Chapter 1. Single Particles in a Fluid 1.1 Motion of Solid Particles in a Fluid: Drag Force

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Chapter 1. Single Particles in a Fluid

1.1 Motion of Solid Particles in a Fluid: Drag Force

1) Drag Force, F_D:

Net force exerted by the fluid on the spherical particle(diameter x) in the direction of flow

F_D= C_D

(

^π4 d²_p

)

^ρ^f2^U² Area Kinetic exerted by energy of friction unit mass fluid

where C_D : Drag coefficient cf. for pipe flow

^τ_w= f ρ

fU²

2 → F_w= f ( π DL) ρ

fU² 2

where _f: Fanning friction factor

CD vs. Rep - Figure 1.1

Re_p= d_pUρ_f μ

where U : the relative velocity of particle with respect to fluid

- For Rep < 1 (creeping flow region)

F_D= 3πd_pμU

Stokes(1851)' law ₌

(

^Re²⁴^p

) ⁽

⁴^π ^d²^p

⁾

^ρ^f²^U²

∴C_D= 24 Re_p cf.f = 16

Re

- For 500 < Re_p< 2×10⁵

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Fluid in continuum

Particle Particle

Fluid molecules

Fluid molecules Particle C_D≈0.44

- For intermediate range

C_D= 24

Re_p(1+ 0.15 Re^0.687_p )

or Table 1.1

2) Motion in a Gas: Non-continuum Effect Mean-free path of fluid

λ = 1

2n_mπd²_m

where

n_m : number concentration of molecules d_m : diameter of molecules

For air at 1 atm and 25^oC ^λ = 0.0651μm

Continuum regime transition regime free-molecule regime

Knudsen number, Kn

Kn = λ d_p

- Continuum regime : _Kn～0 (<0.1) - Transition regime : Kn～1 (0.1～10) - Free molecule regime : _Kn～∞ (>10)

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F_g F_D

U_T

Corrected drag force

F_D= 3πd_pμU C_c

where C_c : Cunningham correction factor

C_c= 1+Kn[2.5 14 + 0.8 exp ( - 0.55/Kn)]

d_p, μm C_c 0.01

0.05 0.1 1.0 10

22.7 5.06 2.91 1.168 1.017 In air at 1atm and 25^oC

1.2 Particle Falling Under Gravity Through a Fluid

1) Terminal Settling Velocity,U_T

The velocity of free falling particle when

F_D= F_g- F_B

∴_C_D

(

^π⁴ ^d²^p

)

^ρ²^f^U² ⁼ ^π⁶^{( ρ}^p^{- ρ}^f^)d³^p^g

∴U_T=

[

⁴³ ^gd^C^D^p

(

^ρ^p^ρ^{- ρ}^f ^f

)]

^1/2

For Stokes law regime

3πd_pμU C_c = π

6( ρ_p- ρ_f)d³_pg

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∴U_T= ( ρ_p-ρ_f)gd²_pC_c 18μ

d_p, μm U_T, cm/s^a 0.1

0.5 1.0 5.0 10.0

8.8×10^-5 1.0×10^-3 3.5×10^-3 7.8×10^-2 0.31

aFor unit density particle in air at 1 atm and 25^oC

Worked Example(additional)

Example. In 1883 the volcano Krakatoa exploded, injecting dust 32km up into the atmosphere. Fallout from this explosion continued for 15 months. If one assumes settling velocity was constant and neglects slip correction, what was the minimum particle size present? Assume particles are rock spheres with a specific gravity of 2.7.

∴U_T= ( ρ_p-ρ_f)gd²_pC_c

18μ = 2.7⋅980⋅d²_p⋅1 18⋅1.81⋅10^{- 4} = 32⋅10³⋅10²

15⋅30⋅24⋅3600 = 0.0823 cm/s

∴ dp= 3.19μm

2) Terminal Settling Velocity for NonStokesian particles In Newton's regime

U_T= 1.74

(

^d^p^(ρ^p^ρ^{- ρ}^f ^f^)g

)

^1/2

For intermediate regime : Trial and error(Numerical) or

C_DRe²_p ⇒ 4 3

d³_pρ

f(ρ_p-ρ_f)g

μ² ≡K (constant)

↓

log C_D= log K - 2 log Re_p

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Log CD

Log Re_p Log K

Slope=-2

Log Re_p Slope’=1

Log K’

Log Re’p

From Re_p,

U_T= Re_pμ d_pρ_f

Worked Example 1.6

* _d_p? for given _U_T

C_D

Re_p ⇒ 4 3

gμ( ρ_p- ρ_f) U³_Tρ²

f

≡K'

log C_D= log Re_p+ log K'

From the Figure above,

d_p= Re_pμ U_Tρ

f

Worked Example 1.5

http://www.processassociates.com/process/separate/termvel.htm ☞_U_T

http://www.processassociates.com/process/separate/termdiam.htm☞d_p

3) Transient Response to Gravitational Field

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t U

τ U_T

Particle diameter, μm Relaxation time, s 0.01

0.1 1.0 10

100

6.8×10^{- 9} 8.8×10^{- 8} 3.6×10^{- 6} 3.1×10^{- 4} 3.1×10^{- 2}

Relaxation time for Unit Density Particles at Standard Conditions

In general,

Particle motion in gravity

m_pdU

dt = F_g-F_B-F_D ρp

π 6 d³_pdU

dt = π

6 ( ρ_p- ρ_f)d³_pg - C_D

(

^π4 d²_p

)

^ρ^f2^U²

For Stokes' law regime, _F_D_{= 3πd}_p_μU Rearranging,

τ dUdt = τg -U

where τ≡ ( ρ_p- ρ_f)d²_pC_c 18μ

Relaxation time Integration yields

U =U_T[ 1 - exp ( - t/τ)]

* 여기서 ^τ는 외부의 변화에 대처하는 입자의 기민성과 관련한다.

Table 1.2

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Powders Dynamic shape factor sphere

cube

cylinder(L/D=4) axis horizontal axis vertical bituminous coal quartz

sand talc

1.00 1.08

1.32 1.07 1.05-1.11

1.36 1.57 2.04

Dynamic shape factors of powders

1.3 Nonspherical particles

* _C_D for nonspherical particles - Figure 1.3

1) Dynamic shape factor, χ

χ = F_D 3πd_vμU

따라서 비 구형 입자의 항력은 동적 형상인자의 값을 알면 구할 수 있다.

2) Equivalent Diameters:

Spheres with the same properties - Volume-Equivalent diameters

Surface-area equivalent diameters - Terminal velocity-equivalent diameters

Stokes diameter :d_St=

[

^(ρ^18μU^p^-ρ^f^T^)g

]

^1/2

Aerodynamic diameter :d_a=

[

^(ρ^18μU⁰^-ρ^f^T^)g

]

^1/2

- Shape factors

Sphericity, ψ

ψ=

(

^d^d^St^v

)

²

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d_v=5μm ρp=4000kg/m³ χ=1.36

d_s=4.3μm ρp=4000kg/m³

d_a=8.6μm ρp=1000kg/m³

U_T=2.2mm/s U_T=2.2mm/s U_T=2.2mm/s

Irregular particles and its equivalent spheres

1.S Migration of Particles by Other External Force Fields

입자는 외부력 장(external force field)하에서 영향을 받아 움직인다.

이 때 움직임을 migration, 그 속도를 migration velocity라 부른다.

In Stokes' law regime

F_ext= F_D

(

⁼ ^3πμd^C^c ^p ^U^mig

)

where U_mig : migration or drift velocity in the fields e.g., For gravitational field,

F_ext= π

6 d³_p(ρ_p- ρ)g → U_T= ρ

pgd²_p 18μC_cρ

p

(ρ_p-ρ_f)

terminal settling velocity Flux by migration

J = C U_mig

where _C: particle concentration

1) Centrifugal migration

F_c= m_p

(

^{1 -}^ρ^ρ^p^f

)

^U^r²^t ^{= m}^p

(

^{1 -}^ρ^ρ^p^f

)

^rω²

↑ ↑

Acceleration of centrifugation, cf. g

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∴U_cf= ( ρ_p-ρ_f)d²_pU²_tC_c 18μr

단순 중력에 의한 것보다 더 높은 가속도를 얻을 수 있으므로 작은 입자의 침강을 보다 빠르게 얻을 수 있다.

2) Electrical Migration

F = q E = n_eeE

where q : charge of particles

_E : strength of electric field _e : charge of electron

(elementary unit of charge) n_e : number of the units

U_e= n_eeEC_c 3πμd_p

전기적 방법에 의한 입도 측정이나 전기집진기의 원리가 된다.