Mitigation of Preferential Concentration due to electric charge in the

dispersed phase

2

Overview

• Introduction

• Simplifying assumptions / Numerical method

• Measures of preferential accumulation

– Stokes number dependence

– Dependence on Re ?

• Charged particle simulations

• Conclusions

3

The problem in physical space

• Dispersed phase flows

– continuous phase (fluid)

– dispersed phase (particles)

4

Dispersed phase flows

• Incompressible, homogeneous, isotropic

• Stationarity obtained using artificial forcing

• One-way coupling

– Particles do not influence fluid motion

• Point particles

– Particle wakes are not resolved

– Particle diameter << kolmogorov length scale

• Particle motion is governed only by “drag”

– Gravitational force not modelled

– Particle collisions not modelled (dilute suspension)

– Particle density >> fluid density

• Particles are “stochastic” for the purpose of charged particle simulations

Fluid: Particles:

5

Governing equations

0 u

p

epp

P

p

m

Ftuttxu

dt

du )()),((

1

687.0

2

Re15.01

1

18 pf

p

fp

pe qEF 0. vqE

Fluid: Particles:

*Symbols have usual meanings

Fup

uut

u

2.

Large-scale forcing function added to maintain stationary

turbulence

Modified Stokes drag law (Valid for Rep <= 800)

6

Numerical scheme (Fluid)

jxij

N

Njfjf euxu )(ˆ)(

2

12

)](.[)()(ˆ)(ˆ

22

uuut

uf

0)(ˆ. u

*V. Eswaran and S.B. Pope, Computers and Fluids, Vol. 16(3), pp. 257-278, 1988

Fluid (pseudo-spectral method):

• dealiasing by 2/3rd rule

• temporal discretization using RK3

• stochastic forcing scheme* to sustain kinetic energy

)()(. SEi

SqE v 0. (suppose)

Particle:

7

Numerical Method (summary)

– The turbulence is limited to homogeneous, isotropic case (HIT) in a periodic cube.

– Particles are not resolved.

– Force on particles is due to Stokes drag.

– One way coupling between fluid and particles

8

Simulation parameters

k

pkSt

308 pray rQ

ray

p

Q

QRa

• Mono-sized particles

• number of particles (Np): 100000

• particle stokes numbers

– Stk :0.2 - 20

• Same charge on all particles (Ra = 0.8, γ = 0.05)

• space charge densities (μC/m3): 5, 10, 25, 50, 100

Stokes Number

Rayleigh Number

where,

9

Points to note -

• All simulations for a given Re, are restarted from same fluid realisation.

• Statistics are collected only after fluid has reached stationary state.

• Particle distribution is assumed to reach stationary state when the positions are completely de-correlated from initial position.

Tu

rbu

len

t kin

eti

c en

erg

y

Part

icle

rm

s ve

loci

ty

10

Evidence of preferential concentration

Reλ = 24.24, St = 0.25

*S. Scott, Ph.D. thesis, Imperial College London, 2006

Reλ = 24.24, St = 4.00

11

Clustering at different scales• Clustering occurs broadly at 2 scales –

– Dissipative scales

• particles are centrifuged out of coherent eddies and accumulate in low-vorticity regions.

– Inertial range

• clustering is a multi-scale phenomenon.

• Eddies larger than Kolmogorov length scale play a part in clustering.

12

Measures of Accumulation• Dissipative range measures

– D ( Fessler et al., 1994 )

– Dc ( Wang and Maxey, 1993 )

– Dn

• Inertial (multi-scale) measures

– RDF ( Sundaram and Collins, 1997 )

– D2

• Fluid-particle correlation measures

– <n’e’>, correlation between number density and enstrophy

– ln, number density correlation length scale

13

D2 measure

• Correlation integral, C(r) : number of particles within range r of any given particle

• D2 is slope of curve log( C(r) ) vs log( r )

• D2 is equal to the spatial dimension for uniform distribution (equal to 3 for a 3D distribution)

r

14D2 data for different cases compared to literature

D2 – probability to find 2 particles at a distance less than a given r: P(r) ~ rD2

15

Binning of particles

h

h – scale used for binning particles

n – number density i.e no. of particles / bin volume

<nc> – mean number density i.e total particles / volume of cube

16

D, Dc : deviation from poisson distribution• Dc*, D** : Deviation from

poisson distribution

Pc: probability of finding cells with given number of particles

k : number of particles in a cell

pN

k

uc kPkPD

0

2))()((

nc

D poisson

*L.P. Wang and M.R. Maxey, J. Fluid Mech., Vol. 256, pp. 27-68, 1993

**J.R. Fessler, J.D. Kulick and J.K. Eaton, Phys. Fluids, Vol. 6(11), pp. 3742-3749, 1994

knc

u nck

ekP

!)(

17

‘D’ measure of accumulation

• Bin size used is corresponding to peak value of ‘D’.

Re = 24

18

<n’e’> – correlation between number density and enstrophy

19

Observations

• D and Dc measures clearly depend on bin-size

– Dependence of Re is attributed to less number of ‘smaller particle structures’ at higher Re.

• D2 measure looks at probability of finding particles in shells around a given particle

– Shows nearly no dependence on Re

• <n’e’> and ‘ln’ capture distribution of particle number density

– Show dependence on Re

Destruction using Lorentz forces*

*A. Karnik and J. Shrimpton, ILASS 2008, Sept 8-10, 2008

21

Particle position, fluid velocity

Reλ = 24.24, St(k) = 1.0, Qv=5μC/m3

*S. Scott, Ph.D. thesis, Imperial College London, 2006

Reλ = 24.24, St(k) = 1.6, Qv=100μC/m3

22

Evidence of preferential concentration destruction

Reλ = 24.24, St(k) = 1.0, Qv=5μC/m3

*S. Scott, Ph.D. thesis, Imperial College London, 2006

Reλ = 24.24, St(k) = 1.6, Qv=100μC/m3

23

Evidence of preferential concentration destruction

Reλ = 24.24, St(k) = 1.0, Qv=5μC/m3

*S. Scott, Ph.D. thesis, Imperial College London, 2006

Reλ = 24.24, St(k) = 1.6, Qv=100μC/m3

24

Parametric study of bulk charge density levels

*St = 0.25 for all plots

• Space charge density of 25-50 µC/m3 is sufficient to destroy preferential accumulation

• With increasing Reynolds number, greater charge density is required to significantly destroy accumulation

25

Effect of Stokes Numbers

Reλ = 24.2 Reλ = 81.1

• Charged particle systems continue to exhibit same trends with Reynolds and Stokes numbers as the charge-free case.

26

Schematic of a spray released from a charged injection atomizer

20

22

0

0

tan4tan41

d

z

d

z

QQz

tan20 zd

rz

• The charge level found in this study (50 μC/m3) corresponds to an area about 2 cm from the nozzle tip

• d0= 500 μm, Q0= 0.5 C/m3, θ = 45o

27

Conclusions

• Preferential accumulation is maximum at St ~ 1.0 based on kolmogorov scale, for all the measures used in this study.

• While ‘ln’ shows clear dependence on Re, D2 is insensitive to Re.

• Bulk charge density level of 50 μC/m3 is sufficient to significantly destroy preferential accumulation. This has been consistently observed using different sensors for preferential accumulation.

– The required charge density level mentioned above is attainable within 2 cms from tip of a nozzle in practical charge injection atomizers.