Settling Velocities of Particulate Systems†
Settling Velocities of Particulate Systems†
F. Concha Department of Metallurgical Engineering University of Concepción.1
Abstract This paper presents a review of the process of sedimentation of individual particles and suspensions of particles. Using the solutions of the Navier-Stokes equation with boundary layer approximation, explicit functions for the drag coefficient and settling velocities of spheres, isometric particles and arbitrary particles are developed. Keywords: sedimentation, particulate systems, fluid dynamics
Discrete sedimentation has been successful to 1. Introduction establish constitutive equations in processes using Sedimentation is the settling of a particle, or sus- sedimentation. It establishes the sedimentation prop- pension of particles, in a fluid due to the effect of an erties of a certain particulate material in a given fluid. external force such as gravity, centrifugal force or Nevertheless, to analyze a sedimentation process and any other body force. For many years, workers in to obtain behavioral pattern permitting the prediction the field of Particle Technology have been looking of capacities and equipment design procedures, an- for a simple equation relating the settling velocity of other approach is required, the so-called continuum particles to their size, shape and concentration. Such approach. a simple objective has required a formidable effort The physics underlying sedimentation, that is, the and it has been solved only in part through the work settling of a particle in a fluid is known since a long of Newton (1687) and Stokes (1844) of flow around time. Stokes presented the equation describing the a particle, and the more recent research of Lapple sedimentation of a sphere in 1851 and that can be (1940), Heywood (1962), Brenner (1964), Batchelor considered as the starting point of all discussions of (1967), Zenz (1966), Barnea and Mitzrahi (1973) the sedimentation process. Stokes shows that the set- and many others, to those Turton and Levenspiel tling velocity of a sphere in a fluid is directly propor- (1986) and Haider and Levenspiel (1989). Concha tional to the square of the particle radius, to the grav- and collaborators established in 1979 a heuristic itational force and to the density difference between theory of sedimentation, that is, a theory based on solid and fluid and inversely proportional to the fluid the fundamental principles of mechanics, but with viscosity. This equation is based on a force balance more or less degree of intuition and empirism. These around the sphere. Nevertheless, the proposed equa- works, (Concha and Almendra 1979a, 1979b, Concha tion is valid only for slow motions, so that in other and Barrientos 1982, 1986, Concha and Christiansen cases expressions that are more elaborate should be 1986), first solve the settling of one particle in a fluid, used. The problem is related to the hydrodynamic then, they introduce corrections for the interaction force between the particle and the fluid. between particles, through which the settling veloc- Consider the incompressible flow of a fluid around ity of a suspension is drastically reduced. Finally, the a solid sphere. The equations describing the phenom- settling of isometric and non-spherical particles was ena are the continuity equation and Navier Stokes treated. This approach, that uses principles of parti- equation: cle mechanics, receives the name of discrete approach v = 0 ∇ · to sedimentation, or discrete sedimentation. ∂v ρ + v v = p + µ 2v +ρg † ∂t Convective∇ · force −∇ ∇ Accepted:July 22nd, 2009 Diffusive force 1 Edmundo Larenas 285, Concepción Chile, E-mail: [email protected] (0.1)
ⓒ 2009 Hosokawa Powder Technology Foundation 18 KONA Powder and Particle Journal No.27 (2009) where v and p are the fluid velocity and pressure dimensionless form known as drag coefficient CD : field, ρ and µ are the fluid density and viscosity and FD CD = 2 2 (2.2) g is the gravity force vector. (1/2ρf u )(πR ) Unfortunately, Navier Stokes equation is non-linear where ρf is the fluid density. Substituting (2.1) into and it is impossible to be solved explicitly in a general (2.2), the drag coefficient of the sphere in Stokes flow form. Therefore, methods have been used to solve it is obtained: in special cases. It is known that the Reynolds num- 24 CD = (2.3) ber, Re = ρf du/µ where ρf , d and u are the fluid Re density, the particle diameter and velocity respective- ly, is an important parameter that characterizes the Macroscopic balance on a sphere in Stokes flow flow. It is a dimensionless number representing the Consider a small solid sphere submerged in a ratio of convective to diffusive forces in Navier Stokes viscous fluid and suspended with a string. If the equation. In dimensionless form, Navier-Stokes equa- sphere, with density greater than that of the fluid, tion becomes: is in equilibrium, the balance of forces around it is
1 ∂v∗ 1 1 2 1 zero. The forces acting on the particles are: gravity St ∂t + ∗v∗ v∗ = Ru ∗p∗ + Re ∗ v∗ F r ek ∗ ∇ · − ∇ ∇ − Fg, that pulls the sphere down, (2) buoyancy Fb, that (0.2) is, the pressure forces of the fluid on the particle that where the stared terms represent dimensionless vari- pushes the sphere upwards and (3) the string resis- ables defined by: v∗ = v/u0; p∗ = p/p0 ; t∗ = t/t0 ; tance Fstring, that supports the particle from falling. ∗ = L and u0 , p0 , t0 and L are characteristic The force balance gives: ∇ ∇ velocity, pressure, time and length in the problem, 0 = Fstring + Fg + Fb and St, Ru, Re and F r are the Struhal, Ruark, (2.4) ρpVpg +ρf Vpg Reynolds and Froud numbers and ek is the vertical − unit vector: 0 = Fstring ρpVpg + ρfVpg (2.5) 2 − t0u0 ρu0 Strouhal St = , Ruark Ru = , F = (ρ ρ ) V g ∆ρV g (2.6) L p0 string p − f p ≡ p ρu L u2 ∆ρ Reynolds Re = 0 , Froude F r = 0 (0.3) µ Lg If the string is cut, forces become unbalanced and, ac- When the Reynolds number is small (Re →0), for cording to Newton’s law, the particle will accelerate. example Re<10-3, convective forces may be neglected The initial acceleration can be obtained from the new in the Navier Stokes equation, obtaining the so called force balance, where the string resistance is absent. Stokes Flow. In dimensional form Stokes Flow is rep- The initial acceleration is: resented by: ρpVpa(t = 0) = ∆ρVpg v = 0 ∇ · ∆ρ ∂v 2 a(t = 0) = g ρ = p + µ v + ρg (0.4) ρ (2.7) ∂t −∇ ∇ p Once the particle is in motion, a new force, the drag, appears representing the resistance opposed by the 2 Hydrodynamic Force on a Sphere in Stokes fluid to the particle motion. This force F is propor- Flow D tional to the relative solid-fluid velocity and to the rel- Do to the linearity of the differential equation in ative particle acceleration. Since the fluid is at rest it Stokes Flow, the velocity, the pressure and the corresponds to the sphere velocity and acceleration. hydrodynamic force in a steady flow are linear func- Once the motion starts, the drag force is added and tions of the relative solid-fluid relative velocity. For the balance of forces becomes: the hydrodynamic force, the linear function, depend ρ V a(t) = ∆ρV g 6πµRu(t) (1/2)ρ V a(t) p p p − − p p (2.8) on the size and shape of the particle (6πR for the Net weight Drag force sphere) and on the fluid viscosity ( µ ). Solving the 3 ρpVp a (t) = ∆ρgV p 6πµRu(t) (2.9) boundary value problem and neglecting Basset term 2 − of added mass, yields (Happel and Brenner 1964): The term(1/2) ρpVp that was added to the mass ρpVp in the first term of equation (2.8) is called added mass F = 6πµRu (2.1) D − induced by the acceleration. It is common to write the hydrodynamic force in its Doe to the increase in the velocity u(t) with time, the
KONA Powder and Particle Journal No.27 (2009) 19 last term of (2.9) increases while the first term dimin- force is zero. This result is due to the fact that the ishes, and at a certain time becomes zero. The veloci- friction drag is zero in the absence of viscosity and ty becomes a constant called terminal velocity that the form drag depends on the pressure distribu- u = u which is a characteristic of the solid-fluid tion over the surface of the sphere and this distribu- ∞ system. From (2.9) with a(t) = 0, tion is symmetric leading to a zero net force. 2 ∆ρR2g 1 ∆ρd2g u = = (2.10) ∞ 9 µ 18 µ 4. Hydrodynamic Force on a Sphere This expression receives the name of Stokes Equation in Prandtl’s Flow and is valid for small Reynolds numbers. For intermediate values of the Reynolds Number, Sedimentation dynamics inertial and viscous forces in the fluid are of the same Equation (2.9) is the differential equation for the order of magnitude. In this case, the flow may be settling velocity of a sphere in a gravity field. It can divided into two parts, an external inviscid flow far be written in the form: from the particle and an internal flow near the parti- 2 18µ 2∆ρ cle, where the viscosity plays an important role. This u˙(t) + 2 u(t) g = 0 3 ρpd − 3ρp (2.11) picture form the basis of the Boundary-Layer Theory Its solution is: (Meksyn 1961, Rosenhead 1963, Golstein 1965, Schli- 1 ∆ρd2g 2 µ chting 1968). u(t) = 1 exp 2 t (2.12) 18 µ − −3 18ρpd In the external inviscid flow, Euler’s equations are The term inside the exponential term multiplying the applicable and the velocity and pressure distribution time t is called Stokes number and the term outside may be obtained from equations (3.2) and (3.3). The the parenthesis is the terminal velocity, as we already region of viscous flow near the particle is known as mentioned in (2.10). the boundary layer and it is there where a steep ve- locity gradient permits the non-slip condition at the solid surface to be satisfied. 3. Hydrodynamic Force on a Sphere The energy dissipation produced by the viscous in Euler’s F low flow within the boundary layer retards the flow and, When the Reynolds number tends to infinity (Re→ at a certain point, aided by the adverse pressure gra- ∞ ), viscous forces disappear and the Navier Stokes dient, the flow reverses its direction. These phenom- equation becomes Euler’s Equation for Inviscid Flow. ena force the fluid particles outwards and away from v = 0 the solid particle producing the phenomena called ∇ · ∂v boundary layer separation, which occurs at an angle of ρ v v = p + ρg (3.1) ∂t ∇ · −∇ separation given by (Lee and Barrow, 1968): In this case, the tangential component of the velocity 0.1 θ = 214Re− para 24 < Re < 10.000 (4.1) at the particle surface is also a linear function of the s relative solid-fluid velocity, but the radial component For Re=24 the value of the angle of separation is is equal to zero: θs=155.7 and for Re=10.000 it is θs=85.2. For Reyn- 3 olds numbers exceeding 10.000, the angle of separa- uθ(θ) = senθ u and u = 0 2 r (3.2) tion diminishes slowly from θs=85.2° to 84° and then Now, the pressure is given by a non-linear function maintains this value up to Re 150.000 (Tomotika called Bernoulli equation (Batchelor 1967): 1937, Fage 1937, Amai 1938, Cabtree et al 1963). 2 2 Due to the separation of the boundary layer, the p(θ) + 1/2ρf u = p + 1/2ρf u = constante θ region of closed streamlines behind the sphere con- 2 1 2 uθ tains a standing ring-vortex, which first appears at p(θ) p = ρf u 1 (3.3) − 2 − u Reynolds number of Re 24. Taneda (1956) deter- Substituting (3.2) into (3.3), the dimensionless pres- mined that beyond Re 130 the ring-vortex began sure, called pressure coefficient and defined by to oscillate and that at higher Reynolds numbers the 2 Cp = (p(θ) p)/(1/2)ρf u , may be expressed in fluid in the region of closed streamlines broke away − Euler’s flow by: and was carried downstream forming a wake.
9 2 The thickness “ δ ” of the boundary layer, defined Cp = 1 sen θ (3.4) − 2 as the distance from de solid surface to the region For an inviscid stationary flow, the hydrodynamic where the tangential velocity uθ reaches 99% of the
20 KONA Powder and Particle Journal No.27 (2009) value of the external inviscid flow, is proportional to to the total interaction force to be negligible. There- Re-1/2 and, at the point of separation, may be written fore, between Re 10.000 to Re 150.000 the drag in the form: coefficient is approximately constant at CD 0.44. δ δ = 0 (4.2) R Re1/2 For Reynolds numbers greater than Re 150.000, McDonald (1954) gives a value of δ0 = 9.06 . the flow changes in character and the boundary layer The separation of the boundary layer prevents the becomes turbulent. The increase in kinetic energy of recovery of the pressure at the rear of the sphere, the external region permits the flow in the boundary resulting in an asymmetrical pressure distribution layer to reach further to the back of the sphere, shift- with a higher pressure at the front of the sphere. Fig. ing the separation point to values close to θ s 110° 4.1 shows the pressure coefficient of a sphere in and permitting also the base pressure to rise. The terms of the distance from the front stagnation point effect of these changes on the drag coefficient is a over the surface of the sphere in an inviscid flow and sudden drop and after that, a sharp increase with the in boundary layer flow. The figure shows that the Reynolds number. Fig. 5.2 shows a plot of standard pressure has an approximate constant value behind experimental values of the drag coefficient versus the the separation point at Reynolds numbers around Re Reynolds number (Lapple and Shepherd 1940, Perry 150.000. This dimensionless pressure is called base 1963, p. 5.61) where this effect is shown. It shows the pressure and has a value of p∗ 0.4 (Fage 1937, variation of the drag coefficient of a sphere for dif- b ≈ − Lighthill 1963, p.108, Goldstein 1965, pp. 15 y 497, ferent values of the Reynolds number, and confirms 1/2 Schlichting 1968, p. 21). that for Re →0, C Re− and that for Re>>1, D ∝ The asymmetry of the pressure distribution ex- CD 0.43. → plains the origin of the form-drag, the magnitude of which is closely related with the position of the point 5. Drag Coefficient for a Sphere in the Range of separation. The farther the separation points from 0 1 0.5 0 0 0.5 1 1.5 2 2.5 3 -0.5 Pressure coefficient Cp -1 -1.5 Angle in radians Fig. 4.1 Pressure coefficient as a function of the distance from the front stagnation point over the surface of a sphere in an inviscid and Fig. 5.1 Physical model for the flow in boundary layer around a sphere a boundary layer flow. (Concha and Almendra 1979). KONA Powder and Particle Journal No.27 (2009) 21 Consider a solid sphere of radius R with an at- π = 2πa2 p(θ)senθ cos θdθ tached boundary layer of thickness δ submerged in 0 a flow at high Reynolds number (Abraham 1970). As- Since the values of p (θ) is different before and after sume that the system of sphere and boundary layer the separation point, separate the integrals into two has a spherical shape with a radius equal to a, which parts, from 0 to θs and from θs to π. can be approximated by a = R + δ , as shown in 2 θs π FD = 2πa p (θ) senθ d (senθ) + p (θ) senθ d (senθ) Fig. 3.1 (Abraham 1970, Concha and Almendra 0 θs 1979). Substituting the values of p (θ) from (5.2) and from Outside the spherical shell of radius a, and up to (5.3) and integrating we obtain: the point of separation θ = θs , the flow is inviscid 2 2 1 2 9 4 1 2 F = πa ρ u sen θ sen θ p∗sen θ and therefore the fluid velocity and the pressure dis- D f 2 s − 16 s − 2 b s tributions are given by: � (5.6) 3 Substituting a = R + δ and defining the function uθ (θ) = usinθ , for 0 θ θs (5.1) 2 ≤ ≤ f (θs, pb∗)in the form: 2 1 2 uθ 1 2 9 4 1 2 p(θ) = ρfu 1 , for 0 θ θs (5.2) f (θ , p∗) = sen θ sen θ p∗sen θ (5.7) 2 − u ≤ ≤ s b 2 s − 16 s − 2 b s Beyond the separation point, a region exists where we can write (5.6) in the form: 2 the pressure is constant and equal to the base pres- 2 2 δ FD = ρf u πR 1 + f (θs, pb∗) (5.8) sure pb : R In terms of the drag coefficient we have: p (θ) = p , for θ θ π (5.3) 2 b s ≤ ≤ δ CD = 2f (θs, p∗) 1 + (5.9) Since the effect of viscosity is confined to the interior b R of the sphere of radius a, the total drag excerpted by and, defining the new parameter C0 in the form: the fluid on a, consists of a form drag only, Then: C0 = 2f (θs, pb∗) (5.10) FD = p(θ) cos θdS (5.4) Sa and using equation (4.2) we obtain: 2 The element of surface of the sphere of radius a is: δ0 CD = C0 (θs, p∗) 1 + (5.11) b Re1/2 dS = a2senθdθdφ (5.5) Calculating the value of f (θs, pb∗) for θs = 84◦ and where φ is the azimuthally coordinate. Replacing p∗ 0.4 , we obtain f (84, 0.4) = 0.142 and b ≈ − − into (5.4) results: from (5.10), C = 0.284 0.28 . Using the value of 0 ≈ 2π π δ0 = 9.06 , we finally obtain: FD = 0 0 p (θ) senθ cos θdθ dφ � 1.E+03 1.E+02 D C 1.E+01 1.E+00 Drag Coefficient 1.E-01 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Reynolds number Re Fig. 5.2 Drag coefficient versus Reynolds number for a sphere. The continuous line is a simulation with equation (5.12). Circles are standard data from Lapple y Shepherd (1940 See table 4.1). See also Perry (1963) p. 5.61). 22 KONA Powder and Particle Journal No.27 (2009) 2 9.06 Re < 5 105. The merit of this last equation is its CD = 0.28 1 + (5.12) × Re1/2 fundamental foundation. Expression (5.12) represents the drag coefficient for a sphere in boundary layer flow (Concha and Almen- 6. Sedimentation Velocity of a Sphere dra 1979). A comparison with experimental data from Lapple and Shepherd (1940) is shown in Fig. 5.2. We have seen that when a particle settles at termi- Several alternative empirical equations have been nal velocity u , a balance is established between ∞ proposed for the drag coefficient of spherical par- drag force, gravity and buoyancy: ticles. See older articles reviewed by Concha and Al- Fdrag + Fgravity + Fbuoyancy = 0 mendra (1979), Zigrang and Sylvester (1981), Turton F = (F + F ) , and Levenspiel (1986), Turton and Clark (1987) and drag − gravity buoancy ≡ Haider and Levenspiel (1989), and the more recent net weight of the particle work of Ganguly (1990), Thomson and Clark (1991), FD = (ρpVp( g) + ρf Vpg) ∆ρVpg (6.1) Ganser (1993), Flemmer et al (1993), Darby (1996), − − ≡ Nguyen et al (1997), Chabra et al (1999) and Tsakala- In (6.1)∆ρ = ρs ρf is the solid-fluid density differ- − kis and Stamboltzis (2001). ence. This equation shows that the drag force for a It is worthwhile to mention the work of Brauer and particle in sedimentation is known beforehand, once Zucker (1976): its volume and the density difference to the fluid are 3 1/2 3 24 3.73 4.83 10− Re known. For a spherical particle, Vp = 4/3πR , so CD = 0.49 + + × Re Re1/2 − 1 + 3.0 10 6Re3/2 that: × − (5.13) 4 3 FD = πR ∆ρg (6.2) and that of Haider and Levenspiel (1989: 3 24 0.6459 0.4251 and the drag coefficient: CD = Re 1 + 0.1806Re + 1+6880.95/Re (5.14) FD 4 ∆ρdg CD = 2 2 2 (6.3) who presented� alternative empirical equations for the 1/2ρf u πR ≡ 3 ρf u ∞ ∞ drag coefficient of spherical particles in the range of where the sphere diameter is d = 2R and u is the ∞ Reynolds numbers less than 260.000. Both empirical terminal settling velocity of a sphere in an infinite flu- equations, (5.13) and (5.14), give better approxima- id. tions than Concha and Almendra’s equation (5.12) Since the Reynolds number for the motion of one for Reynolds numbers in the range of 5 103 < particle in an infinite fluid is defined by: × 1.E+03 Standard experimental curve Brauer and Sucker 1.E+02 Haider and Levenspiel D 1.E+01 1.E+00 Drag Coefficient C 1.E-01 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Reynolds Number Re Fig. 5.3 Comparison of the Drag Coefficient for a sphere, simulated by Brauer and Zucker (1976) and by Haider and Levenspiel (1989), and standard experimental points from Lapple and Shepherd (1940). KONA Powder and Particle Journal No.27 (2009) 23 du ρf Re = ∞ ∞ µ , (6.4) Re = d∗u∗ (6.9) f ∞ ∞ combining it with the drag coefficient yields two di- Replacing (5.11) and (6.9) into (6.7) we obtain: mensionless numbers (Heywood 1962): 2 3 δ0 2 2 d∗ = C0 1 + 1/2 (u∗ d∗) 2 4 ∆ρρf g 3 Re 3 ρf 3 (u d ) ∞ ∞ ∗ ∗ CDRe = 2 d = u ∞ ∞ 3 µ CD 4 ∆ρµf g ∞ 3/2 f 1/2 d∗ u∗ d∗ + δ0 (u∗ d∗) = 0 ∞ ∞ − 1/2 (6.5) C0 Concha and Almendra (1979a) defined the character- Solving these algebraic equation, explicit expressions istic parameters P and Q of the solid-liquid system: are obtained for the dimensionless settling velocity 1/3 1/3 u∗ of a sphere of dimensionless size d∗ and for the 2 ∞ 3 µf 4 ∆ρµf g P = Q = dimensionless size d∗ of a sphere settling at dimen- 4 ∆ρρ g 3 ρ2 (6.6) f f sionless velocity u∗ (Concha y Almendra 1979a): ∞ so, that equations (6.5) may be written in the form: 1/2 2 3 3 2 d 3 Re u 1 δ0 4 3/2 (6.10) 2 3 u∗ = 1 + 1/2 d∗ 1 CDRe = = d∗ ∞ = ∞ u∗ 4 d∗ 2 ∞ C0 δ0 − ∞ P CD Q ≡ ∞ 1/2 2 (6.7) 3/2 1 2 4δ0 (6.11) d∗ = C0u∗ 1 + 1 + 1/2 u∗− Expression (6.7) defines a dimensionless size d∗ and 4 C ∞ ∞ 0 a dimensionless velocity u∗ , which are characteristics of a solid-liquid system: Equations (6.10) and (6.11) are known as Concha and d u Almendra’s equations for a sphere. These two equa- d∗ = u∗ = ∞ (6.8) P ∞ Q tions are general for spheres settling in a fluid at any Since there is a direct relationship between the Drag Reynolds number. Introducing the values of Coefficient and the Reynolds Number, see for exam- C0 = 0.28 and δ0 = 9.06, the following final equa- ple equations (4.2) to (6.4), there must be a similar re- tions are obtained: 2 2 lationship between the dimensionless groups CDRe 20.52 3/2 1/2 u∗ = d 1 + 0.0921d∗ 1 (6.12) and Re/CD . Table 7.1 gives that relationship. ∗ − 2 Multiplying the two terms in equation (6.7), we can 2� 3/2 1/2 d∗ = 0.07u∗ 1 + 1 + 68.49u∗− (6.13) observe that the Reynolds number may be written in terms of the dimensionless size and velocity: Using equation� (6.12) the values of column 1.E+03 1.E+02 1.E+01 1.E+00 Dimensionles Velocity u* 1.E-01 Spheres Correlation 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Dimensionless Diameter d* Fig. 6.1 Dimensionless velocity versus dimensionless diameter for the sedimentation of spheres according to equation (6.12) of Concha and Almendra. Circles are standard data from Lapple y Shepherd (1940) in table 3.1. 24 KONA Powder and Particle Journal No.27 (2009) d∗sim and u∗sim are obtained. The plot of these study the interaction of particles in a suspension dur- data is shown in Fig. 6.1. ing sedimentation. These types of studies were dis- cussed in Tory (1996). In a recent approach, Quispe et al. (2000) used the 7. Sedimentation of a Suspension of Spheres tools of lattice-gas and cellular automata to study the In a suspension, the spheres surrounding a given sedimentation of particles and the fluid flow through sphere, as it settles, hinder its motion. This hindrance an ensemble of settling particles. They were able to is due to several effects. In the first place, when the obtain some important macroscopic properties of the particle changes its position, it can find the new site suspensions. Unfortunately, none of these works has occupied by another particle, and will collide with it yielded a sufficiently general and simple relationship changing its trajectory. The more concentrate the between the variables of the suspension and its set- suspension is the greater the chance of collision. The tling velocity to be used for practical purposes. result is that hindrance is a function of concentration. Concha and Almendra (1979b) assumed that the On the other hand, the settling of each particle of the same equation used for individual spherical particles suspension produces a back flow of the fluid. This is valid for a suspension of particles. Then, the sedi- back flow will increase the drag on a given particle, mentation of a suspension can be described by equa- retarding its sedimentation. Again, an increase in tion (6.12) replacing the parameters P and Q by concentration will increase the hindrance. It is clear P (ϕ) and Q (ϕ) , parameters depending on the vol- that in both cases, the hindrance depends on the frac- ume fraction of solids. Write equation (6.12) it in the tion of volume occupied by the particles and not on form: their weight and therefore the appropriate parameter 1/2 2 20.52 3/2 to measure hindrance is the volumetric concentration U ∗ = 1 + 0.0921D∗ 1 (7.1) D∗ − rather than the percent by weight. w h e r e Several theoretical works have been devoted to Table 7.1 Drag coefficient versus Reynolds Number and dimensionless velocity versus dimensionless diameter LAPPLE AND SHEPHERD (1940) CONCHA AND ALMENDRA 2 2 1/3 1/3 Re/CD d*=(CDRe ) u*=(Re/CD) CD sim d* sim u* sim 1.00E-01 2.40E+02 2.4 4.17E-04 1.34E+00 7.47E-02 246.158212 1.00E-01 4.35E-04 2.00E-01 1.20E+02 4.8 1.67E-03 1.69E+00 1.19E-01 126.541954 2.00E-01 1.73E-03 3.00E-01 8.00E+01 7.2 3.75E-03 1.93E+00 1.55E-01 86.1544439 3.00E-01 3.89E-03 5.00E-01 4.95E+01 12.4 1.01E-02 2.31E+00 2.16E-01 53.4219699 4.00E-01 6.88E-03 7.00E-01 3.65E+01 17.9 1.92E-02 2.62E+00 2.68E-01 39.1775519 5.00E-01 1.07E-02 1.00E+00 2.65E+01 26.5 3.77E-02 2.98E+00 3.35E-01 28.337008 6.00E-01 1.53E-02 2.00E+00 1.46E+01 58.4 1.37E-01 3.88E+00 5.15E-01 15.359281 7.00E-01 2.08E-02 3.00E+00 1.04E+01 93.6 2.88E-01 4.54E+00 6.61E-01 10.8703803 8.00E-01 2.70E-02 5.00E+00 6.90E+00 172.5 7.25E-01 5.57E+00 8.98E-01 7.1456645 9.00E-01 3.39E-02 7.00E+00 5.30E+00 259.7 1.3E+00 6.38E+00 1.10E+00 5.48098455 1.00E+00 4.16E-02 1.00E+01 4.10E+00 410.0 2.4E+00 7.43E+00 1.35E+00 4.18275399 1.10E+00 5.00E-02 2.00E+01 2.55E+00 1.02E+03 7.8E+00 1.01E+01 1.99E+00 2.56366185 1.20E+00 5.91E-02 3.00E+01 2.00E+00 1.80E+03 1.5E+01 1.22E+01 2.47E+00 1.97242199 1.30E+00 6.89E-02 5.00E+01 1.50E+00 3.75E+03 3.3E+01 1.55E+01 3.22E+00 1.45718355 1.40E+00 7.93E-02 7.00E+01 1.27E+00 6.22E+03 5.5E+01 1.84E+01 3.81E+00 1.21474559 1.50E+00 9.04E-02 1.00E+02 1.07E+00 1.07E+04 9.3E+01 2.20E+01 4.54E+00 1.01719408 1.60E+00 1.02E-01 2.00E+02 7.70E-01 3.08E+04 2.6E+02 3.13E+01 6.38E+00 0.75367474 1.70E+00 1.14E-01 3.00E+02 6.50E-01 5.85E+04 4.6E+02 3.88E+01 7.73E+00 0.64953579 1.80E+00 1.27E-01 5.00E+02 5.50E-01 1.38E+05 9.1E+02 5.16E+01 9.69E+00 0.55286511 1.90E+00 1.41E-01 7.00E+02 5.00E-01 2.45E+05 1.40E+03 6.26E+01 1.12E+01 0.5045975 2.00E+00 1.55E-01 1.00E+03 4.60E-01 4.60E+05 2.17E+03 7.72E+01 1.30E+01 0.46342473 2.10E+00 1.69E-01 2.00E+03 4.20E-01 1.68E+06 4.76E+03 1.19E+02 1.68E+01 0.40494085 2.20E+00 1.84E-01 3.00E+03 4.00E-01 3.60E+06 7.50E+03 1.53E+02 1.96E+01 0.38029197 2.30E+00 1.99E-01 5.00E+03 3.85E-01 9.63E+06 1.30E+04 2.13E+02 2.35E+01 0.35634822 2.40E+00 2.15E-01 7.00E+03 3.90E-01 1.91E+07 1.79E+04 2.67E+02 2.62E+01 0.34392446 2.50E+00 2.31E-01 1.00E+04 4.05E-01 4.05E+07 2.47E+04 3.43E+02 2.91E+01 0.33303434 2.60E+00 2.48E-01 2.00E+04 4.50E-01 1.80E+08 4.44E+04 5.65E+02 3.54E+01 0.31702494 2.70E+00 2.65E-01 3.00E+04 4.70E-01 4.23E+08 6.38E+04 7.51E+02 4.00E+01 0.31005856 2.80E+00 2.83E-01 5.00E+04 4.90E-01 1.23E+09 1.02E+05 1.07E+03 4.67E+01 0.3031495 2.90E+00 3.01E-01 7.00E+04 5.00E-01 2.45E+09 1.40E+05 1.35E+03 5.19E+01 0.29950474 3.00E+00 3.19E-01 1.00E+05 4.80E-01 4.80E+09 2.08E+05 1.69E+03 5.93E+01 0.29627397 3.10E+00 3.38E-01 2.00E+05 4.20E-01 1.68E+10 4.76E+05 2.56E+03 7.81E+01 0.29145983 3.20E+00 3.57E-01 3.00E+05 2.00E-01 1.80E+10 1.50E+06 2.62E+03 1.14E+02 0.2893397 3.30E+00 3.76E-01 5.00E+05 8.40E-02 2.10E+10 5.95E+06 2.76E+03 1.81E+02 0.28722112 3.40E+00 3.95E-01 7.00E+05 1.00E-01 4.90E+10 7.00E+06 3.66E+03 1.91E+02 0.28609695 3.50E+00 4.15E-01 1.00E+06 1.30E-01 1.30E+11 7.69E+06 5.07E+03 1.97E+02 0.28509658 3.60E+00 4.35E-01 3.00E+06 2.00E-01 1.80E+12 1.50E+07 1.22E+04 2.47E+02 0.28293691 3.70E+00 4.56E-01 KONA Powder and Particle Journal No.27 (2009) 25 d u The quotient between these two equations is: D∗ = and U ∗ = (7.2) P (ϕ) Q (ϕ) u∗ 1/2 Para Re = fp− (ϕ) fq (ϕ) (7.8) It is convenient to express the properties of a sus- ∞ → ∞ u∗ To find functional forms∞ for the functions fp (ϕ)and, pension, such as the viscosity and density , as the fq (ϕ), experimental values for the settling velocities product of the property of the fluid with a function of u and u (ϕ) are needed. concentration. Assume that P (ϕ) and Q (ϕ) can be ∞ related with P and Q in that form, then: Functional form for fp (ϕ) and fq (ϕ) Several authors have presented expressions for the P (ϕ) = P f (ϕ) and Q (ϕ) = Qf (ϕ) (7.3) p q velocity ratio u/u . See Concha and Almendra ∞ Replacing (7.3) into (7.2), and using the definitions (1979b). Richardson and Zaki (1954), made the most (6.7) of d∗ and u∗ , results in: comprehensive and most cited work on the relative d∗ u∗ particle-fluid flow under gravity. We will use their D∗ = and U ∗ = (7.4) fp (ϕ) fq (ϕ) data in this paper. With these definitions equation (7.1) may be written Richardson and Zaki (1954) performed careful in the form: sedimentation and liquid fluidization test with mono- 1/2 2 20.52 3/2 3/2 sized spherical particles in a wide range of particles u∗ = fp (ϕ) fq (ϕ) 1 + 0.0921fp− d∗ 1 d∗ − sizes and fluid densities and viscosities. They ex- (7.5) pressed their result in the form: This expression, known as Concha and Almendra’s u/u = (1 ϕ)n for Re 0 and u/u = (1 ϕ)m equation for a suspension of spheres, permits the calcu- ∞ − → ∞ − for Re lation of the settling velocity of a sphere of any size → ∞ and density when it forms part of a suspension with (7.9) volume fraction ϕ . To perform the calculations it is From (7.7), (7.8) and (7.9), we can write f (ϕ) 2 n necessary to determine the parameters p and f − (ϕ) f (ϕ) = (1 ϕ) p q − fq (ϕ). 1/2 m (7.10) f − (ϕ) f (ϕ) = (1 ϕ) p q − Asymptotic expressions for the sedimentation and, by solving this algebraic set we obtain: velocity (2/3)(m n) (1/3)(4m n) fp = (1 ϕ) − and fq = (1 ϕ) − For small values of the Reynolds number, Re→0, − − the following expressions may be derived from (6.12) (7.11) and (7.5), which reduce the settling equation to the In Table 7.1 the characteristics of these particles expression indicated: and fluid are given and values for u/u obtained ∞ from their experimental results are shown. 3/2 3/2 0.0921d∗ fp− << 1 Using equation (7.6) and (7.7) and the calculated ⇒ 2 0.0921 2 2 values in Table 7.1, the best values for m and n u∗ = 20.52 d∗ fp− (ϕ) fq (ϕ) 2 were: 3/2 0.0921d∗ << 1 n = 3.90 and m = 0.85 (7.12) ⇒ 2 0.0921 2 Then, u∗ = 20.52 d∗ ∞ 2 (7.6) 2.033 0.167 fp (ϕ) = (1 ϕ)− , fq (ϕ) = (1 ϕ)− In these expressions, the symbols u∗ and u∗ indi- − − ∞ cate the settling velocity of a particle in an infinite (7.13) medium and the velocity of the same particle in a Fig. 6.1 shows a plot of the dimensionless settling suspension respectively. The quotient between these velocity versus Reynolds number for spheres, accord- two terms is: ing to the experimental data of Richardson and Zaki u∗ 2 (1954) and the simulation of Concha and Almendra For Re 0 = fp− (ϕ) fq (ϕ) (7.7) ∞ → u∗ (1979) with equations (7.5) and (7.13). Data of u/u ∞ ∞ With a similar deduction we can write for high Reyn- calculated from Richardson and Zaki (1954) are in olds numbers, equations (6.12) and (7.5) in the form: Table 7.2 and Fig. 7.1. 3/2 3/2 0.0921d∗ f − >> 1 If all data of Table 7.1 are plot in the form U ∗ ver- p ⇒ 1/2 1/2 sus D∗ with the definitions (7.4), Fig. 7.2 is ob- u∗ = 20.52x0.0921xd∗ fp− (ϕ) fq (ϕ) tained. 3/2 1/2 0.0921d∗ >> 1 u∗ = 20.52x0.0921xd∗ ⇒ ∞ On the other hand, Fig. 7.3 shows the settling ve- 26 KONA Powder and Particle Journal No.27 (2009) (7.5) and (7.13). Data of D with the definitions (7.4), figure 7.2 is obtained. Table 7.1 ExperimentalTable data 7.1 of RichardsonExperimental data and of Richardson Zaki (1954) and Zaki (1954) Experimental data from Richardson and Zaki (1954) 3 2 N° d (cm) Us (g/cm ) Pf x10 (g/cms) Uf (g/cm3) vst (cm/s) Reoo n+1 n P 0.0181 1.058 20.800 1.034 0.00206 0.000185 4.90 3.90 Q 0.0181 1.058 20.800 1.034 0.00206 0.000185 4.79 3.79 K 0.0096 2.923 62.000 1.208 0.01390 0.000216 4.75 3.75 L 0.0096 2.923 62.000 1.208 0.01390 0.000216 4.65 3.65 F 0.0358 1.058 20.800 1.034 0.00807 0.001430 4.92 3.92 G 0.0358 1.058 20.800 1.034 0.00807 0.001430 4.89 3.89 H 0.0096 2.923 20.800 1.034 0.04550 0.002180 4.76 3.76 I 0.0096 2.923 20.800 1.034 0.04550 0.002180 4.72 3.72 J 0.0096 2.923 20.800 1.034 0.04550 0.002180 4.69 3.69 R 0.0230 2.623 62.000 1.208 0.06590 0.002950 4.85 3.85 S 0.0230 2.623 62.000 1.208 0.06590 0.002950 4.80 3.80 T 0.0128 2.960 1.890 2.890 0.03307 0.064700 4.84 3.84 M 0.0128 2.960 1.890 2.890 0.03307 0.064700 4.72 3.72 C 0.0181 1.058 1.530 1.001 0.06400 0.078900 4.76 3.76 A 0.0181 1.058 1.530 1.001 0.06640 0.078900 4.90 3.90 B 0.0181 1.058 1.530 1.001 0.06640 0.078900 4.79 3.79 X 0.1029 2.976 112.900 1.221 0.89100 0.099500 5.30 4.30 Y 0.1029 2.976 112.900 1.221 0.89100 0.099500 5.20 4.20 D 0.0253 1.058 2.910 0.935 0.14750 0.120000 4.94 3.94 E 0.0253 1.058 2.910 0.935 0.14750 0.120000 4.90 3.90 N 0.0096 2.923 1.612 2.170 0.23400 0.030200 4.74 3.74 O 0.0096 2.923 1.612 2.170 0.23400 0.030200 4.65 3.65 2 0.0253 2.78 6.075 1.135 0.82700 0.391000 4.65 3.65 5 0.0253 1.06 1.000 1.000 0.19400 0.490000 4.53 3.53 8 0.0230 2.623 1.890 2.890 0.34900 1.227000 4.450 3.450 9 0.0230 2.623 1.890 2.890 0.34900 1.227000 4.520 3.520 12 0.0230 2.623 1.890 2.890 0.34900 1.227000 4.140 3.140 10 0.0230 2.623 1.612 2.170 0.65250 2.021000 4.300 3.300 11 0.0230 2.623 1.612 2.170 0.65250 2.021000 4.350 3.350 13 0.0230 2.623 1.612 2.170 0.65250 2.021000 4.240 3.240 6 0.0510 2.745 6.075 1.135 2.89000 2.745000 4.22 3.22 14 0.1029 2.976 10.960 1.153 6.03000 6.530000 4.300 3.300 15 0.1029 2.976 10.960 1.153 6.03000 6.530000 4.070 3.070 16 0.1029 2.976 10.960 1.153 6.03000 6.530000 4.000 3.000 4 0.0253 2.78 1.000 1.000 3.55000 8.971000 3.59 2.59 3 0.1029 10.6 15.010 0.875 19.60000 11.750000 3.72 2.72 1 0.1029 2.976 1.890 2.890 1.16000 18.180000 3.800 2.800 0.1029 2.976 1.890 2.890 1.16000 18.180000 3.640 2.640 0.1029 2.976 1.890 2.890 1.16000 18.180000 3.860 2.860 0.1029 2.976 1.839 2.745 2.48000 38.260000 3.340 2.340 0.1029 2.976 1.839 2.745 2.48000 38.260000 3.560 2.560 0.1029 2.976 1.839 2.745 2.48000 38.260000 3.500 2.500 0.0510 2.745 1.000 1.000 8.10000 41.720000 3.11 2.11 0.1029 2.745 1.000 1.0 7.35000 14.450000 3.78 2.78 0.1029 2.745 1.000 1.000 7.35000 14.450000 3.78 2.78 0.4200 2.89 15.010 0.875 31.90000 78.250000 3.34 2.34 0.1029 10.6 3.810 0.818 36.15000 79.800000 3.08 2.08 0.2466 11.25 15.010 0.875 58.10000 80.350000 3.39 2.39 0.3175 7.73 15.010 0.875 54.70000 101.200000 3.17 2.17 0.4200 2.89 6.075 1.135 34.05000 267.000000 2.58 1.58 0.1029 10.6 1.000 1.000 47.50000 488.700000 2.43 1.43 0.4200 2.89 1.000 1.000 48.60000 2041.000000 2.33 1.33 0.3175 7.73 1.000 1.000 79.70000 2530.000000 2.36 1.36 0.6350 7.74 1.000 1.000 112.70000 7150.000000 2.38 1.38 locity u∗ versus d∗ for suspensions of spheres in wa- q = vs (1 ϕ) vr (7.14) − − 17 ter at 20℃ for different values of the concentration ϕ Fig. 7.3 divides the u∗ d∗ plane into three regions: × . This figure can be used to visualize the state of flow a porous bed, between the d∗ axis and the line of con- of particulate system. stant concentration ϕ = 0.585(Wen and Yu 1966 and The relationship between the volume average ve- Barnea and Mednick 1975 demonstrated that this locity q, also known as spatial velocity or percolation concentration corresponds to the minimum fluidiza- velocity, to de solid velocity vs and the relative solid tion velocity), a second region of fluidized bed be- fluid velocity vr is given by: tween 0.585 ϕ 0 and a third region of hydraulic ≤ ≤ or pneumatic transport, for values of velocities above KONA Powder and Particle Journal No.27 (2009) 27 Table 7.2 Values of u uf from the experimental results of Richardson and Zaki (1954) Table 7.2 Values of u/u∞ from the experimental results of Richardson and Zaki (1954) u/uoo Reoo 0.000 0.010 0.050 0.100 0.150 0.200 0.300 0.400 0.500 0.585 0.00019 0.0021 0.96156 0.81869 0.66305 0.53056 0.41884 0.24882 0.13639 0.06699 0.03239 0.00019 0.0021 0.96263 0.82333 0.67078 0.54013 0.42925 0.25877 0.14428 0.07229 0.03568 0.00026 0.0139 0.96301 0.82502 0.67361 0.54365 0.43310 0.26249 0.14725 0.07433 0.03696 0.00026 0.0139 0.96398 0.82926 0.68075 0.55256 0.44287 0.27202 0.15497 0.07966 0.04035 0.00144 0.0081 0.96137 0.81786 0.66165 0.52884 0.41698 0.24705 0.13501 0.06606 0.03182 0.00144 0.0081 0.96166 0.81911 0.66375 0.53142 0.41978 0.24971 0.13709 0.06745 0.03267 0.00217 0.0455 0.96292 0.82460 0.67290 0.54277 0.43213 0.26156 0.14650 0.07381 0.03663 0.00217 0.0455 0.96330 0.82629 0.67574 0.54631 0.43601 0.26532 0.14953 0.07589 0.03794 0.00217 0.0455 0.96359 0.82756 0.67788 0.54898 0.43894 0.26817 0.15184 0.07748 0.03896 0.00295 0.0659 0.96205 0.82080 0.66655 0.53489 0.42354 0.25330 0.13992 0.06935 0.03384 0.00295 0.0659 0.96253 0.82290 0.67007 0.53925 0.42829 0.25785 0.14354 0.07179 0.03537 0.06473 0.0331 0.96214 0.82122 0.66725 0.53576 0.42449 0.25420 0.14064 0.06983 0.03414 0.06473 0.0331 0.96330 0.82629 0.67574 0.54631 0.43601 0.26532 0.14953 0.07589 0.03794 0.07579 0.0640 0.96292 0.82460 0.67290 0.54277 0.43213 0.26156 0.14650 0.07381 0.03663 0.07863 0.0664 0.96156 0.81869 0.66305 0.53056 0.41884 0.24882 0.13639 0.06699 0.03239 0.07863 0.0664 0.96263 0.82333 0.67078 0.54013 0.42925 0.25877 0.14428 0.07229 0.03568 0.09916 0.8910 0.95770 0.80207 0.63569 0.49717 0.38308 0.21574 0.11119 0.05077 0.02278 0.09916 0.8910 0.95867 0.80619 0.64242 0.50531 0.39172 0.22357 0.11701 0.05441 0.02488 0.11990 0.1475 0.96118 0.81702 0.66026 0.52712 0.41512 0.24529 0.13363 0.06515 0.03127 0.11990 0.1475 0.96156 0.81869 0.66305 0.53056 0.41884 0.24882 0.13639 0.06699 0.03239 0.30240 0.2340 0.96311 0.82544 0.67432 0.54454 0.43407 0.26343 0.14801 0.07484 0.03728 0.30240 0.2340 0.96398 0.82926 0.68075 0.55256 0.44287 0.27202 0.15497 0.07966 0.04035 0.39091 0.8270 0.96398 0.82926 0.68075 0.55256 0.44287 0.27202 0.15497 0.07966 0.04035 0.49082 0.1940 0.96514 0.83438 0.68941 0.56344 0.45489 0.28392 0.16477 0.08657 0.04484 1.22741 0.3490 0.96592 0.83781 0.69524 0.57081 0.46308 0.29214 0.17164 0.09151 0.04811 1.22741 0.3490 0.96524 0.83481 0.69013 0.56436 0.45591 0.28493 0.16561 0.08717 0.04524 1.22741 0.3490 0.96893 0.85124 0.71833 0.60031 0.49625 0.32629 0.20109 0.11344 0.06319 2.02024 0.6525 0.96738 0.84428 0.70632 0.58490 0.47885 0.30819 0.18531 0.10153 0.05490 2.02024 0.6525 0.96689 0.84212 0.70261 0.58017 0.47353 0.30275 0.18064 0.09807 0.05254 2.02024 0.6525 0.96796 0.84689 0.71080 0.59063 0.48530 0.31486 0.19108 0.10584 0.05787 2.75371 2.8900 0.96816 0.84775 0.71230 0.59256 0.48747 0.31711 0.19304 0.10732 0.05890 6.52757 6.0300 0.96738 0.84428 0.70632 0.58490 0.47885 0.30819 0.18531 0.10153 0.05490 6.52757 6.0300 0.96962 0.85430 0.72364 0.60718 0.50406 0.33454 0.20841 0.11908 0.06721 6.52757 6.0300 0.97030 0.85738 0.72900 0.61413 0.51200 0.34300 0.21600 0.12500 0.07147 8.98150 3.5500 0.97431 0.87560 0.76118 0.65644 0.56105 0.39701 0.26632 0.16609 0.10250 11.7571 1.9600 0.97303 0.86978 0.75083 0.64272 0.54501 0.37902 0.24921 0.15177 0.09143 18.2520 1.1600 0.97225 0.86622 0.74452 0.63441 0.53537 0.36836 0.23923 0.14359 0.08522 18.2520 1.1600 0.97382 0.87335 0.75718 0.65113 0.55483 0.38999 0.25961 0.16043 0.09810 18.2520 1.1600 0.97167 0.86355 0.73983 0.62826 0.52825 0.36056 0.23201 0.13774 0.08084 38.0915 2.4800 0.97676 0.88690 0.78150 0.68366 0.59324 0.43404 0.30260 0.19751 0.12771 38.0915 2.4800 0.97460 0.87695 0.76359 0.65965 0.56482 0.40128 0.27044 0.16958 0.10525 38.0915 2.4800 0.97519 0.87965 0.76843 0.66611 0.57243 0.40996 0.27885 0.17678 0.11095 41.3100 8.1000 0.97902 0.89742 0.80067 0.70970 0.62448 0.47115 0.34033 0.23165 0.15634 75.6315 7.3500 0.97245 0.86710 0.74610 0.63648 0.53776 0.37100 0.24169 0.14559 0.08673 75.6315 7.3500 0.97245 0.86710 0.74610 0.63648 0.53776 0.37100 0.24169 0.14559 0.08673 78.1029 31.900 0.97676 0.88690 0.78150 0.68366 0.59324 0.43404 0.30260 0.19751 0.12771 79.8642 36.150 0.97931 0.89880 0.80320 0.71317 0.62868 0.47622 0.34558 0.23651 0.16052 83.5212 58.100 0.97627 0.88463 0.77739 0.67813 0.58666 0.42637 0.29497 0.19078 0.12222 101.241 54.700 0.97843 0.89466 0.79562 0.70281 0.61618 0.46117 0.33006 0.22221 0.14831 267.187 34.050 0.98425 0.92215 0.84665 0.77354 0.70288 0.56919 0.44615 0.33448 0.24918 488.775 47.500 0.98573 0.92928 0.86014 0.79263 0.72681 0.60047 0.48168 0.37113 0.28432 2041.20 48.600 0.98672 0.93405 0.86925 0.80561 0.74321 0.62227 0.50692 0.39777 0.31046 2530.48 79.700 0.98642 0.93262 0.86650 0.80170 0.73825 0.61565 0.49921 0.38958 0.30237 7156.45 112.70 0.98623 0.93166 0.86468 0.79909 0.73496 0.61127 0.49414 0.38422 0.29710 concentration ϕ = 0. in the form: 1/2 1/2 2 2 δ0fp (ϕ)fq (ϕ) CD = C0fp(ϕ)fq− (ϕ) 1 + 1/2 Drag Coefficient for a suspension of spheres Re From equations (6.10) and (7.5) we deduce that: (7.19) ˜2 2 and substituting the values of the parameters: δ0 = δ0fp(ϕ)fq(ϕ) (7.15) 1/2 1/2 2 ˜1/2 ˜2 1/2 2 3/2 2 δ0fp (ϕ)fq (ϕ) C0 δ0 = C0 δ0fp (ϕ) (7.16) CD = 0.28fp(ϕ)fq− (ϕ) 1 + Re1/2 therefore, the parameters of the Drag Coefficient are: ˜ 2 C0 = C0fp(ϕ)fq− (ϕ) (7.17) 8. Sedimentation of Isometric Particles ˜ 1/2 1/2 18 δ0 = δ0fp (ϕ)fq (ϕ) (7.18) The behavior of non-spherical particles is different and the Drag Coefficient of the sphere can be written than that of spherical particles during sedimentation. 28 KONA Powder and Particle Journal No.27 (2009) 1.0 0.010 0.050 0.9 0.100 0.150 0.200 oo 0.8 u / 0.300 u 0.7 0.6 0.400 0.5 0.500 0.4 0.585 0.3 Razón de velocidades 0.2 0.1 n=3.90 y m=0.85 0.0 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 Número de Reynolds Reoo Fig. 7.1 Dimensionless settling velocities versus Reynolds number, together with data obtained from Richardson and Zaki (1954) experimental results. 1.E+03 1.E+02 * 1.E+01 U 1.E+00 FI=0.01 1.E-01 FI=0.05 FI=0.10 1.E-02 FI=0.15 1.E-03 FI=0.20 Dimensionless velocity FI=0.30 1.E-04 FI=0.40 FI=0.50 1.E-05 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 Dimensionles size D * Fig. 7.2 Dimensionless velocity U* for suspensions of spheres of any size and density versus the dimensionless diameter D*, with experimental data of Richardson and Zaki (1954). While spherical particles fall in a vertical trajectory, that velocities in Stokes flow for isometric particles non-spherical particles rotate, vibrate and follow spi- may be described with the following expression: ral trajectories. Several authors have studied the sedi- 2 up ψ ∆ρd g mentation of isometric particles, which have a high = 0.843log with u = e (8.1) u 0.065 e 18µ degree of symmetry with three equal mutually per e f pendicular symmetry axes, such as the tetrahedron, where de is the volume equivalent diameter, that is, octahedron and dodecahedron. Wadell (1932, 1934), the diameter of a sphere with the same volume as the Pettyjohn and Christiansen (1948) and Christiansen particle, and ue is its settling velocity and Barker (1965) show that isometric particles fol- In the range of 2.000 Re 17.000, the same au- ≤ ≤ low vertical trajectories at low Reynolds numbers, but thors derived the following equation for the settling rotate and vibrate and show helicoidally trajectories velocity: for Reynolds numbers between 300 and 150.000. 4 ∆ρdeg ue = , (8.2) Pettyjohn and Christiansen (1948) demonstrate 3 ρf CD KONA Powder and Particle Journal No.27 (2009) 29 1.E+03 1.E+02 1.E+01 0.585 1.E+00 0.500 0.001 0.400 1.E-01 0.000 0.300 Dimensionless velocity u* 0.200 1.E-02 0.150 0.100 0.005 1.E-03 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 Dimensionless diameter d* Fig. 7.3 Simulation of the dimensionless velocity u＊ versus the dimensionless size d＊ for spherical particles settling in a suspen- sion in water at 20℃ with the volume fraction of solids a parameter. 1.E+07 Cubes 1.E+06 Tetrahedrons Octahedrons 1.E+05 Cube octahedrons D Max. sphericity cylinders C 1.E+04 Spheres 1.E+03 1.E+02 Drag coefficient 1.E+01 1.E+00 1.E-01 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Reynolds Number Re Fig. 8.1 Plot of the drag coefficient versus de Reynolds number for the settling of isometric particles according to Pettyjohn and Christiansen (1948), and Barker (1951). with the drag coefficient C D given by: C D ＝ where λ is the quotient between the solid and fluid (5.31 4.88ψ)/(1.433 0.43). The value of 1.433 in − × densities λ = ρp/ρf . Data from Pettyjohn and Chris- the denominator of this equation is a factor that takes tiansen (1948) and from Barker (1951) are plotted in the theoretical value of CD = 0.3 to the average ex- Fig. 8.1. Fig. 8.2 gives details of the higher Reyn- perimental value CD = 0.43. olds end. As we have already said, for Re＞300, the particles begin to rotate and oscillate, which depends on the Drag coefficient and sedimentation velocity particle density. To take into account these behaviors, All the results obtained for spherical particles Barker (1951) introduced the particle to fluid density (Concha y Almendra 1979a, 1979b), may be used to ratio as a new variable in the form: develop functions for the drag coefficient and sedi- (5.31 4.88ψ) mentation velocity of isometric particles. C (ψ, λ) = λ1/18 − , (8.3) D 0.62 Assume that equation (5.11) and (6.10) are valid 30 KONA Powder and Particle Journal No.27 (2009) 1.E+02 Cubes Tetrahedrons Octahedrons Cube octahedrons D 1.E+01 Max. sphericity cylinders C Spheres 1.E+00 Drag coefficient 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Reynolds Number Re Fig. 8.2 Details for Fig 4.17 at the high Reynolds number end. (Pettyjohn and Christiansen 1948, and Barker 1951). for isometric particles, with values of C0 and δ0 as To determine the functions fA , fB , fC and fD we functions of the sphericity ψ and of the density quo- will use the correlations presented by Pettyjohn and tient λ (Concha y Barrientos 1986): Christiansen (8.13) and(8.15), and by Barker (1951). 2 δ (ψ,λ) From (8.9) and (8.10) we can write: C (ψ, λ) = C˜ (ψ, λ) 1 + 0 (8.4) 1 D 0 Re1/2 2 2 ψ − fA (ψ) fB (ψ) fC (λ) fD (λ) = 0.843 log 0.065 1/2 2 ˜2 1 δ0 (ψ,λ) 4 3/2 u∗ = 1 + 1/2 d∗ 1 (8.11) p 4 d∗ C˜ (ψ,λ)δ˜2(ψ,λ) 0 0 − 1/18 5.31 4.88ψ fA (ψ) fC (λ) = λ − (8.12) (8.5) 0.62 where the Reynolds number is defined using the vol- From (8.10) and (8.12) we deduce that: ume equivalent diameter. 5.31 4.88ψ 1/18 fA (ψ) = − fC (λ) = λ (8.13) To determine the equation’s parameters assume 0.62 that: Since in Stokes regime the density does not influ- ˜ (8.6) ences the flow, equation (8.11) implies that: C0 (ψ, λ) = C0fA (ψ) fC (λ) 2 1/36 ˜ fC (λ) fD (λ) = 1 fD (λ) = λ− (8.14) δ0 (ψ, λ) = δ0fB (ψ) fD (λ) (8.7) ⇒ where C0 and δ0 are the same parameters of a therefore 1/2 sphere. 5.31 4.88ψ ψ − fB (ψ) = 0−.62 0.843 log 0.065 (8.15) Assume that we can approximate the velocity of iso- × metric particles at low Reynolds number, Re 0, in The following figures show the drag coefficients of → the same way as for spherical particles. Then: isometric particles and the dimensionless settling ve- 2 2 locities versus the dimensionless particle size. de∗ de∗ u∗ = and u∗ = e 2 p 2 (8.8) The experimental data used in the previous correla- C0δ0 C˜0 (ψ, λ) δ (ψ, λ) 0 tions are 655 points including spheres, cubes-octahe- Taking the quotient of these terms and substituting drons, maximum sphericity cylinders, octahedrons (8.6) and (8.7), results in: and tetrahedrons in the following ranges: u∗ ue 0.1cm KONA Powder and Particle Journal No.27 (2009) 31 1.E+07 1.E+06 Cube octahedrons =0.906 ) 1.E+05 ( D 1.E+04 1.E+03 1.E+02 Drag coefficient C 1.E+01 1.E+00 1.E-01 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Reynolds Number Re Fig. 8.3a Simulation with Concha and Barrientos’ equation (8.4) and experimental values for isometric particles from Pettyjohn and Christiansen (1948) and Barker (1951) for cube octahedrons. 1.E+07 1.E+06 Octahedron =0.846 1.E+05 ) ( D 1.E+04 1.E+03 1.E+02 Drag coefficient C 1.E+01 1.E+00 1.E-01 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Reynolds Number Re Fig. 8.3b Simulation with Concha and Barrientos’ equation (8.4) and experimental values for isometric particles from Pettyjohn and Christiansen (1948) and Barker (1951) for octahedrons. (Happel and Brenner 1965; Barker 1951) and the pa- \ f A (\ ) f B (\ ) rameters of Concha and Barrientos (1986) Sphere 1.000 1.0000 1.0000 Alternative equations for isometric particles were Cube octahed 0.906 1.4334 0.8826 proposed by Ganser (1993): Octahedron 0.846 1.9057 1.2904 Cube 0.806 2.2205 1.5468 1 24 0.6567 CD = 1 + 0.1118 (K1K2Re) Tetrahedron 0.670 3.2910 2.3108 K1 Re Max sph cylin 0.875 1.6774 1.0966 0.4305K K2Re + 1 2 (8.16) 3305 + K1K2Re 32 KONA Powder and Particle Journal No.27 (2009) 1.E+07 1.E+06 Cubes =0.806 1.E+05 ) ( D 1.E+04 1.E+03 1.E+02 Drag coefficient C 1.E+01 1.E+00 1.E-01 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Reynolds Number Re Fig. 8.3c Simulation with Concha and Barrientos’ equation (8.4) and experimental values for isometric particles from Pettyjohn and Christiansen (1948) and Barker (1951) for cubes. 1.E+07 1.E+06 Tetrahedron =0.670 1.E+05 ) ( D 1.E+04 1.E+03 1.E+02 Drag coefficient C 1.E+01 1.E+00 1.E-01 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Reynolds Number Re Fig. 8.3d Simulation with Concha and Barrientos’ equation (8.4) and experimental values for isometric particles from Pettyjohn and Christiansen (1948) and Barker (1951) for tetrahedrons. where particles of arbitrary shape. 1 − 2 1 2 1/2 ˜ δ0(ψ,λ,ϕ) K1 = + ψ− , CD (ψ, λ, ϕ) = C0 (ψ, λ, ϕ) 1 + Re1/2 (9.1) 3 3 0.5743 1.8148( log ψ) 1/2 2 (8.17) ˜2 K2 = 10 − 1 δ0 (ψ,λ,ϕ) 4 3/2 up∗ (ψ, λ, ϕ) = 4 d 1 + ˜1/2 ˜2 d 1 C0 (ψ,λ,ϕ)δ0 (ψ,λ,ϕ) ∗ − ∗ 9. Sedimentation of Particles of Arbitrary Shape (9.2) Concha and Christiansen (1986) extended the va- where ψ, λ and ϕ are the hydrodynamic sphericity of lidity of equations (6.4) and (6.5) to suspensions of the particles, the density ratio of solid and fluid and KONA Powder and Particle Journal No.27 (2009) 33 1000 100 * u 10 1 0.1 Sphericity=1.0 Sphericity=0.9 Sphericity=0.8 0.01 Sphericity=0.7 Dimensionless velocity Sphericity=0.6 0.001 Sphericity=0.5 Sphericity=0.4 0.0001 0.01 0.1 1 10 100 1000 10000 100000 Dimensionless size d * Fig. 8.4 Dimensionless size versus dimensionless velocity for isometric particles according to Concha and Barrientos equation (8.5). the volume fraction of solid in the suspension. Modified drag coefficient and sedimentation ve- Similarly as in the case of isometric particles, they locity ˜ ˜ assumed that the functions C0 and δ0 may be written A unified correlation can also be obtained for the in the form: drag coefficient and the sedimentation velocity of ir- regular particles forming a suspension. Defining C˜ (ψ, λ, ϕ) = C f (ψ) f (λ) f (ϕ) f 2 (ϕ) (9.3) 0 0 A C p q− CDM , ReM , deM∗ and upM∗ in the following form: Re ˜ 1/2 1/2 (9.4) CD (ψ, λ) δ0 (ψ, λ, ϕ) = δ0fB (ψ) fD (λ) fF (ϕ) fp (ϕ)fq (ϕ) CDM = ReM = 2 2 2 fA (ψ) fC (λ) fp (ϕ) fB (ψ) fD (λ) fp (ϕ) 5.31 4.88ψ with (9.5) d (ψ, λ) fA (ψ) = 0−.62 e∗ (9.9) 1/2 deM∗ = 2/3 5.31 4.48ψ ψ − 1/2 2 1/2 2 f (ϕ) fA (ψ) f (ψ) fC (λ) f (λ) fB (ψ) = 0−.62 0.843 log 0.065 (9.6) p × × B × × D × (9.10) 1/36 up∗ (ψ, λ) 1/18 (9.7) u∗ = fC (λ) = λ fD (λ) = λ− pM f (ψ) f (λ) f (ϕ) B D q (9.11) 2.033 0.167 f (ϕ) = (1 ϕ)− , f (ϕ) = (1 ϕ)− (9.8) p − q − Fig. 9.1 and 9.2 show the unified correlations for the data from Concha and Christiansen (1986). Ganser (1993) proposed an empirical equation for Hydrodynamic sphericity the drag coefficient of non-spherical non-isometric Concha y Christiansen (1986) found it necessary particles, including irregular particles, similar to that to define a hydrodynamic shape factor to be used with given earlier for spherical particles (8.16), but with the above equations, since the usual methods to different values for the parameters K1. measure sphericity did not gave good results. They 1 24 0.6567 CD = 1 + 0.1118 (K1K2Re) defined the effective hydrodynamic sphericity of a par- K1 Re 2 ticle as the sphericity of an isometric particle having 0.4305K1Kz 2Re + (9.12) the same drag (volume) and the same settling velocity 3305 + K1K2Re as the particle. where 1 0.5743 The hydrodynamic sphericity may be obtained 1 dp 2 1/2 − 1.8148( log ψ) K = + ψ− ,K = 10 − 1 3 de 3 2 by performing sedimentation, or fluidization experi- ments, calculating the drag coefficient for the particle (9.13) using the volume equivalent diameter and obtaining In equation (9.13) for K1,de and dp are the volume the sphericity (defined for isometric particles) that fit equivalent and the projected area equivalent diam- the experimental value. eters of the irregular particle respectively. 34 KONA Powder and Particle Journal No.27 (2009) 1.0E+03 DM C 1.0E+02 1.0E+01 1.0E+00 Modified Drag Coefficient 1.0E-01 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 Modified Reynolds Number ReM Fig. 9.1 Unified drag coefficient versus Reynolds number for quartz, limestone and sand particles (The same data of Fig. 4.20 (Concha y Christiansen 1986). 100.0 eM * u 10.0 1.0 Limestone Dimensionles velocity Quartz Sand Concha and Almendra 0.1 29 1 01 001 0001 Dimensionless size d *eM Fig. 9.2 Unified sedimentation velocity versus size for limestone, quartz and sand particles (The same data of Fig. 4.20 (Concha y Christiansen 1986). Finally, it is interesting to mention the work of Yin Acknowledgments et al (2003) who analyzed the settling of cylindrical particles analytically, and obtained, by linear and This paper was financed by INNOVA Project CM0117 and angular momentum balances, the forces and torques AMIRA Project p996. applied to the particle during its fall. Using Ganser’s equation for the drag coefficient, they solved the dif- References ferential equations of motion numerically obtaining results close to those measured experimentally by Abraham, F.F. (1979): Functional dependence of the drag them. coefficient of a sphere on Reynolds Number. Phys. Fluid, 13, pp.2194-2195. KONA Powder and Particle Journal No.27 (2009) 35 Barker H. (1951): The effect of shape and density on the namics”, Prentice-Hall, Inc, Englewood Cliffs N.J., USA, free settling rates of particles at high Reynolds Numbers, p.220. Ph.D. Thesis, University of Utah, Table 7, pp.124-132; Heywood, H. 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(1965): “Low Reynolds Hydrody- 36 KONA Powder and Particle Journal No.27 (2009) Author’s short biography Fernando Concha Arcil Fernando Concha Arcil obtained a Batchelor degree in Chemical Engineering at the University of Concepción in Chile in 1962 and a Ph.D. in Metallurgical Engi- neering from the University of Minnesota in 1968. He is professor at the Depart- ment of Metallurgical Engineering, University of Concepción. His main research topic is the application of transport phenomena to the field of Mineral Processing, He has published four books: Modern Rational Mechanics, Design and Simulation of Grinding-Classification Circuits and Filtration and Separation, all in Spanish, and Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory in English. He also has published more than 100 technical papers in inter- national journals. For his work in sedimentation with applications to hydrocyclones and thickeners he received the Antoine Gaudin Award 1998, presented by the American Society for Mining Metallurgy and Exploration, SME. KONA Powder and Particle Journal No.27 (2009) 37