cbe 150a – transport spring semester 2014 fixed and fluidized beds

CBE 150A – Transport Spring Semester 2014

Fixed and Fluidized Beds


Goals

• Describe forces that act on a bed of particles.

• Describe how pressure drop and bed height (or void fraction) vary with fluid velocity.

• Apply basic equations to compute pressure drop across the bed, the bed height and the diameter of the bed.

• List advantages and disadvantages of fluidized beds.


Flow Through a Bed of Particles


Response to Superficial Flow

Fluid does not impart enough drag to overcome gravity and particles do

not move. Fixed Bed.

At high enough velocities fluid drag plus buoyancy overcomes the

gravity force and the bed expands. Fluidized Bed.

Low Velocity

High Velocity

p for Increasing u0

Until onset of fluidization p increases, then becomes constant.

Bed Length for Increasing u0

L is constant until onset of fluidization and then begins to increase.


Response to Superficial Velocities


Fixed Bed

How do we calculate the pressure drop across a fixed bed?

Start with the MEB:

fbf

hgLp

24

2V

D

Lfh f

For pipe flow we determined:


Pressure Drop

For now make the following assumptions:

• Horizontal Bed (or small L)Gravity not important.

• Particles pack uniformly giving rise to continuous flow channels

• Bed can be modeled as bundle of small pipes.

• Flow is laminar (f = 16/Re).


Laminar Flow

2

164

2V

D

L

Re

p

f

fD

VL

2

32

?

?

What are the proper velocity and diameter?


Velocity

Lb S = Volume of Bed

Lb S = Volume Available for Flow

For a unit length of bed:

SuSu 0Mass

Balance

0uu


Diameter

Since this is not true pipe flow must use hydraulic radius.

perimeterwetted

flowforareasectionalcrossDh 4

areasurfacewetted

flowforavailablevolumeDh 4

Multiply by L/L


Diameter

sb

bh aSL

SLD

14

as is the ratio of particle surface area to volume.

The denominator above is then the particle volume multiplied by as or the particle surface area.

ps DR

Ra

643

34

2

For a sphere:

ph DD

16

4


Laminar Flow

3

2

20 172

pD

Lup

In actuality the above equation does not account for the tortuous path

through the bed and L is much longer. Experimental data show that a

numerical constant of 150 should replace the 72.

Blake-Kozeny equation. Assumes < 0.5 and Rep < 10.

fp

p

uDRe 0

1

1

3

2

20 1150

pD

Lup


Turbulent Flow

One cannot use the Hagen-Poiseuille approximation when flow is turbulent. After substituting in Dh and velocity correction

3

20 13

pD

Lufp

Experimentally:

000,1pRe

Burke-Plummer Equation

3

20 175.1

pD

Lup


Intermediate Flow

Ergun Equation

75.1150

1

3

20

p

p

ReL

D

u

p

Note: equation can be used with gases using average gas density between inlet and outlet.

3

20

3

2

20 175.11150

p

b

p

b

D

Lu

D

Lup


Fixed Bed “Friction Factor”


Irregular ShapesTo increase surface area and liquid solid contact, many particles are often of irregular shape. In that case the particle is treated as a sphere by introducing a factor called sphericity s which allows calculation of an equivalent diameter.

particleparticle

p

particle

spheres VS

D

a

a

/

6

Where Dp is the diameter of a sphere of the same volume as the particle


Example: Cube

3

26

aV

aS

What is diameter of sphere of volume a3?

aD

Da

p

p

31

33

6

6

81.0

666

63131

a

as


Sphericity

Note entries for cubes and cylinders. For convenience, some just calculate a nominal (average) diameter and assign a sphericity of unity.

For greatest contact area want lower sphericity.


Adsorbent Mesh Sizes

6 X 8 Mesh dp = (0.132 + 0.0937) / 2 = 0.113 in (0.0094 ft)


Irregular Shapes

So the final Ergun equation is:

3

20

3

2

220 175.11150

ps

b

ps

b

D

Lu

D

Lup


Fluidization (Refinery Application)


Fluidization (Drug Application)


Fluidization

At fluidization, the gravity force on the particles in the bed must be balanced (Fk = 0) by the drag, buoyancy, and pressure forces.

0121 gLppSF fpbk

Substituting the Ergun equation for the pressure drop.

75.1

1150

03

20

fpsps

ffp uDD

ug


Minimum Fluidization Velocity

This equation can be used to calculate the minimum fluidization velocity umf if the void fraction mf at incipient fluidization is known.

75.1

11503

2

fmfps

mf

mfps

mfffp uDD

ug


Void Fraction at Min. Fluidizationmf depends on the shape of the particles. For spherical

particles mf is usually 0.4 – 0.45.


Minimum Fluidization

What if mf (and maybe s) is unknown?

Wen and Yu found for many systems:

14

13 mfs


Bed Length at Minimum Fluidization

Once we obtain the minimum void fraction

ballpongPingmfBedTube

ballspongPingmfBed S

ML

,, 1

LBed

STube


Example

A packed bed is composed of cubes 0.02 m on a side. The bulk density of the packed bed, with air, is 980 kg/m3. The density of the solid cubes is1500 kg/m3.

• Calculate the void fraction () of the bed. • Calculate the effective diameter (Dp) where Dp is the diameter of a sphere

having the equivalent volume.• Determine the sphericity of the cubes.• Estimate the water flow rate (m3/sec) required for minimum fluidization of the solid using water at 38 C and a tower diameter of 1.0 m.


35.01500

98011

:

3

3

mkg

mkg

V

VV

andVV

VV

VVV

WWWandVVVknowWe

FractionVoid

solids

bed

bed

solids

bedbedbed

solidssolidsbedbed

fluidfluidsolidssolids

solidssolidsfluidfluidbedbed

solidsfluidbedsolidsfluidbed


mDD

Da

diameterEffective

pp

p

025.06

02.0

6

33

33

81.0

666

63131

a

a

Sphericity

s


2

45

3

23

3

2

3

222

3

2

10748.9445.0025.081.0

99475.175.1

445.014

1

495980.93

9943

1500

75.11150

mf

mf

mfps

mff

mfmfs

fmfps

mf

mfps

mfffp

um

kg

m

um

kg

D

u

sm

kg

s

m

m

kg

m

kg

uDD

ug

VelocityonFluidizatiMimimum

LHS

RHS Term No. 1


GPM) (884.5 /sm 0558.0071.0*4

(1.0) flow Volumetric

233.0071.0

4959159710748.90

1597

445.0025.081.0

001.0693.0445.011501150

322

2232

45

3

322322

s

mm

s

ft

s

mu

sm

kgu

sm

kgu

m

kg

usm

kg

m

usm

kgcp

D

u

mf

mfmf

mf

mf

mfps

mfmf

RHS Term No. 2

Final Equation

cbe 150a – transport spring semester 2014 fixed and fluidized beds

Documents

transport spring semester

bed of particles

volume of bed

bed length

bed height

horizontal bed

ll slide

continuous flow channels