Forces Acting on Single Particles in a Fluid

Transcription

1 Micro- and Nanoarticle Technology orces Acting on Single Particles in a luid Dr. K. Wegner - Lecture March 019

2 Particle Science and Technology Particle ormation action Crystallization Preciitation Condensation Modiication Grinding Agglomeration Coating Drying Transort, storage, handling Characterization Change o state/ surrounding matrix Mixing Disersion luidization Searation H. Schubert, Handbuch der mechanischen Verahrenstechnik, Wiley-VCH, 00. M. Rhodes, Introduction to Particle Technology, nd ed., J. Wiley, 008. Micro- and Nanoarticle Technology - S019

3 Particle motion relative velocity Understanding the motion o articles in luids is imortant or the design o equiment such as clariiers, cyclones, sorters, neumatic conveyors, luidized beds, or mills. u ωω V, u c v rel luid, η Particle has translational and angular velocity c c, ω Particle is exeriencing the relative velocity: v rel u c Micro- and Nanoarticle Technology - S019

4 orces that can act on a article in a luid orces due to article motion: low orce in direction o vrel : drag orce D low orce erendicular to v rel : buoyancy orce B, dynamic M Torsional moment: ield orces, e.g.: Gravitational orce: G V g Electrostatic orces: el Q E Micro- and Nanoarticle Technology - S019

5 orce, μn v.d. Waals orce Electrostatic orce Gravitational orce Particle size, μm K. Borho, R. Polke, K. Wintermantel, H. Schubert, K. Sommer, Chem. Ing. Tech. 6, 79, Micro- and Nanoarticle Technology - S019 5

6 Comressive orces, e.g.: Navier-Stokes: or stagnant luid: grad u P V grad g u + η 0 grad g P B, static V g u Buoyancy Inertial orces, e.g.: Inertial orce: Centriugal orce: I C V V c ϖ ( ϖ R) urther: orces due to collisions, wall contact, riction, diusion (concentration or thermal gradients), Micro- and Nanoarticle Technology - S019 6

7 orces acting on a single article in luid low luid low Pressure gradient B, static (static buoyancy) B, dynamic (dynamic buoyancy) orce ield (inertial orce) I c v rel (e.g. electrostatic orce) c u D (drag orce) (gravitational orce) G Micro- and Nanoarticle Technology - S019 7

8 Assumtions to simliy the orce balance: Rigid articles Geometric similarity o articles, described by a characteristic length d (ideally sheres or use equivalent diameter) Continuous luid No inluence o walls or suraces Incomressible Newtonian luid (, η const.) Homogeneous, linear steady-state low Drag orce D Newton s law o riction: or a shere: D (laminar or weakly turbulent low) D c D () A rel Micro- and Nanoarticle Technology - S019 8 P v πd vrel α( ) ; Drag coeicient; shere: α v rel ν d

9 Drag coeicient or a single shere Based on exerimental data shere Micro- and Nanoarticle Technology - S019 9

10 low around a shere Creeing motion Stokes law < 1: Predominant viscous orces, inertia negligible; Stokes law No sli condition D α π η d vrel η v d ( ) rel d Boundary layer thickness at equator δ Velocity gradients in boundary layer can result in orces on neighboring articles Micro- and Nanoarticle Technology - S019 10

11 low ield around a shere Stokes law Intermediate regime Newton s-law regime Suercritical regime At P,crit : transition to turbulent low in boundary layer U: transition oint ( Umschlagsunkt ); A: break-o / searation oint ( Ablöseunkt ) Micro- and Nanoarticle Technology - S019 11

12 low ield around a shere Newton s law (square law) regime 10 < <.5 x 10 5 crit : redominant inertial orces α( ) 0. 5 D π 16 v rel d Stagnation oint: v rel Micro- and Nanoarticle Technology - S019 1

13 Pressure distribution in Newton s law regime Stagnation area on the ront side o the shere: A. Back side: A π d v rel π 0. 5 d 0. v rel orce resulting rom total ressure: D P A π v 0. 5 d drag coeicient: α( ) (.6 ( 0.) ) v 0.5 d Micro- and Nanoarticle Technology - S019 1 π 0 rel rel average on ront side average on back side

14 Aroximation ormulas Stokes regime ( <1): α Intermediate regime (1< <1000; e.g. Schiller & Naumann, 19): α 687 ( ) Newton s law regime (1000< <.5x10 5 ): α 0.5 Entire regime ( <,crit ; e.g. Haider & Levensiel, 1989): α ( ) Haider A., Levensiel O. (1989) Powder Technol. 5, 6, Micro- and Nanoarticle Technology - S019 1

15 Drag coeicients shere - cylinder Drag coeicient c D Cylinder Shere ynolds-# Micro- and Nanoarticle Technology - S019 15

16 Drag coeicients or non-sherical articles or the volume-equivalent diameter d V : 0.5 < 0.5 :, Gr : c c D D ( ) ( ) 8 K A + B + C, Gr crit : c D ( ) cd Gr, Note: The -# must be ormed with the diameter o the volume-equivalent shere d V, Gr : ynolds-# limit or a low regime (, Grenze ) Micro- and Nanoarticle Technology - S019 16

17 c D, Gr d V : equal-volume shere diameter Micro- and Nanoarticle Technology - S019 17

18 Settling velocity o a rigid shere at steady state orce balance or a settling shere at steady state: G α ( ) v d g ( ) ( ) G D D B B, static α rel 0 π ( ) π 6 rel v g d ( ) 1 d seciic orce or load actor Lastvielaches The seciic orce n is the ratio between the drag orce acting on the individual article and the weight o this article. low orces dynamic orces n D gravitational orce buoyancy static orces G A Micro- and Nanoarticle Technology - S019 18

19 n α v g d ( ) rel 1 or sheres settling at steady state Or, introducing the roude-# r n α ( ) r 1 rel v g d inertia gravitational orces or Stokes and Newtons law regimes with well deined drag coeicient, the settling velocity (terminal velocity) o a article with diameter d can be calculated directly (and vice versa) using the concet o the seciic orce n. Micro- and Nanoarticle Technology - S019 19

20 Use dimensionless numbers to develo a diagram relating relative velocity with article diameter or given materials: Substitute v rel d ν n α( ) 1 ν g d g ( ) d α ν ( ) Ar Archimedes-# deends only on luid and article roerties! Substitute d v ν rel vrel g ν ( ) α( ) Ω Omega-# or Lijatschenko-# Micro- and Nanoarticle Technology - S019 0

21 Ar-Ω Diagram or a rigid shere settling at steady state (n1). Also called Grassmann diagram. E.g.: Determine settling velocity or a shere with d P at known luid roerties by calculating Ar and obtaining Ω rom the diagram. Micro- and Nanoarticle Technology - S019 1

22 Motion o drolets and bubbles at steady state Alications: e.g. srays, bubble columns, lotation, Internal circulation Drolets and bubbles are treated similarly to rigid sheres but can be deormed, ragmented and can have internal circulation. luidic shere: No low attachment at the surace. Shear leads to inner circulation and reduction o drag orce comared to the rigid shere. Micro- and Nanoarticle Technology - S019

23 or sherical bubbles: 0 < P < 0.5 : cd 16 / P with K being the luid constant: < P < 1. K : cd 18.7 /P K g ν γ ( ) We We r Ω Ar Weber-# : We d v v γ rel inertial orces surace tension With increasing Weber-# the bubbles tend to deorm rom sherical to ellitical. Peebles,.N., Garber, H.J., Chem. Eng. Prog. 9, 88 (195). Micro- and Nanoarticle Technology - S019

24 Drag coeicient C D Drolets Bubbles Rigid articles The drag coeicient increases or ellitical or deormed and umbrella-like bubbles and drolets. Source: VDI Wärmeatlas, 10 th ed. Lda7 (006). (Use diameter o volume-equivalent shere or ) Micro- and Nanoarticle Technology - S019

25 Table: Drag coeicients and dimensionless numbers or steady-state settlingand rising velocities o rigid sheres, drolets and bubbles. Micro- and Nanoarticle Technology - S019 5

26 Accelerated rigid sheres In general (accelerated shere, no dyn. buoyancy or rotation): G D B n 1 n α ( ) rel v g d ( ) member: at steady state balanced orces n1; Now: α( ) Ar Ω Ar Ω n n α ( ) or non-sherical articles to be ormed with equivalent diameter d V Micro- and Nanoarticle Technology - S019 6

27 Ar-Ω diagram or a rigid shere. n<1: accelerated shere n>1: decelerated shere Micro- and Nanoarticle Technology - S019 7