Dust Resuspension Modeling

Paul W. Humrickhouse and Brad J. Merrill

Fusion Safety Program

2nd IAEA RCM on Dust in Fusion Devices,21-23 June 2010

Overview

• Motivation and Objective• Aerosol deposition vs Resuspension• Particle-surface interactions• Resuspension model evolution• Model theory and description

- VZFG- RRH / Rock n’ Roll

• Numerical Implementation• Code and Model Comparison• Summary and Future Work

Introduction

• Radioactive/Toxic/Reactive dust presents a variety ofsafety issues

• Present work seeks to assess the extent of resuspensionand subsequent transport in an accident, e.g. LOVA

• MELCOR is a systems code that solves thermal hydraulicsand other things, including aerosols

- Developed at Sandia National Lab for LWR severe accidentmodeling

- Modified by INL for fusion applications and qualified forITER safety analysis

• MELCOR will deposit aerosols on surfaces, but has nomodel or mechanism to resuspend them

Deposition vs Resuspension

• This shortcoming is not limited to MELCOR• Deposition does not really treat particle surface interactions

- Body forces on particles steer them toward a surface

∂n∂ t

= u ·∇n = D∇2n−∇ ·vn

- They are simply assumed to stick

• Resuspension requires modeling interactions between theparticle, surface, and boundary layer flow

Particle-surface interaction

• Assumed that particles adhere to surfaces due to Van derWaals forces

F ∼ πγR

Resuspension as a static force balance

• The simplest picture of resuspension is that it occurs whendrag, or lift, forces exceed adhesive forces

• Particles are small: they reside in the viscous sublayer(u+ = y+), where u+ = u/u∗, y+ = ρyu∗/µ, andu∗ = (τw/ρ)1/2

• Drag on a sphere:

FD =12

ρU2p πR2CD =

πCDρ3u4∗R

4p

2µ2

• For CD = 24/Rep,FD = 3πρu2

∗R2p

• Force ratio (JKR, FA = 3πγRp):

FD

FA=

ρu2∗Rp

γ

Shortcomings of static force balance

• It’s an “all or nothing” approach:

- Zero resuspension occurs for FD/FA < 1- Complete resuspension for FD/FA > 1

• There is no time component to this...

- resuspension occurs completely and instantaneously

• Resuspension occurs when such a model predicts thereshould be none

Effects of turbulence

• The viscous sublayer is not really static, but affected byturbulent bursts

- These have a characteristic size and frequency,

`∼ ν

u∗

1t∼ u2

∗ν

• The latter was identified with a resuspension rate constant

Which forces are important?

• The above example considered only drag vs. adhesiveforces:

FD = 3πρu2∗R

2p

• The first models incorporating the turbulent burst idea usedlift forces:

FL ∼ µu3∗R

3p

• Thus, for small particles, drag is likely to dominate• It was later realized that drag was much more likely to

instigate movement by rolling due to the drag moment

- This approach has been adopted in both approches tofollow

Kinetic models

• The temporal character of resuspension was proposed tobe analagous to molecular desorption

- Resuspension rate constant given by an Arrenhius law:

p = f0exp(− Q

2〈PE〉

)• f0 is a “typical” frequency of particle/surface deformation

- Determined by experimental measurements of the lift forceenergy spectrum

f0 =

√〈 ˙f 2〉/〈f 2〉

2π≈ 0.00658

(u2∗

ν

)• Q/2〈PE〉= f (FD,FL,FA, ...) differs for each of two models

considered next

Vainshtein, Ziskind, Fichman, and Gutfinger (VZFG)

• Formulates Q/2〈PE〉 in terms of moments

- MA/MD turns out to be equivalent to Faτ/FD

• Forces are integrated to get potential and maximum energies

• Particle oscillates like a linear spring with stiffness χe:

χe =9

10 (6πγ)1/3 k2/3R2/3

VZFG (2)• Consideration of the particle equation of motion leads to

FD =χe

2R2 x3

• Integrating with respect to x gives PE ∼ x4, PE ∼ F4/3D

• Based on the maximum displacement from the JRK model,

Faτ = 9.3γ4/3R2/3

k1/3 , k =43

(1−ν2

1E1

+1−ν2

2E2

)• Drag on a stationary sphere on a plane in shear flow:

FD = 9.6µ2R2u2

∗ρ

Faτ

FD≈ ργ4/3

k1/3µ2R4/3u2∗

p = f0exp

(−(

Faτ

FD

)4/3)

Surface roughness

• In reality, particles do not adhere to smooth surfaces• They are presumed to rest on asperities, characterized by

their own radius of curvature ra

• Non-dimensionalform: r′a = ra/R

• Assumed r′a� R, sor′a should be used tocalculate adhesiveforce

• ra presumed to be lognormally distributed, with probabilitydensity ϕ(r′a):

ϕ(r′a)=

1

(2π)1/2

1r′a

1ln(σ ′a)

exp

(−(ln(r′a)− ln

(r̄′a))2

2(ln(σ ′a))2

)

VZFG Summary

Faτ

FD=

ργ4/3r2/3a

k1/3µ2R2u2∗

p = 0.0033(

u2∗

ν

)exp

(−(

Faτ

FD

)4/3)

ϕ(r′a)=

1

(2π)1/2

1r′a

1ln(σ ′a)

exp

(−(ln(r′a)− ln

(r̄′a))2

2(ln(σ ′a))2

)Fraction of partices remaining fR on a surface at time t:

fR (t) =∫

∞

0exp[−p(r′a)

t]

ϕ(r′a)

dr′a

Rock n’ Roll model (RNR)

• Based on prior RRH (Reeks, Reed, and Hall) model

- RRH originally proposed the Arrhenius law and energybalance

• Considers both drag and lift effects on rolling

• Primary differences between VZFG and RNR:

- RNR assumes a lognormal distribution of adhesive forces,rather than asperity radii used to calculate adhesive forces

- Considers both drag and lift effects on rolling- Treats contact with multiple asperities- Includes Gaussian distribution of drag forces- Spring oscillations are linear, in the direction of fluctuating

force

Rock n’ Roll (2)

Rock n’ Roll model (3)

p = f0exp

(−(fa−〈F〉)2

2〈f 2〉

)/12

[1+ erf

((fa−〈F〉)/

√2〈f 2〉

)]

• fa is the adhesive force, lognormally distributed:

ϕ(f ′a)=

1

(2π)1/2

1f ′a

1ln(σ ′a)

exp

(−(ln(f ′a)− ln

(f̄ ′a))2

2(ln(σ ′a))2

)

• 〈f 2〉 is the covariance of the fluctuating lift force f , givenexperimentally by √

〈f 2〉= 0.2〈F〉

• 〈F〉 is the mean aerodynamic force

Rock n’ Roll model (4)

• 〈F〉 contains both lift and drag contributions:

F(t) =12

FL +Ra

FD

• with the “geometric factor” R/a stated to be ~100• Mean lift and drag forces are given experimentally by

〈FL〉= 20.9ρf ν2f

(Ru∗νf

)2.31

〈FD〉= 32ρf ν2f

(Ru∗νf

)2

Numerical Implementation (1)

• Need to discretize in time and adhesive force (and particleradius)

- Force distribution evolves in time as loosely bound particlesare resuspended

∆Fn+1R,i −∆Fn

R,i

∆t=−p

(r′a)×(

θ ×∆Fn+1R,i +(1−θ)×∆Fn

R,i

)• Fortran code has been written to solve the problem

- Will be implemented as a MELCOR subroutine

• Structure is in place to do so

Numerical Implementation (2)

• Discretized MELCOR implementation will require:

- nr particle radius intervals- na adhesive force intervals- each for ns total surfaces

• nr×na×ns may present significant memory requirements• Possible solution: take all resuspending particles from one

bin

Size Distribution Approximation

x =lnr′a− lnr̄′a

lnσ ′a

VZFG implementation

0.0 2.0 4.0 6.0

Friction velocity (m/s)

0.0

0.2

0.4

0.6

0.8

Fraction remaining (FR)

Vainshtein's results

Approx. Eq. 18

Eq. 18

1.0

RNR implementation

0 1 10

Friction velocity (m/s)

0.0

0.2

0.4

0.6

0.8

Fraction remaining (FR)

Hall Data

Rock ‘n Roll R&H Rock 'n Roll Eq. 18

1.0

VZFG vs RNR

0.1 1 10

Friction velocity (m/s)

0.0

0.2

0.4

0.6

0.8

Fraction remaining

Rock'n Roll

Hall 10 data

Vainshtein

1.0

RnR - 20 micron, graphite particles

• Left: 23 µm alumina, σ ′a = 19, fred = 37• Right: 13 µm graphite, σ ′a = 19, fred = 1.55

Adhesion Distribution Measurements

• Reeks et al. take r′a = 1/37, σ ′a = 10.4 (i.e., from differentexperiments)

Influence of adhesion distribution

• Case 1: r′a = 1/37, σ ′a = 2.55• Case 2: r′a = 1/592, σ ′a = 49

Varying adhesion distributions in the RnR model

Time dependence• Prior experimental work has shown a resuspension rate∼ 1/t at long times

• Reeks et al. demonstrate this with RnR model• Our work shows similar behavior for the VZFG model

Adhesion distribution and time dependence• 1/t dependence occurs only for a sufficiently wide spread

in the lognormal distribution

“One bin at a time” approximation

• Agrees reasonably well with the “exact” solution for bothmodels

Influence of Gaussian Drag Force Distribution

p = f0exp

(−(fa−〈F〉)2

2〈f 2〉

)————————————————/

12

[1+ erf

((fa−〈F〉)/

√2〈f 2〉

)]

• Assumed by NRG code SPECTRA

Influence of particle size

• Alumina parameters (with σ ′a = 10.4, fred = 37), 10 µm and23 µm particles

Influence of Geometric Factor

• G = R/a, where a measures the distance betweenasperities

• Smaller a allows for easier rolling (i.e., more readilyresuspended by drag)

• Reeks et al. use G = 100, not abundantly clear why

Summary and Future Work

• Both VZFG and RNR resuspension models have beenimplemented numerically

• Results agree well with those published

• Outlook

- Models need to be implemented in MELCOR- Appropriateness of simplifying assumptions needs to be

verified for a broader range of flow parameters- Relative strenghts and weaknesses of competing models

need to be assessed

• Comparison to experimental data if available

• Biggest challenge: a quantitative result, given evolvingPFC characteristics