me 437/537-particle g. ahmadi - clarkson university
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G. AhmadiME 437/537-Particle
●● Hydrodynamic ForcesHydrodynamic Forces and and MomentsMoments
●● Diffusion MechanismsDiffusion Mechanisms●● Particle Adhesion and Particle Adhesion and
DetachmentDetachment●● Particle ChargingParticle Charging
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
Lift
ufup
)dydusgn(|
dydu|)uu(d615.1F
f2/1
fpf22/1
)Saff(L −ρν=
SaffmanSaffman (1965, 1968)(1965, 1968)
G. AhmadiME 437/537-Particle
Drag
Gravity
guuu pfp
m)(C
d3dt
dmc
+−πµ
=
Equation of Motion Equation of Motion
G. AhmadiME 437/537-Particle
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 −
G. AhmadiME 437/537-Particle
dydU
0 µ=τdydUu 2* ν=
1dydU
=+
+
++ = yu 5y0 ≤< +
Turbulent stress is negligibleTurbulent stress is negligible
3)Saff(L d807.0F ++ =
G. AhmadiME 437/537-Particle
Fick’sFick’s LawLawdxdcDJ −=
cDctc 2∇=∇⋅+∂∂ v
d3kTC
D c
πµ=
Diffusion Diffusion EquationEquation
DiffusivityDiffusivity
G. AhmadiME 437/537-Particle
Similarity MethodSimilarity Method
Separation of Variable Separation of Variable MethodMethod
Integral MethodIntegral Method
G. AhmadiME 437/537-Particle
●● 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
G. AhmadiME 437/537-Particle
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ 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
G. AhmadiME 437/537-Particle
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
G. AhmadiME 437/537-Particle
**3* P21P1a −+−=
dW23
PPA
*
π−=
31
2A
*
K4dW3
aa
⎟⎟⎠
⎞⎜⎜⎝
⎛ π=
3/1*****JKR* )P21P1(PaPM −+−==
42.0M JKR*max =
G. AhmadiME 437/537-Particle
EFE q= neq =
d3qCZu cp
πµ==
Coulomb ForceCoulomb Force
Particle Particle MobilityMobility
G. AhmadiME 437/537-Particle
BoltzmannBoltzmann Equilibrium Equilibrium Charge DistributionCharge Distribution
}d
n05.0exp{d24.0)n(f
2
−π
= m02.0d µ>
)m(d,d36.2n µ≈=
G. AhmadiME 437/537-Particle
∞→+ε−ε
+=∞ tase4
Ed]2
)1(21[n
2
p
p
cgscgs ]tden)kTm
2(1ln[e2
dkTn 2i
2/1
i2 ∞
π+=
G. AhmadiME 437/537-Particle
(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
G. AhmadiME 437/537-Particle
(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
G. AhmadiME 437/537-Particle
(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−×=