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Understanding Fixed and Fluidized Beds: Calculations, Equations, and Applications

Learn about forces acting on particle beds, how pressure drop varies with fluid velocity, applied equations for computations, and the advantages and disadvantages of fluidized beds. Discover the response to different superficial flow velocities in fixed and fluidized beds, calculation methods for pressure drop, and considerations for irregular particle shapes. Explore applications in refineries and drug production, as well as the mathematics behind minimum fluidization velocity and void fraction determinations.

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Understanding Fixed and Fluidized Beds: Calculations, Equations, and Applications

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

  10. Diameter 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 = (0.132 + 0.0937) / 2 = 0.113 in (0.0094 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.

  29. LHS RHS Term No. 1

  30. RHS Term No. 2 Final Equation

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