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Forces on Particles and Bubbles Title Forces on Particles and Bubbles M. Sommerfeld Mechanische Verfahrenstechnik Zentrum für Ingenieurwissenschaften Martin-Luther-Universität Halle-Wittenberg D-06099 Halle (Saale), Germany www-mvt.iw.uni-halle.de Martin-Luther-Universität Halle-Wittenberg Content of the Lecture BBO equation and particle tracking Forces acting on particles moving in fluids Drag force Pressure, virtual mass and Basset Transverse lift forces Electrostatic force Thermophoretic and Brownian force Importance of the different forces Particle response time and Stokes number Behaviour of bubbles and forces Particle response to oscillatory flow filed Martin-Luther-Universität Halle-Wittenberg Equation of Motion 1 The equation of motion for particles in a quiescent fluid was first derived by Basset (1888), Bousinesque (1885), and Oseen (1927) BBO-equation. A rigorous derivation of the equation of motion for non-uniform Stokes flow was performed by Maxey and Riley (1983). The BBO-equation without the Faxen terms (due to curvature of the velocity field) is given by: d u 18 µ m Du d u P = F − − P − ∇ + ∇τ + F − P mP 2 mP (uF uP ) ( p ) 0.5 mF d t ρP DP ρP Dt d t Importance of d uF d uP the different t * − * ρ µ m d t d t * (u − u ) forces ??? + 9 F F P dt + F P 0 + m g ∫ * 1 2 P π ρP DP 0 (t − t ) t Accounts for d Derivative along D Substantial : : initial condition dt particle path Dt derivative 1: drag force 2: pressure term 3: added mass 4: Basset force (with initial condition) 5: gravity force Martin-Luther-Universität Halle-Wittenberg Equation of Motion 2 The calculation of particle trajectories requires the solution of several partial differential equations: particle location particle velocity particle angular velocity d x p d u d ω = u p p = p mp = ∑ Fi Ip T dt dt dt The consideration of heat and mass transfer requires the solution of two additional partial differential equations for droplet diameter and droplet temperature. Analytical solutions for the different forces acting on the particle are only available for small particle Reynolds numbers (Stokes regime). For higher particle Reynolds numbers empirical correlations are needed to calculate the different forces. In the flowing the different forces for describing the particle motion are introduced and discussed (see also Sommerfeld 2000, 2008 and 2010). Martin-Luther-Universität Halle-Wittenberg Drag Force 1 The drag force is the most important one and includes the friction and form drag or pressure drag. For highly viscous flow (low particle Reynolds number) an analytic solution can be obtained for the drag force (Stokes 1851): F = 3 π µ D (u − u ) D P F P At higher particle Reynolds numbers however, empirical correlations are needed which are expressed as a drag coefficient: F C = D D ρ v2 A 2 rel P This results in the following expression for the drag force: ρ π F = F D2 C (u − u ) u − u D 2 4 P D F P F P Martin-Luther-Universität Halle-Wittenberg Drag Force 2 Fitting of drag coefficient data for spherical solid particles: Stokes regime Standard drag correlation (Stokes 1851): (Schiller and Naumann, 1933): 24 C = 24 24 D = + 0.687 = ReP CD (1 0.15 ReP ) fD ReP ReP 2 measurements 10 Stokes regime correlation Newton regime: 101 Newton regime [-] CD ≈ 0.44 D C 100 Critical Reynolds number -1 10 -1 0 1 2 3 4 5 6 ρ D (u − u ) 10 10 10 10 10 10 10 10 F P F P ReP = Re P [-] µF Martin-Luther-Universität Halle-Wittenberg Drag Force 3 Summary of the drag coefficient obtained in different experimental studies (Crowe et al. 2012): Martin-Luther-Universität Halle-Wittenberg Drag Force 4 The average drag coefficient for non-spherical particles (for a certain stable orientation) is determined from empirical correlations fitted to experimental data (Haider and Levenspiel 1989; Thompson and Clark 1991). 1000 24 B C CD = (1+ A ReP )+ ReP D 1+ 100 φ = 0.025 ReP φ = 0.05 [ - ] φ = 0.1 D φ = 0.2 c 10 φ = 0.3 φ = 0.5 φ = 0.7 1 φ = 0.9 A = exp (2.3288 − 6.4581 φ + 2.4486 φ2 ) φ = 1.0 -1 0 1 2 3 4 5 B = 0.0964 + 0.5565 φ 10 10 10 10 10 10 10 Re P [ - ] C = exp (4.905 −13.8944 φ +18.4222 φ2 −10.2599 φ3 ) D = exp (1.4681+12.2584 φ − 20.7322 φ2 +15.8855 φ3 ) sphericity S Martin-Luther-Universität φ = V Halle-Wittenberg SPartikel Drag Force 5 RarefactionThe importance effects: of rarefaction effects may be estimated on the basis of the Knudsen number (Crowe 2006): λ π γ Ma (u − u ) Kn = = P = F P P Ma P Molecules DP 2 ReP a Mean free path of the gas molecules: µF 8 p λ = c = Particle Mol π ρ 0.499 cMol ρF F Classification of the different regimes: Continuum: KnP < 0.015 Slip flow: 0.015 < KnP < 0.15 Transitional: 0.15 < KnP < 4.5 Free molecule: KnP > 4.5 Martin-Luther-Universität Halle-Wittenberg Drag Force 6 Cunningham correction for rarefaction effects (Davies 1945): C 1 1 D = = < CD,Stokes Cu 0.55 ReP 0.25 1+ Kn 2.514 + 0.8 exp− P 0.1< Kn <1000 Kn P P 1 [ - ] 0.1 D, Stokes 0.01 / C D C 1E-3 1E-4 1E-3 0.01 0.1 1 10 100 1000 Kn [ - ] Martin-Luther-Universität Halle-Wittenberg Drag Force 7 Other effects on the drag force: Turbulence of the surrounding fluid reduces the critical Reynolds number to about 1000. Surface roughness also causes a reduction of the critical Reynolds number. The porosity of particles results in a reduction of the drag coefficient. With increasing particle concentration the drag is considerably increased (hydrodynamic interaction). Particle motion in the vicinity of walls (normal and parallel). Martin-Luther-Universität Halle-Wittenberg Pressure Force Force on the particle due to a local pressure gradient and the shear stress in the flow: mP FP = (− ∇ p + ∇ τ) ρP With the Navier-Stokes equation one finds that: D u − ∇ + ∇ τ = ρ F − p F g D t This yields the total pressure force as: ρ D u F = m F F − g The second term is p P the buoyancy force ρP D t Martin-Luther-Universität Halle-Wittenberg Virtual Mass and Basset Force 1 For higher particle Reynolds numbers the virtual (added) mass and the Basset (history) force may be expressed as: mP d FA = 0.5 CA ρF (u F − u P ) ρP d t d The second term in the (u − u ) ρ µ m t d t F P (u − u ) Basset force accounts = F F P τ + F P 0 FB 9 CB 1 2 d for the initial slip (Reeks π ρ D ∫ (t − τ) t P P 0 & McKee 1984) Coefficients given by Odar and Hamilton (1964): 0.132 Acceleration number C = 2.1− A A2 + 0.12 C 2 ReP < 60 u F − u P 0.52 AC = CB = 0.48 + d u − u (A +1)3 D F P C P d t Martin-Luther-Universität Halle-Wittenberg Virtual Mass and Basset Force 2 Recent studies of Michaelides and Roig (2011) showed that the added mass coefficient is constant (CA =1) and the Basset coefficient should be expressed in dependence of the Strouhal number: 0.82 2.5 CB = 2.0 −1.0533 [1− exp(− 0.14 ReP Sr ) ] The Strouhal number describes the behaviour of oscillatory flows. In this context the fluid time scale is the reciprocal value of the characteristic fluid oscillations. In turbulent flows this time scale corresponds to the integral time scale of turbulence. 1 T Sr = = I St τP Consequently the Strouhal number is the reciprocal value of the particle turbulent Stokes number. Martin-Luther-Universität Halle-Wittenberg Other Forces Other important forces (field or external forces) are: Gravity force: = Fg mp g Centrifugal and Coriolis force (appears only if the equation of motion is written in polar co-ordinates): 2 vP,ϕ vP,r vP,ϕ FZ = mP FC = mP rP rP Electrostatic force: Fe = qP E Thermophoretic force Magnetic force Martin-Luther-Universität Halle-Wittenberg Electrostatic Forces 1 In an electrostatic precipitator, for example, the particles are charged by an ion-bombardment created by a negative corona discharge in the vicinity of a charging wire (Löffler 1988). The charging of the particles is caused by two mechanisms: t Field charging occurs 2 qp (t) = n ⋅e = π ε0 p DP E1 due to the convective t + τq motion of the ions and is 2 relevant for particles larger qmax = n e = π ε0 p DP E1 than about 0.5 to 1 µm. D − 1 (conducting particles: p = 3, D ∞) p = 1+ 2 D + 2 (non-conducting particles p = 1.5 ÷ 2) D: relative dielectric constant of particle ε0: absolute dielectric constant in vacuum Diffusion charging is the e: elementary charge result of the thermal motion E1: field strength in the charging region of the ions and is relevant for particles with a diameter k ⋅T D D c N e2 t = π ε P + P 0 smaller than about 0.2 µm. qP (t) 4 0 ln 1 e 2 8 ε0 k T Martin-Luther-Universität Halle-Wittenberg Electrostatic Forces 2 Calculated flow field, electrostatic potential and particle trajectories in an electrostatic precipitator (Böttner and Sommerfeld 2003) Uav = 0.5 m/s VH = 60 kV Particle charge: 0.66 ⋅ qmax Channel: 300 mm x 600 mm 0.15 Separation wires: 150 mm 0.10 Y [ m ] 0.05 Particles 12 – 20 µm 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.15 X [ m ] 0.10 Particles 0.9 – 1.5 µm Y [ m ] 0.05 0.00 0.0 0.2 0.4 0.6 Martin0.8 -Luther1.0 -Universität X [ m ] Halle-Wittenberg Thermophoretic Force Thermophoretic force due to temperature gradient (LBM): κF 6 π 1.17 + 2.18 Kn κP 2 1 FT = dPart ρF νF ∇TF κ κ κ T F, P: thermal (1+ 3.42 Kn) 1+ 2 F + 4.36 Kn F conductivity of κ P fluid and particles dT/dh=1K/mm hot cold Molecules dT/dh=10 K/mm Particle DP = 100 nm Martin-Luther-Universität Halle-Wittenberg Brownian Motion 1 Brownian particle motion (LBM): 216⋅k ν T = ζ Boltz F F FB mP 2 5 ρ ρF dP ∆t ζ Random number (0 ≤ ζ ≤ 1) π P i CCun ρF Only drag force drag force and Brownian motion Martin-Luther-Universität Halle-Wittenberg
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