Fixed bed and fluidized bed Ref: BSL, McCabe & Smith Why fixed (or fluidized) bed? Expensive Catalyst enzyme (immobilized) Large Surface area Used in reaction/adsorption/ elution (for example) Goal: Expression for pressure drop, try some examples

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)

Fixed bed What are important parameters? (For example, for adsorption of a protein from a broth) rate of adsorption (faster is better) saturation concentration (more is better) From the product requirement (eg X kg per day), density and product concentration in broth ==> volumetric flow rate

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

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)

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)

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

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

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)

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

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

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

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

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

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?

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

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

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

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

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)

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

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

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...

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

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 = 0.002 Pa-s (assumed) What is the pressure drop needed to operate this column?

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

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

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

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

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

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

Fluidized bed: Operation 80 Range of operation depends on Ar 40 R 100 104 108 Ar

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