Fixed bed and fluidized bed
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Transcript of Fixed bed and fluidized bed
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed and fluidized bed
Why fixed (or fluidized) bed? Expensive Catalyst enzyme (immobilized) Large Surface area
Used in reaction/adsorption/ elution (for example)
Goal: Expression for pressure drop, try some examples
Ref: BSL, McCabe & Smith
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed
Filled with particles Usually not spherical
To increase surface area To increase void fraction
To decrease pressure drop For analytical calculation, assume all particles are
identical Usable, because final formula can be modified by a
constant factor (determined by experiment)
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed
What are important parameters? (For example, for adsorption of a protein from a
broth) rate of adsorption (faster is better) saturation concentration (more is better)
From the product requirement (eg X kg per day), density and product concentration in broth ==> volumetric flow rate
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed
Sphericity Volume of particle = Vp
Surface Area of particle = Ap
Surface Area of sphere of same volume (Vs =Vp) = As
Sphericity = As/Ap
May be around 0.3 for particles used in packed beds lower sphericity ==> larger surface area
Assume quick adsorption (rate of adsorption is high) Calculate the surface area of particles needed for
operation
As, Vs
Ap,Vp
Sphericity <=> specific surface area <=> average particle diameter
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed Specific surface area
= Ap /Vp
Minimal value for sphere Some books use S to denote area (instead of A) Assume all the particles are identical
==> all particles have exactly same specific surface area
Tarus saddlePall Ring
Rings (Raschig,etc)
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed What is the pressure drop we need, to force the fluid through
the column? (i.e. what should be the pump spec)
We know the volumetric flow rate (from adsorption equations, productivity requirements etc)
We know the area per particle (we assume all particles are identical). And the total area for adsorption (or reaction in case of catalytic reactor).
Hence we can calculate how many particles are needed Given a particle type (eg Raschig ring) , the approximate
void fraction is also known (based on experimental results)
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed What is void fraction? Volume of reactor = VR
Number of particles = Np
Volume of one particle = Vp
Volume of all the particles = Vp * Np = VALL-PARTICLES
R ALL PARTICLES
R
V V
V
VOIDS
R
VVoid fraction
V
R P P
R
V V N
V
1RP
P
VN
V
Knowing void fraction, we can find the reactor volume needed Alternatively, if we know the reactor volume and void
fraction and the Vp, we can find the number of particles
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed To find void fraction experimentally Prepare the adsorption column (or reactor....) and fill it
with particles Fill it with water Drain and measure the quantity of water (= void volume) Calculate void fraction
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed Since we know Vp, Np, , we can find VR
Choose a diameter and calculate the length (i.e. Height) of the column (for now) In normal usage, both the terms ‘height’ and ‘length’ may be used
interchangeably (to mean the same thing) Adsorption rate, equilibrium and other parameters will also
influence the determination of height & diameter To calculate the pressure drop
Note: columns with large dia and shorter length (height) will have lower pressure drop
What can be the disadvantage(s) of such design ? (tutorial)
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed To calculate the pressure drop
You want to write it in terms of known quantities Length of column, void fraction, diameter of particles, flow rate of fluid, viscosity
and density Obtain equations for two regimes separately (turbulent and laminar) Consider laminar flow
Pressure drop increases with velocity viscosity inversely proportional to radius
Actually, not all the reactor area is available for flow. Particles block most of the area. Flow path is not really like a simple tube
Hence, use hydraulic radius
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar flow)
To calculate the pressure drop, use Force balance
Force P Area2
Area where flow occurs = 4
D 2
4
DForce P
Resistance : due to Shear Find Contact Area Find shear stress
Contact areaForce
Until now, we haven’t said anything about laminar flow. So the above equations are valid for both laminar and turbulent flows
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar Flow)
Find contact area
Wetted Area= ppN A =
1 p
R
p
VA
V
= 1 pR
p
AV
V
To calculate the shear stress, FOR LAMINAR FLOW
max42 avgVV
R R
r R
dV
dr
8 avgV
D
2
max 21
rV V
R
max 2 avgV V
Here V refers to velocity for flow in a tube
However, flow is through bed, NOT a simple tube
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar Flow)
Find effective diameter (i.e. Use Hydraulic radius), to substitute in the formula
Also relate the velocity between particles to some quantity we know
To find hydraulic radius ( and hence effective dia)
RFlowvolume V
Wetted Area= ppN A =
1 p
R
p
VA
V
4H
Flow AreaD
ContactPerimeter
Hydraulic diameter*
4*
Flow Area Column Height
ContactPerimeter Column Height
4Flowvolume
wetted area
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar Flow)
4
1H
p
p
DA
V
8 avg
H
V
D
8 1
4
pavg
p
AV V
2 1 pavg
p
AV V
Vavg is average velocity of fluid “in the bed”, between particles Normally, volumetric flow rate is easier to find
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar Flow)
Can we relate volumetric flow rate to Vavg ?
Use a new term “Superficial velocity” (V0)
0
Volumetric flowrateV
Column Area 0 2
4
QV
D
I.e. Velocity in an ‘empty’ column, that will provide the same volumetric flow rate
Can we relate average velocity and superficial velocity?
0avg
VV
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar Flow)
2 1gp
pav
AV V
0
2
2 1 p
p
AVV
2
4
DForce P
02
2
2 1 1
4
p
p pR
p
AV V AD
P VV
Force balance: Substitute for etc.
Contact areaForce
2
4R
DV L
Volume of reactor (say, height of bed = L)
2
0
2
2
2
2
4
1
4
2
p
p
AV
PV
D DL
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar Flow)
Pressure drop
2
2
0 2
2
2
4
1
4
2
p
p
AV V
P LD D
2
2
0
3
2 1
p
p
ALV V
P
Specific surface area vs “average diameter”
p
p
A
V
Define “average Dia” of particle as
6p
p
p
DA
V
Some books (BSL) use Dp
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar Flow)
Pressure drop
2
2
0
3
62 1
pDLVP
2
02 3
72 1
p
LV
D
However, using hydraulic radius etc are only approximations Experimental data shows, we need to multiply the pressure requirement by ~ 2 (exactly 100/48)
2
2
0
3
25
6
1
p
p
ALV V
P
In terms of specific surface area
2
02 3
1
150
p
LVP
D
In terms of average particle diameter
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Turbulent Flow)
Pressure drop and shear stress equations2
4
DForce P
Contact areaForce
Only the expression for shear stress changes
f
Re
For high turbulence (high Re),
2=constant
12 avg
fV
21=constant 2 avgV
202
=V
K
0avg
VV
However
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Turbulent Flow)
We have already developed an expression for contact area
Wetted Area= ppN A = 1 pR
p
AV
V
=1
pR
p
VA
V
2
02
1 pR
p
AVK V
V
2
Contact area4
DForce P
Hence, force balance
2
4R
DV L
Volume of reactor (say, height of bed = L)
2
30 1 p
p
AVP K L
V
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Turbulent Flow)
2
03
16
p
VP L
DK
In terms of average particle diameter
2
03
1 p
p
AVP K L
V
In terms of specific surface area
Value of K based on experiments ~ 7/24 What if turbulence is not high? Use the combination of laminar + turbulent pressure drops: valid for all regimes!
2
0Laminar 2 3
150 1
p
LVP
D
20
3
1
7
4Turbulentp
LVP
D
2 20 02 3 3
150 1 7 1
4totalp p
LV LVP
D D
Ergun Equation for
packed bed
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar OR Turbulent Flow)
If velocity is very low, turbulent part of pressure drop is negligible
If velocity is very high, laminar part is negligible
2 20 02 3 3
150 1 7 1
4totalp p
LV LVP
D D
Ergun Equation for
packed bed
0 20
2 2
2
2 1724
1
2
p
p
avg
AV V V
fV
Some texts provide equation for friction factor
212 avg
fV
laminar
212
turbulent
avg
fV
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar OR Turbulent Flow)
0
2
2
20
2
2
0
2 1
1
2
p
p
AV
K
fV
VV
0
4 17
12
p
p
V
AV
For pressure drop, we multiplied the laminar part by 2 (based on data) . For the turbulent part, the constant was based on data anyway.
Similarly...
0
100
48
4 17
12
p
p
AV
Vf
0
25 17
3 12
p
p
AV
V
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed bed - pressure drop calculation (Laminar OR Turbulent Flow)
0
25 17
3 12
p
p
AV
fV
Multiply by 3 on both sides (why?)
0
25 17
3 1
6
2p
V
D
0
150 1 73
4p
fD V
0
150 1 73
4pD Vf
Packed bed friction factor = 3 f
150 13 1
R.75
e ppf f
Eqn in McCabe and Smith
Reynolds number for packed bed
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Example Adsorption of Cephalosporin (antibiotic) Particles are made of anionic resin(perhaps resin coatings on ceramic
particles) void fraction 0.3, specific surface area = 50 m2/m3(assumed) column dia 4 cm, length 1 m feed concentration 2 mg/liter (not necessary to calculate pressure drop, but
needed for finding out volume of reactor, which, in this case, is given). Superficial velocity about 2 m / hr
Viscosity = 0.002 Pa-s (assumed) What is the pressure drop needed to operate this column?
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed Bed What is the criteria for Laminar flow? Modified Reynolds Number Turbulent flow:- Inertial loss vs turbulent loss
Loss due to expansion and contraction Packing uniformity
In theory, the bed has a uniform filling and a constant void fraction Practically, near the walls, the void fraction is more
Edge Center Edge
0.2
0.4
0.8
Ergun Eqn commonly used, however, other empirical correlations are also used
e.g. Chilton Colburn eqn
Re Ren
A Bf C
1p oD V
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed Bed Sphericity vs Void Fraction
0 1
1
~0.4
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fixed Bed Alternate method to arrive at Ergun equation (or similar correlations) Use Dimensional analysis
dependent variableP
( subscript, means fluid density or )fwithout
, , , , , , (i.e. sphericity)p o columnD L V D
2
2( , , , )p p o p
o column
D D V DPf
V L D
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fluidized bed When the fluid (moving from bottom of the column to the top)
velocity is increased, the particles begin to ‘move’ at (and above) a certain velocity.
At fluidization, Weight of the particles == pressure drop (area) Remember to include buoyancy
2
14 s f R
DP V
2
14s f
DL
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fluidized bed: Operation Empirical correlation for porosity
n
t
V
V
Types of fluidization: Aggregate fluidization vs Particulate fluidization
Larger particles, large density difference (SOLID - FLUID) ==> Aggregate fluidization (slugging, bubbles, etc)
==> Typically gas fluidization Even with liquids, lead particles tend to undergo
aggregate fluidization Archimedes number
3
2
f pg DAr
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fluidized bed: Operation Porosity increases Bed height increases Fluidization can be sustained until terminal velocity is reached If the bed has a variety of particles (usually same material, but different
sizes) calculate the terminal velocity for the smallest particle
Range of operability = R Minimum fluidization velocity = incipient velocity (min range) Maximum fluidization velocity = terminal velocity (max range) Other parameters may limit the actual range further
e.g. Column may not withstand the pressure, may not be tall enough etc
R = Vt/VOM
Theoretically R can range from 8.4 to 74
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fluidized bed: Operation
Range of operation depends on Ar
Ar
100 104 108
R
0
80
40
IIT-Madras, Momentum Transfer: July 2005-Dec 2005
Fluidized bed: Operation Criteria for aggregate fluidization
Semi empirical
0.5
20.6 ( )
0.3 ( )
p
s
Dfor liquid
for gas
Particulate fluidization Typically for low Ar numbers More homogenous mixture