1. 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)

2. Fixed bed Assume quick adsorption (rate of adsorption is high)
Calculate the surface area of particles needed for operation
Sphericity <=> specific surface area <=> average particle diameter
Ap, Vp
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
As, Vs

3. 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 saddle
Pall Ring
Rings (Raschig,etc)

4. 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)

5. 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
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

6. 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

7. Fixed bed Since we know Vp, Np, e, 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)

8. 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

9. Fixed bed - pressure drop calculation (Laminar flow)
To calculate the pressure drop, use Force balance
Resistance : due to Shear
Find Contact Area
Find shear stress
Until now, we haven’t said anything about laminar flow. So the above equations are valid for both laminar and turbulent flows

10. Fixed bed - pressure drop calculation (Laminar Flow)
Find contact area
To calculate the shear stress, FOR LAMINAR FLOW
Here V refers to velocity for flow in a tube
However, flow is through bed, NOT a simple tube

11. 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)
Hydraulic diameter

12. Fixed bed - pressure drop calculation (Laminar Flow)
Vavg is average velocity of fluid “in the bed”, between particles
Normally, volumetric flow rate is easier to find

13. Fixed bed - pressure drop calculation (Laminar Flow)
Can we relate volumetric flow rate to Vavg ?
Use a new term “Superficial velocity” (V0)
I.e. Velocity in an ‘empty’ column, that will provide the same volumetric flow rate
Can we relate average velocity and superficial velocity?

14. Fixed bed - pressure drop calculation (Laminar Flow)
Force balance: Substitute for t etc.
Volume of reactor (say, height of bed = L)

15. Fixed bed - pressure drop calculation (Laminar Flow)
Specific surface area vs “average diameter”
Define “average Dia” of particle as
Some books (BSL) use Dp

16. Fixed bed - pressure drop calculation (Laminar Flow)
However, using hydraulic radius etc are only approximations
Experimental data shows, we need to multiply the pressure requirement by ~ 2 (exactly 100/48)
In terms of specific surface area
In terms of average particle diameter

17. Fixed bed - pressure drop calculation (Turbulent Flow)
Pressure drop and shear stress equations
Only the expression for shear stress changes f Re
For high turbulence (high Re),
However

18. Fixed bed - pressure drop calculation (Turbulent Flow)
We have already developed an expression for contact area
Hence, force balance
Volume of reactor (say, height of bed = L)

19. Fixed bed - pressure drop calculation (Turbulent Flow)
In terms of average particle diameter
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!
Ergun Equation for packed bed

20. Fixed bed - pressure drop calculation (Laminar OR Turbulent Flow)
Ergun Equation for packed bed
If velocity is very low, turbulent part of pressure drop is negligible
If velocity is very high, laminar part is negligible
Some texts provide equation for friction factor

21. Fixed bed - pressure drop calculation (Laminar OR Turbulent Flow)
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...

22. Fixed bed - pressure drop calculation (Laminar OR Turbulent Flow)
Multiply by 3 on both sides (why?)
Packed bed friction factor = 3 f
Eqn in McCabe and Smith
Reynolds number for packed bed

23. 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 = Pa-s (assumed)
What is the pressure drop needed to operate this column?

24. Fixed Bed e 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
Ergun Eqn commonly used, however, other empirical correlations are also used
e.g. Chilton Colburn eqn
0.8 e
0.4
0.2
Edge
Center
Edge

25. Fixed Bed
Sphericity vs Void Fraction 1 f
~0.4 1 e

26. Fixed Bed
Alternate method to arrive at Ergun equation (or similar correlations)
Use Dimensional analysis

27. 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

28. Fluidized bed: Operation
Empirical correlation for porosity
Types of fluidization: Aggregate fluidization vs Particulate fluidization
Larger particles, large density difference (rSOLID - rFLUID) ==> Aggregate fluidization (slugging, bubbles, etc)
==> Typically gas fluidization
Even with liquids, lead particles tend to undergo aggregate fluidization
Archimedes number

29. 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

30. Fluidized bed: Operation80
Range of operation depends on Ar 40 R
100
104
108 Ar

31. Fluidized bed: Operation
Criteria for aggregate fluidization
Semi empirical
Particulate fluidization
Typically for low Ar numbers
More homogenous mixture