cbe 150a – transport spring semester 2014 fixed and fluidized beds
TRANSCRIPT
CBE 150A – Transport Spring Semester 2014
Fixed and Fluidized Beds
CBE 150A – Transport Spring Semester 2014
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.
CBE 150A – Transport Spring Semester 2014
Flow Through a Bed of Particles
CBE 150A – Transport Spring Semester 2014
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.
CBE 150A – Transport Spring Semester 2014
Response to Superficial Velocities
CBE 150A – Transport Spring Semester 2014
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:
CBE 150A – Transport Spring Semester 2014
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).
CBE 150A – Transport Spring Semester 2014
Laminar Flow
2
164
2V
D
L
Re
p
f
fD
VL
2
32
?
?
What are the proper velocity and diameter?
CBE 150A – Transport Spring Semester 2014
Velocity
Lb S = Volume of Bed
Lb S = Volume Available for Flow
For a unit length of bed:
SuSu 0Mass
Balance
0uu
CBE 150A – Transport Spring Semester 2014
Diameter
Since this is not true pipe flow must use hydraulic radius.
perimeterwetted
flowforareasectionalcrossDh 4
areasurfacewetted
flowforavailablevolumeDh 4
Multiply by L/L
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
Fixed Bed “Friction Factor”
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
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.
CBE 150A – Transport Spring Semester 2014
Adsorbent Mesh Sizes
6 X 8 Mesh dp = (0.132 + 0.0937) / 2 = 0.113 in (0.0094 ft)
CBE 150A – Transport Spring Semester 2014
Irregular Shapes
So the final Ergun equation is:
3
20
3
2
220 175.11150
ps
b
ps
b
D
Lu
D
Lup
CBE 150A – Transport Spring Semester 2014
Fluidization (Refinery Application)
CBE 150A – Transport Spring Semester 2014
Fluidization (Drug Application)
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
Void Fraction at Min. Fluidizationmf depends on the shape of the particles. For spherical
particles mf is usually 0.4 – 0.45.
CBE 150A – Transport Spring Semester 2014
Minimum Fluidization
What if mf (and maybe s) is unknown?
Wen and Yu found for many systems:
14
13 mfs
CBE 150A – Transport Spring Semester 2014
Bed Length at Minimum Fluidization
Once we obtain the minimum void fraction
ballpongPingmfBedTube
ballspongPingmfBed S
ML
,, 1
LBed
STube
CBE 150A – Transport Spring Semester 2014
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.
CBE 150A – Transport Spring Semester 2014
35.01500
98011
:
3
3
mkg
mkg
V
VV
andVV
VV
VVV
WWWandVVVknowWe
FractionVoid
solids
bed
bed
solids
bedbedbed
solidssolidsbedbed
fluidfluidsolidssolids
solidssolidsfluidfluidbedbed
solidsfluidbedsolidsfluidbed
CBE 150A – Transport Spring Semester 2014
mDD
Da
diameterEffective
pp
p
025.06
02.0
6
33
33
81.0
666
63131
a
a
Sphericity
s
CBE 150A – Transport Spring Semester 2014
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
CBE 150A – Transport Spring Semester 2014
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