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.
4 Response to Superficial Flow Low VelocityFluid does not impart enough drag to overcome gravity and particles do not move. Fixed Bed.High VelocityAt high enough velocities fluid drag plus buoyancy overcomes the gravity force and the bed expands. Fluidized Bed.p for Increasing u0Until onset of fluidization p increases, then becomes constant.Bed Length for Increasing u0L is constant until onset of fluidization and then begins to increase.
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 channelsBed 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 FlowFor a unit length of bed:MassBalance
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 FlowIn 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 FlowOne cannot use the Hagen-Poiseuille approximation when flow is turbulent. After substituting in Dh and velocity correctionExperimentally:Burke-Plummer Equation
14 Intermediate Flow Ergun Equation Note: equation can be used with gases using average gas density between inlet and outlet.
16 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 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: CubeWhat is diameter of sphere of volume a3?
18 SphericityNote 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 Sizes6 X 8 Mesh dp = ( ) / 2 = in ( ft)
20 Irregular ShapesSo the final Ergun equation is:
23 FluidizationAt 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 fractionLBedSTube
28 ExampleA packed bed is composed of cubes 0.02 m on a side. The bulk densityof the packed bed, with air, is 980 kg/m3. The density of the solid cubes is1500 kg/m3.Calculate the void fraction (e) of the bed.Calculate the effective diameter (Dp) where Dp is the diameter of a spherehaving the equivalent volume.Determine the sphericity of the cubes.Estimate the water flow rate (m3/sec) required for minimum fluidization of thesolid using water at 38 C and a tower diameter of 1.0 m.