1. Fixed and Fluidized Beds

2. 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.

3. Flow Through a Bed of Particles

4. Response to Superficial Flow
Low Velocity
Fluid does not impart enough drag to overcome gravity and particles do not move. Fixed Bed.
High Velocity
At high enough velocities fluid drag plus buoyancy overcomes the gravity force and the bed expands. Fluidized Bed.
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.

5. Response to Superficial Velocities

6. Fixed Bed How do we calculate the pressure drop across a fixed bed?
Start with the MEB:
For pipe flow we determined:

7. 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).

8. Laminar Flow ? ?
What are the proper velocity and diameter?

9. Velocity For a unit length of bed: Lb S = Volume of Bed
e Lb S = Volume Available for Flow
For a unit length of bed:
Mass
Balance

10. Diameter Multiply by L/L
Since this is not true pipe flow must use hydraulic radius.
Multiply by L/L

11. Diameter 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.
For a sphere:

12. Laminar Flow
In actuality the above equation does not account for the tortuous path through the bed and DL is much longer. Experimental data show that a numerical constant of 150 should replace the 72.
Blake-Kozeny equation. Assumes e < 0.5 and Rep < 10.

13. Turbulent Flow
One cannot use the Hagen-Poiseuille approximation when flow is turbulent. After substituting in Dh and velocity correction
Experimentally:
Burke-Plummer Equation

14. Intermediate Flow Ergun Equation
Note: equation can be used with gases using average gas density between inlet and outlet.

15. Fixed Bed “Friction Factor”

16. Irregular Shapes
To 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 Fs which allows calculation of an equivalent diameter.
Where Dp is the diameter of a sphere of the same volume as the particle

17. Example: Cube
What is diameter of sphere of volume a3?

18. 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.

19. Adsorbent Mesh Sizes
6 X 8 Mesh dp = ( ) / 2 = in ( ft)

20. Irregular Shapes
So the final Ergun equation is:

21. Fluidization (Refinery Application)

22. Fluidization (Drug Application)

23. Fluidization
At fluidization, the gravity force on the particles in the bed must be balanced (Fk = 0) by the drag, buoyancy, and pressure forces.
Substituting the Ergun equation for the pressure drop.

24. Minimum Fluidization Velocity
This equation can be used to calculate the minimum fluidization velocity umf if the void fraction emf at incipient fluidization is known.

25. Void Fraction at Min. Fluidization
emf depends on the shape of the particles. For spherical particles emf is usually 0.4 – 0.45.

26. Minimum Fluidization What if emf (and maybe Fs) is unknown?
Wen and Yu found for many systems:

27. Bed Length at Minimum Fluidization
Once we obtain the minimum void fraction
LBed
STube

28. 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 is
1500 kg/m3.
Calculate the void fraction (e) 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.

31. LHS
RHS Term No. 1

32. RHS Term No. 2
Final Equation