FLOW THROUGH GRANULAR BEDS AND PACKED COLUMN Lecturer

FLOW THROUGH GRANULAR BEDS AND PACKED COLUMN Lecturer: Dr Hairul Nazirah bt Abdul Halim

Introduction � In most of technical processes, liquids or gases flow through beds of solid particles. � Example: i) A single fluid flow through a bed of granular solid such as ionexchange and catalytic reactors. ii) two phase countercurrent flow of liquid and gas through packed columns.

Flow of single fluid through a granular bed Single fluid flow through a granular bed or porous medium involves in; � fixed bed reactor � filtration � adsorption

Specific surface and Voidage � The general structure of a bed of particles can often be characterized by: i) the fractional voidage of the bed ii) the specific surface area of the bed � Voidage/porosity (ε) -The fraction of the volume of the bed not occupied by solid material. It is dimensionless and given by;

� Specific surface area of the particles (av) - The surface area of a particle divided by its volume. Sp : surface area of a particle in m 2 υp : volume of particle in m 3 For a spherical particle, Dp : diameter in m

� The where: � For volume fraction of particles in the bed: a = the ratio of total surface area in the bed to total volume of bed in m-1. packed bed, Reynolds number for a packed bed can be defined as follows:

Kozeny-Carman equation Pressure drop At a steady state, and negligible gravity effect, The pressure drop is given by; change to Eq. 7. 15 Eq. 7. 16

Kozeny-Carman equation � However, the experiments give an empirical constant of 150 for 72λ 1, which gives the Kozeny-Carman equation for laminar flow. � For flow through beds at particle Reynolds number up to about 1. 0: Eq. 7. 17 � Flow is proportional to the pressure drop and inversely proportional to the fluid velocity.

� At high Reynolds number (Rep > 1000), the Burke. Plummer equation is applied: Eq. 7. 20

� Ergun equation: Eq. 7. 22 fitted data for spheres, cylinders, and crushed solids over a wide range of flow rates (for low, intermediate and high Reynolds numbers).

MOTION OF PARTICLES THROUGH FLUIDS

Mechanics of Particle Motion � For a rigid particle moving through a fluid, there are 3 forces acting on the body: - The external force (gravitational or centrifugal force) - The buoyant force (act parallel with the external force but in opposite direction) - The drag force (acts to oppose the motion and act parallel with the direction of movement but in the opposite direction. )

Equation for One-dimensional Motion of Particle through Fluid � Consider a particle of mass m moving through a fluid under the action of an external force Fe. � Let the velocity of the particle relative to the fluid be u � Let the buoyant force on the particle be Fb � and let the drag be FD � Then, the resultant force on the particle is:

Equation for One-dimensional Motion of Particle through Fluid � The acceleration of the particle is du/dt, and mass (m) is constant: � The external force (Fe ) is expressed as a product of the mass (m) and the acceleration (ae) of the particle from this force:

The buoyant force (Fb) The buoyant force is given by where is the density of the fluid. The drag force (FD) where CD is the drag coefficient, Ap is the projected area of the particle in the plane perpendicular to the flow direction.

� By substituting all the forces in the Eq. (1) � Motion from gravitational force

Terminal Velocity � In gravitational settling, g is constant ( 9. 81 m/s 2) � The drag coefficient (CD) always increases with velocity (u). � The acceleration (a) decreases with time and approaches zero. � The particle quickly reaches a constant velocity which is the maximum attainable under the circumstances. � This maximum settling velocity is called terminal velocity.

� For spherical particle of diameter Dp moving through the fluid, the terminal velocity is given by � Substitution of m and Ap into the equation for ut gives the equation for gravity settling of spheres Frequently used

Drag Coefficient � Drag � The coefficient (CD) is a function of Reynolds number (NRE). drag curve applies only under restricted conditions: i). The particle must be a solid sphere; ii). The particle must be far from other particles and the vessel wall so that the flow pattern around the particle is not distorted; iii). It must be moving at its terminal velocity with respect to the fluid.

Figure 7. 7 Drag coefficients (CD) for spheres and irregular particles

Reynolds Number � Particle � For Thus, Reynolds Number u : velocity of fluid stream Dp : diameter of the particle : density of fluid : viscosity of fluid Re < 1 (Stokes Law applied - laminar flow)

� For 1000 < Re < 200 000 (Newton’s Law applied – turbulent flow) � Newton’s law applies to fairly large particles falling in gases or low viscosity fluids.

Criterion for settling regime � To identify the range in which the motion of the particle lies, � the velocity term is eliminated from the Reynolds number (Re) by substituting ut from Stokes’ law and Newton’s law. � Using Stoke’s Law; Eq. 7. 44

� To determine the settling regime, a convenient criterion K is introduced. Eq. 7. 45 � Substituting Eq. 7. 45 into Eq. 7. 44, Re = K 3/18. � Set Re = 1 and solving for K gives K=2. 6. � If K < 2. 6 then Stokes’ law applies.

� Using Newton’s Law; � Substitution � Thus, � Set Re for ut by criterion K, = 1000 and solving for K gives K = 68. 9. = 200, 000 and solving for K gives K = 2, 360. � Newton’s Law applies for 68. 9 < K < 2360.

THUS; � Stokes’ law range: K < 2. 6 � Newton’s law range: 68. 9 < K < 2, 360 � Intermediate range : when K > 2, 360 or 2. 6 < K < 68. 9, ut is found from; � ut is calculated using a value of CD found by trial from the Fig. 7. 7.

� In general case, the terminal velocity, can be found by trial and error after guessing Re to get an initial estimate of drag coefficient CD. � Normally for this case the particle diameter Dp is known Fig. 7. 7 Drag coefficients (CD) for spheres and irregular particles

Particle settling Free settling � When a particle is at sufficient distance from the wall of the container and from other particle, so that its fall is not affected by them, the process is called free settling. � Terminal velocity is also known as free settling velocity. Hindered settling � When the particles are crowded, they settle at a lower rate and the process is called hindered settling. � The particles will interfere with the motion of individual particles � The velocity gradient of each particle are affected by the close presence of other particles.

Hindered settling � The velocity for hindered setting can be computed by this equation: Correction factor Stokes Law � where, ε is volume fraction of the slurry mixture and Ψp is empirical correction factor. � Bulk density of mixture – � Empirical correction factor -

Hindered settling The Reynolds number is then based on the velocity relative to the fluid is Where the viscosity of the mixture µm is given by;

FLUIDIZATION

Fluidization � Fluidization is a process whereby a granular material is converted from a static solid-like state to a dynamic fluid-like state. � This process occurs when a fluid (liquid or gas) is passed up through the granular material. � The most common reason for fluidizing a bed is to obtain vigorous agitation of the solids in contact with the fluid, leading to an enhanced transport mechanism (diffusion, convection, and mass/energy transfer). See this video: https: //www. youtube. com/watch? v=l. Fhrp. SJZzck

Applications of Fluidization

Fluidized Bed Reactor

�When a gas flow is introduced through the bottom of a bed of solid particles, it will move upwards through the bed via the empty spaces between the particles. �At low gas velocities, aerodynamic drag on each particle is also low, and thus the bed remains in a fixed state.

� Increasing the velocity, the aerodynamic drag forces will begin to counteract the gravitational forces, causing the bed to expand in volume as the particles move away from each other � Further increasing the velocity, it will reach a critical value at which the upward drag forces will exactly equal the downward gravitational forces, causing the particles to become suspended within the fluid. At this critical value, the bed is said to be fluidized and will exhibit fluidic behavior. � See this video: https: //www. youtube. com/watch? v=1 HXcq 54 Nu. NM

� By further increasing gas velocity, the bulk density of the bed will continue to decrease, and its fluidization becomes more violent, until the particles no longer form a bed and are “conveyed” upwards by the gas flow. � When fluidized, a bed of solid particles will behave as a fluid, like a liquid or gas. � Objects with a lower density than the bed density will float on its surface, bobbing up and down if pushed downwards, while objects with a higher density sink to the bottom of the bed � The fluidic behavior allows the particles to be transported like a fluid, channeled through pipes, not requiring mechanical transport

Conditions for Fluidization

Based on the Figure: � If the superficial velocity, VO is gradually increased, the pressure drop will increases, but the particles do not move and the height (L) remains the same. � At a certain velocity, the pressure drop across the bed counterbalances the forces of gravity on the particles or the weight of the bed � At point A = Any further increase in velocity, causes the particles to move � At point B = Further increase in velocity, the particles become separate enough to move about in the bed and true fluidization begins.

� From point B to point C = Once bed is fluidized, the pressure drop across the bed stays constant, but the bed heights continues to increase with increasing velocity. � From point C to B = If the velocity is gradually reduced, the pressure drop remains constant and the bed height decreases. *The pressure drop required for the liquid or the gas to flow through the column at a specific flow rate

Minimum Fluidization Velocity � Minimum velocity of fluidization took place at incipient (beginning) fluidization. � During this stage, the ratio of pressure drop to the vessel height (L) is given by; where is the minimum porosity

� The minimum fluidization velocity by this equation; � For can be obtained roughly spherical particles, is generally between 0. 4 and 0. 45 (commonly taken as 0. 45) which increasing slightly with decreasing particle diameter.

� If the Reynolds number is used, then,