ME 437/537-Particle G. Ahmadi - Clarkson University

G. AhmadiME 437/537-Particle


●● Hydrodynamic ForcesHydrodynamic Forces and and MomentsMoments

●● Diffusion MechanismsDiffusion Mechanisms●● Particle Adhesion and Particle Adhesion and

DetachmentDetachment●● Particle ChargingParticle Charging


Dc

D CC

Ud3F πµ=

]e4.0257.1[d

21C 2/d1.1c

λ−+λ

+=Cunningham Cunningham CorrectionCorrection

Re]Re15.01[24

C687.0

D

+=

µρ

=UdRe


Lift

ufup

)dydusgn(|

dydu|)uu(d615.1F

f2/1

fpf22/1

)Saff(L −ρν=

SaffmanSaffman (1965, 1968)(1965, 1968)


Drag

Gravity

guuu pfp

m)(C

d3dt

dmc

+−πµ

=

Equation of Motion Equation of Motion


guuu pfp

τ+−=τ )(dt

d

ν=

µρ

=πµ

=τ18

CSd18

Cdd3

mC c2

cp2

c

Relaxation Time Relaxation Time

f

p

Sρρ

=

)m(d103)s( 26 µ×≈τ −


dydU

0 µ=τdydUu 2* ν=

1dydU

=+

+

++ = yu 5y0 ≤< +

Turbulent stress is negligibleTurbulent stress is negligible

3)Saff(L d807.0F ++ =


Fick’sFick’s LawLawdxdcDJ −=

cDctc 2∇=∇⋅+∂∂ v

d3kTC

D c

πµ=

Diffusion Diffusion EquationEquation

DiffusivityDiffusivity


Similarity MethodSimilarity Method

Separation of Variable Separation of Variable MethodMethod

Integral MethodIntegral Method


●● van van derder WaalsWaals ForceForce●● JKR Adhesion ModelJKR Adhesion Model●● DMT Adhesion ModelDMT Adhesion Model●● MaugisMaugis--Pollock ModelPollock Model●● Particle Detachment MechanismsParticle Detachment Mechanisms●● Maximum Moment ResistanceMaximum Moment Resistance


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ π

+π+π+=2

AAA

3

2dW3dPW3dW

23P

K2da

JohnsonJohnson--KandallKandall--Roberts (1971) Roberts (1971)

1

2

22

1

21

E1

E1

34K

−

⎥⎦

⎤⎢⎣

⎡ υ−+

υ−=

K2dPa3 =Hertz Model Hertz Model


dWF ADMTPo π=

DerjaguinDerjaguin--MullerMuller--ToporovToporov (1975) (1975)

JKRPo

DMTPo F

34F =

PullPull--Off Force Off Force

31

2A

0 K2dWa ⎟⎟

⎠

⎞⎜⎜⎝

⎛ π=

0a =

Contact Radius Contact Radius at Zero Force at Zero Force

Contact Radius at Contact Radius at Separation Separation


HadWP 2A π=π+ Y3H =

32

0 d~a

21

0 d~a

ElasticElastic

PlasticPlastic


**3* P21P1a −+−=

dW23

PPA

*

π−=

31

2A

*

K4dW3

aa

⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

3/1*****JKR* )P21P1(PaPM −+−==

42.0M JKR*max =


EFE q= neq =

d3qCZu cp

πµ==

Coulomb ForceCoulomb Force

Particle Particle MobilityMobility


BoltzmannBoltzmann Equilibrium Equilibrium Charge DistributionCharge Distribution

}d

n05.0exp{d24.0)n(f

2

−π

= m02.0d µ>

)m(d,d36.2n µ≈=


∞→+ε−ε

+=∞ tase4

Ed]2

)1(21[n

2

p

p

cgscgs ]tden)kTm

2(1ln[e2

dkTn 2i

2/1

i2 ∞

π+=


(40 points) Consider a steady convective-diffusion process with a flow velocity near an absorbing wall. The governing equation is given by

where D is the diffusivity and a is a constant. The boundary conditions are:

, and

i. Use a similarity variable , reduce the governing

equation and boundary conditions to the similarity form. ii. Evaluate the concentration profile and the deposition velocity to the

wall.

ME 437/537 Final Exam Dec. 02

2

22

yCD

xCay

∂∂

=∂∂

0C)y,0(C = 0C),x(C =∞0)0,x(C =

4/1)a/Dx(2y

=η

Problem 1





(35 points) Consider a 12 µm silicon particle that is attached to a silicon wafer in a turbulent air flow with a shear velocity of 2 m/s.

i. Evaluate the drag, the Saffman lift and the hydrodynamic moment acting on the particle in wall units and in SI units.

ii. Evaluate the pull-off force as predicted by the JKR model. iii. Find the contact radius at zero force and at the separation

according to the JKR model. iv. Is the particle going to be removed by the rolling

mechanism? (Assume u+ = y+, and for silicon use WA= 0.0389 J/m2, E = 1.79x1011 N/m2, and Poisson ratio of 0.27. The kinematic viscosity of air is )

Problem 2

s/m105.1 25−×=ν





(25 points) Consider a cloud of 12 µm quartz particles with a concentration of 105 particles per cm3. i. Find the average absolute number of charge for the

equilibrium Boltzmann distribution. ii. Determine the number of particles that will carry 5 positive

charges. How many will carry no charges in this case? iii. Find the mean electrostatic precipitation velocity for a field

of 400 Volt/cm for particles with the average absolute charge distribution.

iv. Find the terminal velocity of these particles and compare with the electrostatic precipitation velocity.

(The density of air is 1.2 kg/m3, the density ratio of quartz particle to air is 2000, and charge of electron is )

Problem 3

.Coul1059.1e 19−×=



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ME 437/537-Particle G. Ahmadi - Clarkson University