Chapter 2 Multiple Particle Systems 2.1 Settling of a Suspension of Particles

Chapter 2 Multiple Particle Systems

2.1 Settling of a Suspension of Particles

Hindered Settling

입자가 모여 있으면 서로의 영향이 침강에 미친다. 이를 간섭침강이라 한다.

Effective viscosity of suspension

μe= μ f ( ε )

Effective density of suspension

ρ_ave= ερ_f+ ( 1 - ε)ρ_p

U_p ,_Uf: actual(interstitial) velocity of particles and fluid, respectively

U_{p s} ,_{U fs}:superficial velocity

U_ps= U_p( 1 - ε )

U_fs= U_fε

where ^ε: voidage or void fraction

∴1 - ε = particle volume

suspension volume = volume concentration of particles = c_v = volume

∴U_{p s} and U _fs: volume flux of particles and fluid (_{m /s⋅} ^m3 fluid or particles

m³ suspension )

☞U_{p s}와 U _fs는 superficial velocity임과 동시에 부피 flux 임!

Corrected terminal settling velocity

U _relT≡U_p- U_f

≠U_T

≡ U_Tεf( ε) Because of no net flow

U_ps+ U_fs= 0

2.2 Batch Settling

(1) Settling Flux as a Function of Suspension Concnetration

U_p( 1 - ε) + U_fε = 0

∴_Up( 1 - ε ) + [ U_p- U_Tεf( ε)]ε = 0

∴U_p= U_Tε²f( ε )

e.g. Richardson and Zaki(1954)

U_p= U_Tεⁿ

where

4.8 - n

n - 2.4 = 0.043Ar^0.57

[

(

)

]

Superficial solid velocity or volumetric solid flux(m/s)

U_ps= U_p( 1 - ε ) = U_Tε²f( ε ) ( 1 - ε )

_{= UT( 1 - ε)ε}ⁿ Settling flux curve (U_{p s} vs.C_v): Figure 2.1

(2) Sharp Interfaces in Sedimentation

C₂ C₁ U_p1

U_p2 U_int

Material Balance over the interface

( U_p1- U _int)C₁= ( U_p2- U _int)C₂

where C =1 - ε, solids fraction

∴_{U in t=} ^{Ups1- Ups2}

C₁- C₂

As ΔC→0,

U = dU_ps dC

The velocity of layer of concentration C

C1=0 C4 C2 C3

U_ps

Slope=velocity of layer of concentration C4

Slope=U_{int, 12}

Slope=Uint, 23

(3) Batch Settling

Supplying information for the design of a thickener Type I Settling

B

A B

S S

A

C_B C_A =0C_BC_S C_A =0C_S

A-B

B-S

A-S

Time, t Height of

interface, h

Variation of heights of interfaces with respect to time

C_A=0 C_B C_B2 C_S U_ps

Velocity of A-B interface

Velocity of B-S interface

Flux vs. concentration

Type II Settling

B

A B

S S

A

C_B C_A =0C_BC_S C_A =0C_S E

A

S

C_A =0C_B E

C_E,max

C_S C_E,max

Recommended web site :

http://www.aem.umn.edu/Solid-Liquid_Flows/video.html

A-B

E_max-S

A-S

Time, t Height of

interface, h

B-E_min

Variation in heights of interfaces with respect to time

C_A=0 C_B C_E,min C_S

U_ps

Velocity of A-B interface

Velocity of E-S interface

C_E,max Velocity of B-E interface

Flux vs. concentration for particle slurry

즉 두 type의 침강은 초기농도에 의존하는 것으로서 후자의 경우 하나의 층이 더 나타나는 것은 _CB의 위치에서 _CS가 변곡점 때문에 가려서 생기는 문제이다. 따라서 농도가 진하여 변곡점 아주 가까이에 있을 때 type II의 침강이 생긴다.

2.3 Continuous Settling

(1) Settling of a Suspension in a Flowing Fluid Thickener vs. Clarifier Figure 2.11

Downward flow

U _{p s} = Q ( 1 - ε )

A + U _Tε²f ( ε )

Total Flux Flux solid due to due to flux bulk flow settling

Define C_v≡1 - ε, particle volume concentration Figure 2.12:

Feed concentration,CF → bottom section concentration, CB

Upward flow

U _{p s} = - Q ( 1 - ε )

A + U _Tε²f ( ε )

Figure 2.13:

Feed concentration,CF → Top section concentration, CT

(2) Real Thickener

C_T C_B

Underflow L, C_L Feed

F, C_F

V, C_V

Feed/ Under(down)flow/ up(over)flow:

F(CF) L(CL) V(CV)

Below feed

U _{p s} = L ( 1 - ε )

A + U _Tε²f ( ε )

Above feed

U _{p s} = - V ( 1 - ε )

A + U _Tε²f ( ε )

C Ups

C U_ps

Underflow flux, slope=L/A

Overflow flux, slope=-V/A

Below feed

Above feed Downward flux

Downward flux

Upward flux Batch flux C_crit

* Critical concentration: C_crit≡ C atU_{p, upward}= 0

(3) Critically Loaded Thickener

C_F= C_crit

∴ _{Up, upward= 0}, _{CB= CF} ,_{CT= CV= 0} and U ps, downflow= FC_F

A = LC_L A

C U_ps

U_ps=FC_F/A =LC_L/A

C_F=C_B

C_crit

C U_ps

U_ps=VC_v/A

=0 C_T and C_v

Underflow flux, L/A

Overflow flux, -V/A

Below feed

Above feed Downward flux

Downward flux

Upward flux Batch flux

C_L Feed flux, F/A

(4) Underloaded Thickener

C_F< C_{c r it}

∴ _{Up, upward= 0}, _{CB< CF} and _{CT= CV= 0}

U ps, downflow= FC_F

A = LC_L A

(5) Overloaded Thickener

C_F> C_{c r it}

∴ _{Up, upward=} ^FCF

A - U ps, downflow, _{CB= CF} and _{CT> CV≠ 0}

U ps, downflow= LC_L A

C Ups

Ups=FCF/A =LCL/A

C U_ps

Ups=VCv/A =0

C_T and C_v

Underflow flux, L/A

Overflow flux, -V/A

Below feed

Above feed CL

Feed flux, F/A

CF

CB Ccrit

Underloaded Thickener

C Ups=LCL/A

CF=CB

CT C

Ups

Ups=VCv/A

Cv

Underflow flux, L/A

Overflow flux, -V/A

Below feed

Above feed Downward flux

Downward flux

Upward flux Batch flux

CL

Feed flux, F/A FCF/A

Overloaded thickener

* Centrifugal Sedimentation _rω² instead of _g

Worked Example 2.4