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).