Chapter 4 Fluid Flow through Packed Bed of Particles

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Chapter 4 Fluid Flow through Packed Bed of Particles

4.1 Pressure Drop - Flow Relationship

(1) Laminar Flow

Fluid flow through a packed bed: simulated by fluid flow through a hypothetical tubes

∴ ^{( - Δ p )}

H = 32μ U

D²

⇒ ^{( - Δ p )}

H_e = K₁μU_i D²_e

Hagen-Poiseille equation

Substituting suitable relations

∴ ^{( - Δ p )}

H = 180 μU

x²

( 1 - ε )² ε³

Carman-Kozeny equation

(2) General Equation for Turbulent and Laminar Flow Ergun equation

( - Δ p )

H = 150 μU

x²

( 1 - ε )²

ε³ + 1.75 ρ_fU² x

( 1 - ε ) ε³

Laminar Turbulent Laminar flow for _{R e * =} ^{x U ρf}

μ ( 1 - ε ) < 1 0

Turbulent flow for R e * = x U ρ_f

μ ( 1 - ε ) > 2000

or

f *= 150

Re * + 1.75

where _{f * ≡} ^{( - Δ p )}

H

x ρfU²

ε³ ( 1 - ε )

Filtrate Filter cake

Slurry

Filter medium

Filtrate Filter cake

Slurry

Filter medium

Batch Continuous

P r e s s u r e Filtration

Filter Press

Shell-and-Leaf Filters Automatic Belt Filters V a c u u m

Filtration Discontinuous Vacuum Filters Rotary Drum Filters Horizontal Belt Filters Centrifugal

Filtration Batch Types Continuous Types Types of liquid filters

Friction factor

Figure 4.1

(3) Nonspherical Particles

x_{s v} (surface-volume diameter) instead of _x

Worked Example 4.1

4.2 Filtration

For filtration theory and practice, see

http://coel.ecgf.uakron.edu/~chem/fclty/chase/

FILTRATION%20FUNDAMENTALS_files/frame.htm

(1) Introduction

Filter media : Canvas cloth, woolen cloth, metal cloth, glass, cloth, paper, synthetic fabrics

Filter aids : To avoid cake plugging

- Diatomaceous silica, perlite, purified woolen cellulose, other inert porous solids

- By either adding slurry (increasing cake permeability) or precoating the filter media surface

(2) Incompressible Cake For cake filter

From laminar part of Ergun equation

( - Δ p )

H = 15 0μ U ( 1 - ε )² x²ε³

where _L : cake thickness

x : surface-volume diameter of particle

* For compressible filter cake,

dp

dL = r_cμU

where r_c: a function of pressure difference By defining cake resistance r_c

r_c= 1 50 ( 1 - ε )² x²ε³ ,

( - Δ p )

H = r_cμU

where _{U =} ¹

A dV

dt

V: volume of slurry fed to filter

Also defining ^φ(volume formed by passage of unit volume filtrate) φ = HA

V , ^dV

dt = A²( - Δ p ) r_cμφV

Including the resistance of filter medium,

since the resistances of the cake and the filter medium are in series,

( - Δ p ) = ( - Δ p_m) + ( - Δ p_c)

↓ ¹

A dV

dt r_cμH_c

By analogy for the filter medium

( - Δ p_m) = 1 A

dV

dt r_mμHm

∴ ( - Δ p ) = 1 A

dV

dt ( r_mμH_m+ r_cμH_c)

Defining equivalent height of filter cake and volume of filtrate

r_mH_m= r_cH_eq and _Heq= ^φVeq

A

where V_eq: volume of filtrate passing to create a cake of thickness _Heq

∴ ¹

A dV

dt = ( - Δ p )A r_cμ ( V + V_eq)φ

Constant rate filtration

1 A

dV

dt = ( - Δ p)A

r_cμ( V + V_eq)φ = constant

Constant pressure filtration Integrating

t

V = r_cφμ

A²( - Δ p )

(

)

Worked Example 4.2

(3) Washing the Cake Figure 4.2