = u1 K   l Eqn 2.7 we find: 1 e3 (-P) K  S 2B  l 3 1 (-P) e uc = 2 2 K  S (1 - e )  l uc = Eqn 2.10 Eqn 2.11 Equation 2.11 is the Carman-Kozeny equation, which is valid for streamline conditions only. Comparison with Darcy's Law indicates that: 3 B= e 2 K  S (1 - e)2 Eqn 2.12 K'' is commonly known as Kozeny's constant and is approximately 5, although it does depend on porosity, shape and other factors. For spheres, where S=6/d. Inserting both these into Eqn 2.11, then uc = 3 2 1 e (-P) d 180 (1 - e)2  l e (-P) d 2 l (1 - e) 3 u c = 0.0055 2 Eqn 2.13 For non-spherical particles, the surface mean diameter (also known as the Sauter diameter) is used for the average particle diameter. Example 1 A toxic dust collected from the baghouse filter of a furnace was subjected to leachate testing in a glass column (lysimeter). A 1 m head of water was applied to the 30 cm diameter column, packed 1 m deep with the dust, and 100 ml of leachate was collected after 24 hours. a. Calculate the permeability coefficient of the dust. b. For comparison, given that the dust was found to have a particle size of 5 μm, work out the specific surface area of spherical particles of that size and estimate the permeability coefficient if Carman-Kozeny type of flow occurred. 2.1.5 STREAMLINE AND TURBULENT FLOW The Carmen Kozeny equation can be extended to cover turbulent conditions in much the same way as the Reynolds number-drag coefficient technique used before. A modified Reynolds number is obtained by taking the same velocity (u1=uc /e) and characteristic linear dimension (d'm = e/SB = e/S(1-e)) as were previously used to derive the Carmen Kozeny equation: Re1 = uc e  e S (1 − e)  Re1 = uc  S (1 − e)  Eqn 2.14 The friction factor, R1/ρu12, to be plotted against the modified Reynolds number can be related to the bed properties as follows: Remember that R1 is the component of the drag force per unit area of particle surface in the direction of motion. In a bed of unit cross sectional area and depth l: ▪ the particle volume = l.(1-e) and ▪ the total surface = S.l.(1-e). Thus the resistance force is R1.S.l.(1-e). The force on the fluid equals the pressure difference, and the fluid area is e, thus (-ΔP)e = R1 S.l.(1-e) Eqn 2.15 e (−P) S (1 − e) l Eqn 2.16 3 (-P) 1 R1 = e 2  u1 S (1 - e) l  u c2 Eqn 2.17 R1 = From which, and using (u1=uc /e): The friction factor is plotted against the modified Reynolds number, using logarithmic scales, to yield a general correlation for the pressure drop-flow rate relationship (Figure 2.1). Figure 2.1: Carmen’s graph of R1/u12 versus Re1 It has been found that the relationship between the friction factor and modified Reynolds number can be described by a general equation: R1 = 5 -1 + 0.4 -0.1 Re1 Re1  u12 Eqn 2.18 thus, including the laminar viscous drag term and a turbulent eddy term (Curve A, Figure 2.1). In the laminar regime (i.e. Re1 < 2 for packed bed flow), the viscous drag term dominates (Eqn 2.19). This is illustrated by combining Eqn 2.17 with the Carmen Kozeny equation to yield: R 1 = 5 -1 Re1  u12 Eqn 2.19 The change from laminar to turbulent flow is very gradual, as turbulence occurs in some pores before others, and in successively smaller pores as the Reynolds number increases. For hollow packings, it has been found that Eqn 2.18 under-represents the friction factor. A better representation is given by (curve B, Figure 2.1) R1 = 5 -1 + -0.1 Re1 Re1  u12 Eqn 2.20 For flow through industrial ring packings, Ergun found that the semi empirical correlation given in Eqn 2.21 and curve C of Figure 2.1 describes the system satisfactorily: Again, term 1 illustrates viscous losses and term 2 kinetic energy losses, hence the equation is relevant across a broad range of modified Reynolds number. R1 = 4.17 -1 + 0.29 Re1  u12 Eqn 2.21 It can (and should) be shown that, by assuming spherical particles and a voidage of 0.4, the Ergun equation is derived from the more commonly quoted form: (-P) (1 - e )2  u c (1 - e)  u c2 = 150 + 1.75 3 2 3 l d e d e Eqn 2.22 Example 2 Water flows through 3.6 kg of glass particles of density 2590 kg/m3 forming a packed bed of depth 0.475 m and diameter 0.0757 m. the variation in frictional pressure drop across the bed with water flow rate in the range 200 to 1200 m3/minute is shown in Table 1. Table 1: experimental data showing the pressure vs superficial velocity relationship a) Demonstrate that the flow is laminar. \ b) Calculate the relevant Reynolds number. 5.4 2.1.6 EFFECT OF BED STRUCTURE The equations presented suggest that the pressure drop across the bed is only affected by the bed characteristics of voidage e and specific surface S (affected by particle size d). However, structure of the bed, and thereby pressure drop across it, is also affected by the particle size distribution, particle shape and way in which the bed is packed. Further, bed packing is affected by nature of the walls and nature of the bed support. Hence, some scatter of data around predictions of above equations can be expected. These are illustrated through effect of K’’ on bed properties. 2.1.6.1 K’’ and Bed Structure: Tortuosity: where K’’ = (l’/l)2 K0 l’/l = tortuosity, a measure of path length through bed K0 = factor dependent of shape of channel = 2.0 for streamlined flow through circular channel = 2.65 for streamlined flow through rectangular channel with 10:1 ratio Wall effects: Particles do not pack as tightly close to the wall so the resistance to flow is less in a small diameter column that in a column of infinite diameter. The following correction factor is used: fw = (1 + 0.5 Sc/S)2 where Sc is the surface of the container per unit volume and the Carmen Kozeny equation is multiplied by this factor. Non-spherical particles: K’’ lies in range 3 to 6. See Figure 2.2. Figure 8.2: Variation in Kozeny constant K’’ with voidage for particle shape Spherical particles: Typically, k’’ = 4.8  0.3 Where the mixture of particle size crosses a larger range, then K’’ decreases and difficulty exists in predicting bed behaviour accurately. Voidage: Deviation from the Carmen Kozeny analysis increases as the fractional voidage increases, especially for e > 0.6. This is explained by flow changing from flow through channels to flow past a series of obstacles. 2.1.7 OTHER USEFUL EQUATIONS (AND REGIONS OF VALIDITY) If we define a term fv, which is the ratio of the pressure drop to the viscous energy: f v= P d 2  3 Lu(1 -  )2 then the Ergun equation can be written as: f v = 150 + 1.75 1 Re (1 -  ) Re  2000 (1 -  ) A number of alternatives have been developed, each applicable in a different regime. The following are available (consult textbook and state them below): Wentz and Thodos: Handley and Heggs: Hicks: Macdonald et al.: . 2.2 Flow through fluidised column TOPICS: 2.2.1 1. 2. 3. 4. 5. 6. 7. Introduction Fluid bed properties and applications Effect of fluidized velocity on pressure drop Minimum fluidization velocity under streamline conditions A general equation for minimum fluidisation velocity Minimum fluidization velocity in terms of settling velocity Bed expansion INTRODUCTION If a fluid passes upwards through a bed of particles, at low flow rates the pressure drop across the bed is the same as for a packed bed. As the flow rate is increased, however, a point is reached when the frictional drag exerted by the fluid on the particle becomes equal to the apparent weight of particles in the bed (actual less buoyancy), and the bed starts to expand, the particles just touching. As the velocity increases, this expansion continues with particle separating from each other and being freely supported in the fluid. The bed is now fluidised. Further increases in fluid velocity causes the bed to expand further, and the pressure difference across the bed remains approximately equal to the weight per unit area of the bed, owing to increased voidage. In a liquid-fluidised system, as the velocity increases, the amount of agitation of the particles increases, with the bed maintaining its uniform character. This is known as particulate fluidisation, or a quiescent bed. In this case, the motion of a particle is chaotic, resembling Brownian movement, and the particle maintains a mean position in the bed. In a gas-fluidised system, the range of velocities for which particulate fluidisation is possible is considerably lower for gases than for liquids. For gases, at higher velocities two phases are formed: the dense (or emulsion) phase and the lean (or bubble) phase. This is called aggregative fluidisation, the bed taking on the appearance of a boiling liquid (or boiling bed). Coalescence of bubbles may occur, and slugs of gas formed that occupy the cross section of the bed can be formed. A particle will traverse the entire bed, with a tendency to clump or move as an aggregate. It has been suggested that the Froude number (umf2/gd, where umf is the minimum velocity required for fluidisation), with values less than unity resulting in particulate fluidisation. 2.2.2 FLUID BED PROPERTIES AND APPLICATIONS Fluidised beds have long been known in laboratories, but only gained industrial importance when they were adopted by the oil industry for catalytic cracking or synthesis of fuels. They are commonly used for reactors due to the following properties: 2.2.2.1 Fluidised bed properties 1. Behaves as a fluid of combined density of fluid and solids 2. Transmits hydrostatic forces – objects float if densities are lower than that of the bed. 3. Intimate mixing 4. Heat transfer is rapid – uniform temperatures are quickly attained. 5. Temperature control for exothermic processes. 2.2.3 EFFECT OF FLUIDISED VELOCITY ON PRESSURE DROP If a porous media consists of individual solid particles and a fluid is made to flow through it, the fluid will exert drag forces upon the particles. The energy lost in the fluid drag (with laminar flow mainly skin friction) is noticeable as a drop in pressure. As long as the solid particles remain in contact, we call the system a “fixed bed” of particles. Let us now consider upward flow of a fluid through a packed bed of solids. At low flowrates, an increase in fluid velocity will result in a corresponding increase in the pressure drop across the bed. If the fluid velocity is increased, a point is reached whereby the drag force on the particles in the bed is equal to the weight of the bed. This velocity is characteristic for the fluid and the solid particles under consideration (i.e characteristic for the system). If the fluid velocity is increased beyond this value, the pressure drop will remain constant and the particles will begin rearranging themselves until they attain their loosest possible packing. We notice this re-arrangement by the expansion of the bed. The bed is then in a state of “incipient fluidisation”. At still higher fluid velocities, the particles become freely suspended in the fluid stream and the bed is said to be in a fluidised state: it has become a “fluidised bed”. As the fluid velocity exceeds the terminal velocity of the individual particles, the particles will begin to dance up and down in the bed, in the same fashion as described for teeter beds. In this fluidised state, the solids have now become the dispersed phase.This can be represented graphically on a double plot of the pressure drop P against the superficial velocity uc. O Figure 1: Pressure drop over fixed and fluidised beds (Coulson & Richardson). If the pressure drop across the bed is plotted against velocity (using logarithmic coordinates), a linear relationship is obtained up to the point where bed expansion takes place. The slope then decreases as the bed expands. The pressure drop then passes through a maximum, and falls slightly before attaining an approximately constant value, independent of velocity. In the absence of channelling, the size and shape of particles alone determines the maximum porosity and pressure drop at fluidisation for a bed of given dimensions. In an ideal system the pressure drop is equal to the buoyant weight of the particles per unit area. The velocity for fluidisation can be calculated from the equations used for flow through packed beds, using the porosity at the onset of fluidisation and the pressure drop equal to the apparent bed weight per unit area. There may be some discrepancy, due to channelling, electrostatic forces, agglomeration and friction. This is done as follows: In a fluidised bed, the total frictional force on the particle must equal the effective bed weight; thus: pressure drop = weight of particles − upthrust on particles bed cross sec tion area For a bed of particles of density ρs, fluidised by a fluid of density ρ to form a bed of depth l - P = Al (1 - e)(  s -  ) g A and voidage e in a vessel of cross section area A: - P = (1 - e)(  s -  ) l g A plot of fluid pressure loss across the bed versus superficial fluid velocity through the bed would have the appearance of Figure 1. The straight-line region OA is the packed bed region. Here the solid particles do not move relative to one another, and their separation is constant. The pressure loss versus fluid velocity relationship in this region is described in general by the Ergun equation. The region CD is the fluidized bed region where the Equation above applies. At point A it will be noticed that the pressure loss rises above the value predicted Equation above. This rise is more marked in powders which have been compacted to some extent before the test and is associated with the extra force required to overcome interparticle attractive forces. This must hold from initial expansion up to the velocity where transport of the solids out of the bed occurs. Transport occurs when the terminal settling velocity of the particles is exceeded. If conditions are streamline, (usually liquid-solid systems) the C-K equation can be used, and for spheres. - P d e 2 (1 - e )  l 3 u c = 0.0055 2 This can be used to determine the relationship between the velocity through the bed, uc, and the bed voidage, e. This, in turn, gives the bed expansion (or bed depth). Substituting for the pressure drop; 2 3 e d ( s -  ) g u c = 0.0055 (1 - e)  Note that the same assumptions as for C-K applies - STREAMLINE FLOW through the bed. Example 3 Glass particles of 6mm diameter are fluidised by water at a velocity of 0.25m/s. What will be the voidage of the bed? (Glass density = 2800 kg/m3; viscosity of water = 1 mNs m-2; Density of water = 1000 kg/m3). Hint: assume very small values of d/dt when calculating the value of n which is a factor that depends on the system. 2.2.3.1 Minimum fluidisation velocity As the upward velocity of the fluid is increased, a point is reached where the particles are just freely supported - the point of incipient (or minimum) fluidisation. The voidage at this point (emf) will not be known accurately but is approximately 0.4. The corresponding minimum fluidisation velocity, umf, is then obtained from: u mf = 0.0055 3 2 emf d (  s -  ) g (1 - emf )  This applies to streamline flow, i.e. low Reynolds numbers, and small particles. More general equations, such as the Ergun equation, can be used to find umf in other cases, by substituting emf for e and for the pressure drop as before. The Reynolds number here is defined by the minimum fluidisation velocity: Re mf = u mf d  It can be shown (and is a good exercise) that the Ergun equation becomes: Ga = 150 1 − e mf e mf 3 Re mf + 1.75 e mf 3 Re mf 2 for a typical value of emf , i.e. 0.4: Ga = 1406 Re mf + 27.3 Re mf 2 from which Remf and umf can be determined explicitly by solving the quadratic equation. This equation, as for flow through packed beds, is generally applicable over all flow conditions, although it has been suggested that the pressure drop is overpredicted. These equations should NOT be used to estimate the bed expansion, therefore. 2.2.3.2 Minimum fluidisation velocity in terms of terminal settling velocity One would expect that the particle properties that determine its terminal settling velocity will also affect the fluidisation velocity. Hence, we would expect some relationship between these to hold. For a known value of the Galileo number and voidage at minimum fluidisation, Re'0 and Re'mf can be calculated from the above equations and that previously determined for the terminal settling velocity. The ratio (Re'0 / Re'mf = u0 / umf) can be plotted as a function of Ga with emf as parameter. For low values of Ga (<3.6) we had that Ga = 18Re'0, and also, that (for Re'mf < 0.003) Re'mf. 0.000712Ga (as the quadratic term can be neglected). while for high values (Ga>105) Ga = Re'02 and also, that (for Re'mf >~200) Re'mf. 0.191Ga (as the linear term can be neglected). It can then be seen that for low Ga and Re'mf , that Re'0 / Re'mf = u0 / umf = 78 while for high values Re'0 / Re'mf = u0 / umf = 9.1 This has the implication that u0/umf is much larger for low values of Ga (generally small particles) than for high Ga numbers. For particulate fluidisation, generally in solid-liquid systems, the theoretical range of fluidising velocities is from the minimum fluidisation velocity to transport at the terminal settling velocity. In the streamline regime, this range is considerably greater than in the turbulent regime. In gas-solid systems, considerably higher velocities can be obtained, since a higher proportion of the gas can pass through the bed as bubbles and effective bypasses the particles, not contributing to their fluidisation. Example 4 4. A catalyst having spherical particles of diameter = 50 microns and density of 2700 kg/m3 is to be used to contact a hydrocarbon vapour in a fluidised reactor at 500oC, and 1 atm pressure. At operating conditions, the fluid viscosity is 1.8 mNs m-2 and its density is 760 kg/m3. Assume the porosity of 0.42. a. Determine the superficial velocity necessary to fluidise the bed. b. The velocity at which transport begins. 2.2.4 BED EXPANSION Liquid fluidised systems usually have a regular expansion of the bed that takes place as the velocity increases from umf to u0. For fluidisation of uniform particles, it has been found that: u c Re c = = en u o Re 0 where the subscript c denotes the fluidising velocity and where n is a factor that depends on the system. Note that the pressure drop across a fluidised bed of a given expansion is directly proportional to the (unfluidised) depth, the fluidising velocity is independent of depth. Assuming that this relationship also holds at the point of incipient fluidisation. u mf uo = Re mf Re 0 = en Hence, by rearranging: n= log(u mf / u o ) log(emf ) = − log(Re 0 / Re mf ) log(emf ) From general relationships for Re'0 and Re'mf in terms of Galileo number, we can write n = 2.51 log (1.83 Ga0.018 - 1.2 Ga-0.016 )13.3 1 + 5.53x 10-5 Ga - 1 (It is considered prudent for students to be able to derive this equation from those previously given) This equation can be used to predict the voidage of a bed at a specific velocity, under both laminar and turbulent conditions. Example: Glass particles of 4mm diameter are fluidised by water at a velocity of 0.25m/s. What will be the voidage of the bed? Glass density = 2500 kg/m3. Ga = 9.52 x 105 u0 = 0.45 m/s n = 2.63 e = 0.77 2.2.5 Relevant Powder and Particle Properties The correct density for use in fluidization equations is the particle density, defined as the mass of a particle divided by its hydrodynamic volume. This is the volume "seen" by the fluid in its fluid dynamic interaction with the particle and includes the volume of all the open and closed pores (see Figure 2). Figure 2: Hydrodynamic volume of a particle Particle density = mass of particle hydrodynamic volume of particle For non-porous solids, this is easily measured by a gas pycnometer or specific gravity bottle, but these devices should not be used for porous solids since they give the true or absolute density, abs , of the material of which the particle is made and this is not appropriate where interaction with fluid flow is concerned. Absolute density = mass of particle volume of solids material making up the particle For porous particles, the particle density p (also called apparent or envelope density) is not easy to measure directly although several methods are given in Geldart (1990). Bed density is another term used in connection with fluidized beds; bed density is defined as: Bed density = mass of particles in a bed volume occupied by particles and the voids between them For example, 600 kg of powder is fluidized in a vessel of cross-sectional area 1 m2 and achieves a bed height of 0.5 m. What is the bed density? Mass of particles in the bed = 600 kg. Volume occupied by particles and voids = 1 x 0.5 = 0.5 m3 Hence, bed density = 600/0.5 = 1200 kg/m3 If the particle density of these solids is 2700 kg/m3, what is the bed voidage? Bed density B is related to particle density p and bed voidage e by B = (1 - e) p. Hence, voidage = 1 - (1200/2700) = 0.555 Another density often used when dealing with powders is the bulk density: it is defined in a similar way to fluid bed density: Bulk density = mass of particles volume occupied by particles and the voids between them The most appropriate particle size to use in equations relating to fluid-particle interactions is a hydrodynamic diameter, ie. an equivalent sphere diameter derived from a measurement technique involving hydrodynamic interaction between the particle and fluid. In practice, however, in most industrial applications sizing is done using sieving and correlations use either sieve diameter, xp or volume diameter, xv. For spherical or near spherical particles xv is equal to xp. For angular particles, xv 1.13 xp. For use in fluidization applications, starting from a sieve analysis the mean size of the powder is often calculated from: mean xp = 1 / (mi/xi) where xi is the arithmetic mean of adjacent sieves between which a mass fraction mi is collected. This is the harmonic mean of the mass distribution, which is equivalent to arithmetic mean of a surface distribution. 2.2.6 Bubbling and Non-bubbling Fluidization Beyond the minimum fluidization velocity bubbles or particle-free voids may appear in the fluidized bed. Figure 3 shows bubbles in a gas fluidized bed. Figure 3: Effect of gas flow on bubbles in a two-dimensional fluidized bed of a Group B powder: (i) lower gas velocity, (ii) higher gas velocity. The equipment shown is a so-called "two-dimensional fluidized bed". This is a favourite tool of researchers looking at bubble behaviour. It consists of a vessel of a rectangular crosssection, whose shortest dimension (in the direction being viewed) is usually only 1 cm or so. At superficial velocities above the minimum fluidization velocity, fluidization may in general be either bubbling or non-bubbling. Some combinations of fluid and particles give rise to only bubbling fluidization and some combinations give only non-bubbling fluidization. Most liquidfluidized systems, except those involving very dense particles, do not give rise to bubbling. Figure 4 shows a bed of glass spheres fluidized by water and exhibiting non-bubbling fluidized bed behaviour. Figure 4: Expansion of a liquid-fluidized bed. (i) just above Umf, (ii) liquid velocity several times Umf. Note the uniform increase in void fraction. Gas-fluidized systems, however, give either only bubbling fluidization or non-bubbling fluidization beginning at Umf , followed by bubbling fluidization as fluidizing velocity increases. Non-bubbling fluidization is also known as particulate or homogeneous fluidization and bubbling fluidization is often referred to as aggregative or heterogeneous fluidization. 2.2.7 Classification of Powders Geldart (1973) classified powders into four groups according to their fluidization properties at ambient conditions. The Geldart Classification of Powders is now used widely in all fields of powder technology. Group A powders, when fluidized by air at ambient conditions, give a region of non-bubbling fluidization beginning at Umf , followed by bubbling fluidization as fluidizing velocity increases Group B powders give only bubbling fluidization under these conditions. Geldart identified two further groups: Group C powders - very fine, cohesive powders which are incapable of fluidization in the strict sense. Group D powders are large particles that are distinguished by their ability to produce deep spouting beds (see Figure 5) Figure 5: A spouted fluidized bed of rice (Group D solids) The fluidization properties of a powder in air may be predicted by establishing in which group it lies. It is important to note that at operating temperature and pressures above ambient a powder may appear in a different group from that which it occupies at ambient conditions. This is due to the effect of gas properties on the grouping and may have serious implications as far as the operation of the fluidized bed is concerned. Table 1 presents a summary of the typical properties of the different powder classes. Figure 6: Geldart's classification of powders Since the range of gas velocities over which non-bubbling fluidization occurs in Group A powders is small, bubbling fluidization is the type most commonly encountered in gasfluidized systems in commercial use. The superficial gas velocity at which bubbles first appear is known as the minimum bubbling velocity Umb. Premature bubbling can be caused by poor distributor design or protuberances inside the bed. Abrahamsen and Geldart (1980) correlated the maximum values of Umb with gas and particle properties using the following correlation:  d p  g0.06    = 2 . 07 exp( 0. 716 F ) u mb   0.347    where F is the fraction of powder less than 45 µm. Umb = superficial gas velocity at minimum bubbling; m/s dp = mean sieve size of a powder, m In Group A powders when Umb > Umf bubbles are constantly splitting and coalescing, and a maximum stable bubble size is achieved. This makes for good quality, smooth fluidization. Figure 7 shows bubbles in a Group A powder in a two-dimensional fluidized bed. Figure 7: Bubbles in a two-dimensional fluidized bed of Group A powder. Through splitting and coalescence, bubbles achieve a maximum stable size, effectively independent of gas velocity or vessel size. In Groups B and D powders Umb= Umf , bubbles continue to grow in size, never achieving a maximum size. This makes for rather poor-quality fluidization since there are large pressure fluctuations. In Group C powders the interparticle forces are large compared with the inertial forces on the particles. As a result, the particles are unable to achieve the separation they require to be totally supported by drag and buoyancy forces and true fluidization does not occur. Bubbles, as such, do not appear; instead, the gas flow forms channels through the powder. Figure 8: An attempt to fluidize a Group C powder produces channels or a discrete plug. Since the particles are not fully supported by the gas, the pressure loss across the bed is always less than apparent weight of the bed per unit cross-sectional area. Consequently, measurement of bed pressure drop is ones means of detecting this Group C behaviour if visual observation is inconclusive. Fluidization of sorts can be achieved with the assistance of a mechanical stirrer or vibration. When the size of the bubbles is greater than about one-third of the diameter of the equipment their rise velocity is controlled by the equipment, and they become slugs of gas. Slugging is attended by large pressure fluctuations. and so it is generally avoided in large units since it can cause vibration to the plant. If the bed is sufficiently shallow slugging is unlikely to occur at any velocity. According to Yagi and Muchi (1952), slugging will not occur provided the criterion in the Equation below is satisfied. This criterion works well for most powders. l mf D  1.9 ( p d p ) Where: L mf = height of bed at incipient fluidisation, m D = bed diameter, m dp = mean sieve size of a powder, m If the bed is deeper than this critical height then slugging will occur when the gas velocity exceeds Ums given by the Equation below (Baeyens and Geldart, 1974). ( U ms = U mf + 0.16 1.34 D 0.175 − l mf ) 2 + 0.07(gD ) 0.5 Where: Umf = superficial gas velocity at minimum fluidization; m/s Ums = minimum velocity for slugging; m/s