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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

Vp

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

- 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

Tarus saddle

Pall Ring

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

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)

- Pressure drop

- Specific surface area vs “average diameter”

- Define “average Dia” of particle as

- Some books (BSL) use Dp

In terms of specific surface area

In terms of average particle diameter

Fixed bed - pressure drop calculation (Laminar Flow)- Pressure drop

- However, using hydraulic radius etc are only approximations
- Experimental data shows, we need to multiply the pressure requirement by ~ 2 (exactly 100/48)

Re

- However

- Pressure drop and shear stress equations

- Only the expression for shear stress changes

- For high turbulence (high Re),

- Volume of reactor (say, height of bed = L)

- We have already developed an expression for contact area

In terms of average particle diameter

In terms of specific surface area

Ergun Equation for packed bed

Fixed bed - pressure drop calculation (Turbulent Flow)- 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!

Fixed bed - pressure drop calculation (Laminar OR Turbulent Flow)

- 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...

Reynolds number for packed bed

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

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

- 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

- 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

- Criteria for aggregate fluidization
- Semi empirical

- Particulate fluidization
- Typically for low Ar numbers
- More homogenous mixture

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