Title orces on articles and Bubbles M. Sommerfeld Mechanische Verfahrenstechnik Zentrum für Ingenieurwissenschaften -06099 Halle (Saale), Germany www-mvt.iw.uni-halle.de
ontent of the Lecture BBO equation and article tracking orces acting on articles moving in fluids rag force ressure, virtual mass and Basset Transverse lift forces Electrostatic force Thermohoretic and Brownian force Imortance of the different forces article resonse time and Stokes number Behaviour of bubbles and forces article resonse to oscillatory flow filed
Equation of Motion 1 The equation of motion for articles in a quiescent fluid was first derived by Basset (1888), Bousinesque (1885), and Oseen (197) BBO-equation. A rigorous derivation of the equation of motion for non-uniform Stokes flow was erformed by Maxey and Riley (1983). The BBO-equation without the axen terms (due to curvature of the velocity field) is given by: m d u d t 18 µ + 9 m µ π m d u d t ( u u ) ( + τ ) m t 0 * d u d t * ( t t ) 1 * dt * u d u + 0.5 m t d t ( u u ) 0 + + m g t Imortance of the different forces??? d dt : erivative along article ath t : Substantial derivative Accounts for initial condition 1: drag force : ressure term 3: added mass 4: Basset force (with initial condition) 5: gravity force
Equation of Motion The calculation of article trajectories requires the solution of several artial differential equations: article location article velocity article angular velocity d x d u d ω u m dt i I T dt dt The consideration of heat and mass transfer requires the solution of two additional artial differential equations for drolet diameter and drolet temerature. Analytical solutions for the different forces acting on the article are only available for small article ynolds numbers (Stokes regime). or higher article ynolds numbers emirical correlations are needed to calculate the different forces. In the flowing the different forces for describing the article motion are introduced and discussed (see also Sommerfeld 000, 008 and 010).
rag orce 1 The drag force is the most imortant one and includes the friction and form drag or ressure drag. or highly viscous flow (low article ynolds number) an analytic solution can be obtained for the drag force (Stokes 1851): 3 π µ u u At higher article ynolds numbers however, emirical correlations are needed which are exressed as a drag coefficient: rel This results in the following exression for the drag force: v ( ) A π 4 ( u u ) u u
rag orce itting of drag coefficient data for sherical solid articles: Stokes regime (Stokes 1851): 4 Standard drag correlation (Schiller and Naumann, 1933): 0.687 4 ( 1 + 0.15 ) f 4 [-] 10 measurements Stokes regime correlation 10 1 Newton regime 10 0 Newton regime: 0.44 ritical ynolds number 10-1 10-1 10 0 10 1 10 10 3 10 4 10 5 10 6 [-] ( u u ) µ
rag orce 3 Summary of the drag coefficient obtained in different exerimental studies (rowe et al. 01):
rag orce 4 The average drag coefficient for non-sherical articles (for a certain stable orientation) is determined from emirical correlations fitted to exerimental data (Haider and Levensiel 1989; Thomson and lark 1991). A ex (.388 6.4581 φ +.4486 φ ) B 0.0964 + 0. 5565 ex ex 4 B ( 1+ A ) φ + 1+ c [ - ] 1000 100 10 3 ( 4.905 13.8944 φ + 18.4 φ 10.599 φ ) 3 ( 1.4681+ 1.584 φ 0.73 φ + 15.8855 φ ) 1 10-1 10 0 10 1 10 10 3 10 4 10 5 [ - ] φ 0.05 φ 0.05 φ 0.1 φ 0. φ 0.3 φ 0.5 φ 0.7 φ 0.9 φ 1.0 shericity S φ S V artikel
rag orce 5 Rarefaction The imortance effects: of rarefaction effects may be estimated on the basis of the Knudsen number (rowe 006): λ π γ Ma ( u u ) Kn Ma a Molecules Mean free ath of the gas molecules: µ 8 λ c 0.499 cmol Mol π lassification of the different regimes: ontinuum: Kn < 0.015 Sli flow: 0.015 < Kn < 0.15 Transitional: 0.15 < Kn < 4.5 ree molecule: Kn > 4.5 article
rag orce 6 unningham correction for rarefaction effects (avies 1945):,Stokes 1 u 1 + Kn 1.514 + 0.8 ex 0.55 Kn < 0.5 0.1< Kn < 1000 /, Stokes [ - ] 1 0.1 0.01 1E-3 1E-4 1E-3 0.01 0.1 1 10 100 1000 Kn [ - ]
rag orce 7 Other effects on the drag force: Turbulence of the surrounding fluid reduces the critical ynolds number to about 1000. Surface roughness also causes a reduction of the critical ynolds number. The orosity of articles results in a reduction of the drag coefficient. With increasing article concentration the drag is considerably increased (hydrodynamic interaction). article motion in the vicinity of walls (normal and arallel).
ressure orce orce on the article due to a local ressure gradient and the shear stress in the flow: m ( + τ) With the Navier-Stokes equation one finds that: + τ u t g This yields the total ressure force as: u m g t The second term is the buoyancy force
Virtual Mass and Basset orce 1 or higher article ynolds numbers the virtual (added) mass and the Basset (history) force may be exressed as: A 0.5 A m d d t ( u u ) d t ( u u ) µ m d t 9 B τ + π d 1 0 ( t τ) t ( u u ) 0 B oefficients given by Odar and Hamilton (1964): The second term in the Basset force accounts for the initial sli (eks & McKee 1984) 0.13 A.1 A + 0.1 B 0.48 + 0.5 ( A + 1) 3 < 60 Acceleration number A u d u u u d t
Virtual Mass and Basset orce cent studies of Michaelides and Roig (011) showed that the added mass coefficient is constant ( A 1) and the Basset coefficient should be exressed in deendence of the Strouhal number: B.0 1.0533 [ ( ) ] 0.8.5 1 ex 0.14 Sr The Strouhal number describes the behaviour of oscillatory flows. In this context the fluid time scale is the recirocal value of the characteristic fluid oscillations. In turbulent flows this time scale corresonds to the integral time scale of turbulence. Sr 1 St T I τ onsequently the Strouhal number is the recirocal value of the article turbulent Stokes number.
Other orces Other imortant forces (field or external forces) are: Gravity force: g m g entrifugal and oriolis force (aears only if the equation of motion is written in olar co-ordinates): Z m v, ϕ r m v,r r v, ϕ Electrostatic force: q e E Thermohoretic force Magnetic force
Electrostatic orces 1 In an electrostatic reciitator, for examle, the articles 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 articles is caused by two mechanisms: ield charging occurs due to the convective motion of the ions and is relevant for articles larger than about 0.5 to 1 µm. iffusion charging is the result of the thermal motion of the ions and is relevant for articles with a diameter ( t) smaller than about 0. µm. q( t) q n e π 4 π ε k T e 0 ε 0 E 1 t t + τ 1 (conducting articles: 3, ) 1+ + (non-conducting articles 1.5 ) : relative dielectric constant of article ε 0 : absolute dielectric constant in vacuum e: elementary charge E 1 : field strength in the charging region q max ln 1+ q n e c N0 e 8 ε k T 0 π ε t 0 E 1
Electrostatic orces alculated flow field, electrostatic otential and article trajectories in an electrostatic reciitator (Böttner and Sommerfeld 003) U av 0.5 m/s V H 60 kv Y [ m ] hannel: 300 mm x 600 mm Searation wires: 150 mm 0.15 0.10 0.05 articles 1 0 µm Y [ m ] 0.15 0.10 0.05 article charge: 0.66 q max 0.00 0.0 0. 0.4 0.6 0.8 1.0 X [ m ] articles 0.9 1.5 µm 0.00 0.0 0. 0.4 0.6 0.8 1.0 X [ m ]
Thermohoretic orce Thermohoretic force due to temerature gradient (LBM): T 6 π ( 1+ 3.4 Kn) κ 1.17 +.18 Kn κ κ 1+ + 4.36 Kn κ d art ν 1 T T κ, κ : thermal conductivity of fluid and articles dt/dh1k/mm hot cold Molecules dt/dh10 K/mm article 100 nm
Brownian Motion 1 Brownian article motion (LBM): 16 k Boltz ν T B ζ m 5 d t π un ζ Random number (0 ζ i 1) Only drag force drag force and Brownian motion
[1/m] Brownian Motion Lagrangian simulation of article Brownian diffusion in homogeneous isotroic turbulence:.0x10 7 1.5x10 7 1.0x10 7 5.0x10 6 S-E k 3πμ d 5nm d 10nm d 50nm d 100nm d 400nm 0.0 0.0.5x10-7 5.0x10-7 7.5x10-7 1.0x10-6 article dislacement, x [m] Boltz f T d f iffusion coefficient, [m²/s] 10-5 10-6 10-7 10-8 10-9 10-10 10-11 Tracking a large number of articles and samling the dislacement Stokes-Einstein / 833 / 500 / 100 10-9 10-8 10-7 10-6 article diameter, d [m]
Sli-Shear Lift orce 1 Illustration of sli-shear lift force: u LS Analytic exression of Saffman (1965, 1968) for small article ynolds numbers: LS,Saff 6.46 4 0.5 0.5 ( µ ) ( u u ) u y
Sli-Shear Lift orce Sli-shear lift for higher article ynolds numbers: LS π 4 LS ( u u ) ω ) Rotation of the fluid: ω rot u u Lift coefficient: LS 4.116 0.5 S f (, ) s Shear ynolds Number: S µ ω orrelation for higher ynolds numbers (Mei 1997): f f ex 0.3314 1 β for : 40 10 + S β 0.5 (, ) ( 1 0.3314 β ) 1 (, ) 0.054 ( β ) 1 for : 40 s s
Sli-Shear Lift orce 3 Lift coefficient as a function of article ynolds number with the non-dimensional shear rate as a arameter (Sommerfeld 010): LS [ - ] 100 Saffman β 0.05 β 0.1 β 0.3 10 β 0.5 1 β 0.5 S Straight lines: Stokes regime 0.1 0.01 0.1 1 10 100 1000 [ - ]
Sli-Shear Lift orce 4 Imortance of sli-shear lift force comared to transverse drag force (Sommerfeld 1996): LS r 6.46 4 π 8 ( µ ) d 0.5 u y 0.5 u u ( v v ) u u du/dy [µm] [1/s] Air Water 1 310 588 LS r 0.17 µ 0.5 u y Stokes drag 0.5 u u 1 0. 66 ( v v ) 1+ Sli velocity ratio u/v: 10 6 1 10 730 186 100 31 59 1000 73 19 0.5 0.5 LS u 1.7 1 r µ y Limiting article diameter: > 0.588 µ 0.5 u y 0.5
Sli-Rotation Lift orce 1 Illustration of sli-rotation lift force: Analytic exression Rubinow and Keller (1961) LR π R 3 ( Ω V) V u u ( ) LR article rotation relative to the fluid Ω 1 u ω ω Lift force for higher article ynolds numbers: LR π Ω LR u u 4 Ω ynolds number of rotation: R µ ( u u ) Ω
Sli-Rotation Lift orce Lift coefficient with non-dimensional relative rate of rotation as a arameter (Sommerfeld 010): Rubinow and Keller (1961) Oesterle and Bui inh (1998) for < 140 (000) Ω LR γ u u 10 R LR 0.45 + γ 0.5 γ 1 γ 3 0.4 0.7 ( γ 0.45) ex ( 0.075 γ ) LR [ - ] 1 Straight lines: Stokes regime 0.1 1 10 100 1000 [ - ]
Sli-Rotation Lift orce 3 Imortance of sli-rotation lift force (Sommerfeld 1996): LR r π 8 3 π 8 1 c v x d u y ω z ( v v ) u u ( u u ) z ω [ 1/ s] Air ( µ m) Water no rotation of the fluid Stokes drag LR r ω 4 µ z ( u u ) ( v v ) 1000 194 45 000 137 3 5000 87 10000 61 15 LR z 0.417 ω r µ 1 Limiting article diameter: >.4 µ 1 ω z
Torque Angular velocity of articles and torque: Rubinow and Keller (1961): I dω dt T π µ Generalised torque: T 5 R The torque-coefficient was found from: 3 Ω Ω ω Exeriments by Sawatzki (1970) irect numerical simulations by ennis et al. (1980) 64 π 1.9 18.4 for : R 3 R + for : 3 < R < 1000 0.5 R < R R 1000 100 10 1 0.1 0.1 1 10 100 1000 R R R Korrelation Gl. 41 und 4 Rubinow und Keller (1961) ennis et al. (1980) Sawatzki (1970)
Effect of Lift orces alculation for a article-laden vertical ie flow (diameter 0 mm) with 400 µm (Lee und urst, 198): Imortance of transverse lift forces Without lift forces With lift forces
article sonse 1 erivation of the article resonse time based on the equation of motion: m du dt π 18 µ u u ( u u ) ( u u ) 4 du dt 4 du dt article resonse time: τ 1 τ 18 µ f ( u u ) f 4 τ f 3 µ u 0.687 ( 1 + 0.15 ) 4 Integration yields: u u t 1 ex τ 0.63 u u τ time
article sonse The resonse of articles to velocity changes in the surrounding fluid may be characterised with the so-called Stokes number (ratio of article resonse time to relevant time scale of fluid): St Analysis of article disersion in a lane mixing layer (rowe et al. 1996): Time scale of vortex develoment in a shear layer τ τ τ U St 18 µ f U
article sonse 3 St << 1 (the articles almost comletely follow the vortex structures) St 1 (the articles become more inertial and accumulate at the rim of the vortex)
article sonse 4 St >> 1 (the articles are not able to follow the vortex structure comletely) Summary of article resonse behaviour
Bubble Behaviour 1 In most of the technical alications bubbles are relatively large ( B > mm) and therefore are generally non-sherical and oscillating. The bubble shaes can be characterised by reresentative non-dimensional arameters (lift et al. 1978): V µ d B e Eo We r g σ B d e Mo g µ 4 3 σ B We V d B e sonse time for bubbles σ τ B B VB r g d ( + 0.5 ) B 18 µ f e B
Bubble Behaviour The drag coefficient of bubbles was obtained from single bubble rise exeriments and was found to be strongly deendent on the liquid tye and the contamination of the liquid. urified liquids: fluid bubble ontaminated liquids: rigid bubble The drag coefficient is an average value
Bubble Behaviour 3 c 4 Rigid bubble 0.687 ( 1+ 0.15 ) 1000 B B < B 5 1.397 c 9.5 10 1000 < < 1530 c.61 B > 1530 B B rag coefficient 100 rigid bubble fluid bubble (resent) fluid bubble Tomiyama Tomiyama et al. (1998) rigid: 4 max 0.687 8 Eo ( 1+ 0.15 ), 3 Eo + 4 c [ - ] 10 1 luid bubble (Lain et al. 00) 16 c B < 1.5 B 14.9 c 1.5 < B < 80 0.78 B 48.1 15 4.756 c 1 1.86 10 B 80 < B < 1530 0.5 B + B c B >.61 1530 0.1 0.1 1 10 100 1000 [ - ] B Tomiyama et al. (1998) fluid: 16 max min 1+ 0.15 b 0.687 ( ) b Eo, g 48 b ( ), f 8 3 σ g b E 0 E 0 + 4
Bubble Behaviour 4 Bubble sli velocity: comarison of exeriments with results from different correlations for the drag coefficient V B [ m/s ] (Bröder and Sommerfeld 007) 0.4 0.3 0. Haberman & Morton (1956) 0.1 rigid bubble fluid bubble (resent) fluid bubble Tomiyama 0.0 0 1 3 4 5 B [ mm ] U b, u'b, v' b [m/s] 0. 0.4 correlations rigid bubble fluid bubble 0.3 Tomiyama Exeriments loo facility 0.1 U b bubble column U b 0.0 0.5 1.0 1.5.0.5 3.0 3.5 4.0 diameter b [mm]
Bubble Behaviour 5 In bubbly flows the added mass force is of great imortance since esecially wobbling bubbles never exhibit stationary rise behaviour (Tomiyama 004) Sherical bubbles B < 1: VM 0. 5 or ellisoidal bubbles the added mass coefficient is a tensor : VM vm,h 0 0 0 vm,h 0 0 0 vm;h or oblate bubbles (asect ratio: E < 1) an analytic solution for the coefficients in horizontal (h) and vertical (v) direction were rovided by Lamb (193): VM,v E E cos 1 1 E E 1 E E cos 1 E 1 cos E E 1 E VM,h for E < 1 1 1 E cos E 1 ( E E)
Bubble Behaviour 6 The magnitude and direction of the transverse lift force strongly deends on the bubble size (Tomiyama 004): A ( UB UL ) x rot UL 3 π e Lift Sherical bubbles L B >> 1: 5 6 Lift 0. Lift min f ( 0.88 tan h ( 0.11 B )), f ( Eoh ) ( Eo ) h for : for : Eo h < 4 4 Eo h Tomiyamacorrelation f 3 ( Eo ) 0.00105 Eo 0.0159 Eo 0.004 Eo 0. 474 h h h h + Eoh σ a h h small bubbles large bubbles
Bubble Behaviour 7 Influence of shear flow on bubble migration calculated by a VO aroach (Tomiyama et al. 1993) Eo 1, Mo 10-3 Eo 10, Mo 10-3 Eo g ( ) f σ g b Mo g µ 4 f ( ) f f σ 3 g etermination of lift forces
Bubble Behaviour 8 low around nearly sherical and large non-sherical bubbles simulated by VO (Bothe et al. 007) Tomiyama correlation A min f Eo ( 0.88 tan h ( 0.11 B )), f ( Eoh ) ( Eo ) h h g ( ) f for : for : σ Eo h g 4 Eo < 4 h h f 3 ( Eo ) 0.00105 Eo 0.0159 Eo 0.004 Eo 0. 474 h h h h +
Bubble Behaviour 9 Lift force on a sherical bubble obtained by Legendre and Magaudet (1997) from resolved numerical simulations. Bubble ynolds-number Non-dimensional d UL VB V shear rate B b µ Sr 1 Sr dy U U L B 10 0.1 b 500 + L,b L,b 6 π L,low L,high ( Sr) 1/ ( 1+ 0. ) b.55 b L,b [ - ] 1 0.1 Leg & Mag, Sr 0.01 Leg & Mag, Sr 0. Tomiyama et al. 0.01 0.1 1 10 100 1000 b [ - ] 1+ 16 / b + 0.5 3/ Sr 1 9 / b +
article sonse 1 Analysis of article resonse in turbulent flow and the imortance of the different forces (Hjelmfelt and Mockros, 1966). Simlified equation of motion for the Stokes regime: m d u d t d u d u t 18 µ u d u d u µ τ τ ( ) + + m d d m u u m 0.5 m 9 τ π d 1 t d t d t t ( t τ) 0 arrangement of the equation: d u d t with: t d u d τ u + a u + c dτ a u + b + c t 0 d u 1 ( t τ) t ( t τ) t t 0 d τ 1 dτ a 18 µ ( + 0.5) b 3 ( 0.5) + c 9 1 µ π ( + 0.5)
article sonse The velocities of fluid and articles are exressed by ourier integrals: u 0 ( ς cosωt + λ sin ωt) dω u 0 ( σ cosωt + ϕ sin ωt) dω f 1 Introducing the velocities into the equation of motion yields the amlitude ratio and the hase angle: η ( 1 + f 1 ) + f 1 f β tan 1+ f1 With: ω ( ω + c 0.5 π ω)( b 1) ω ( a + c 0.5 π ω)( b 1) f ( a + c 0.5 π ω) + ( ω + c 0.5 π ω) ( a + c 0.5 π ω) + ( ω + c 0.5 π ω) As a arameter a modifies Stokes number is used (different from Hjelmfelt and Mockros, 1966) : St τ ω ω 18 µ
article sonse 3 rolets in air ( / 1000, 50 µm): Amlitude ratio η [-] 10 10 1 10 0 10-1 S+G+AM+BH (I) S+G+AM (II) S+G (III) S (IV) Tye II, III, IV 10-10 -3 10-10 -1 10 0 10 1 10 10 3 Stokes number St [-] Increasing ω Amlitude hase angle β [radians] 1.5 1.0 0.5 0.0-0.5-1.0 S+G+AM+BH S+G+AM S+G S hase -1.5 10-3 10-10 -1 10 0 10 1 10 10 3 Stokes number St [-]
article sonse 4 articles in liquid ( /.5, 00 µm): Amlitude ratio η [-] 10 10 1 10 0 10-1 10-10 -3 10-10 -1 10 0 10 1 10 10 3 Stokes number St [-] St τ S+G+AM+BH S+G+AM S+G S ω 18 µ ω hase Amlitude hase angle β [radians] 1.5 1.0 0.5 0.0-0.5-1.0 S+G+AM+BH S+G+AM S+G S -1.5 10-3 10-10 -1 10 0 10 1 10 10 3 Stokes number St [-]
Amlitude ratio η [-] article sonse 5 Air bubbles in water air ( / 0.001, 500 µm): 10 3 10 10 1 10 0 10-1 10-10 -3 10-10 -1 10 0 10 1 10 10 3 Stokes number St [-] St τ ω 18 µ ω S+G+AM+BH S+G+AM S+G S hase Amlitude hase angle β [radians] 1.5 1.0 0.5 0.0-0.5-1.0-1.5 10-3 10-10 -1 10 0 10 1 10 10 3 Stokes number St [-] S+G+AM+BH S+G+AM S+G S
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