Fluidized Bed Theory RK AG by MO8kc34A

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									Fluidized Bed Theory


       Fluidized beds are commonly used in chemical engineering process for multiple

purposes, such as catalytic reactions, solid-gas reactions, combustion of coal, roasting ores,

drying, and adsorption operations (McCabe 3). Even though fluidized beds also contain a

packing material to give good contact between the phases, they have several major differences

from packed beds. First, the packing is supported by the upflowing phases and behaves much

like a liquid (hence the name). The packing phase is in constant motion within the fluidized bed.

Such a unit is more complex than a simple fixed bed and has several advantages which can be

important in industrial applications: (1) the rapid mixing motion in the bed gives a high heat

transfer rate between the bed and the shell of the unit; thus, heat can be easily transferred to or

away from the bed, (2) the bed unit tends to be quite uniform in concentration when compared to

the non-mixed packed bed; this can be an advantage or a disadvantage for a given separation or

chemical reaction, and (3) the packing can flow out of the unit for separate treatment and back

into the unit. (McCabe 145)


       At low gas velocities, the pressure drop through the bed can be described by the Ergun

equation. However, as the flowrate increases a point is reached where the pressure drop becomes

constant and does not change with gas flowrate. This is defined as the point of fluidization. The

point of fluidization is where the particles and the gas find a point of equilibrium. When particles

in the bed become suspended on the upward following gas the system becomes known as a

fluidized bed. Fluidization can only be used with relatively small particles, <300 μm with gases

(Sinnott, Towler 667). In the case of this lab experiment sand and silica are used to test the affect

of particle size on point of fluidization and pressure drop. Visually, the bed does not appear like

a fixed bed but there is random motion of the particles within the bed; if a valve is opened on the
side of the unit the packing will flow out much like a liquid. Further increases in gas flowrate

result in violent mixing within the bed and the formation of large gas bubbles passing through

the bed. Upon decreasing the gas flowrate, the pressure drop versus gas flowrate does not exactly

follow the previous curve as seen in figure 1 below. There is a significant hysterisis effect due to

the frictional forces in the initial packed bed. The reason for this path dependence is due to a

change in the orientation of the packing particles as they become looser than before due to the air

flowing through the bed; this results in a lower friction factor and lower pressure drop.



       The behavior of the particles based on pressure drop with relation to upward superficial

velocity can be seen in figure 1 below.




                 Figure 1: Pressure Drop vs. Superficial Velocity (McCabe 3)


In this figure it can be seen as the pressure drop is increased the superficial velocity increases

somewhat exponentially until it reaches a plateau where the minimum fluidization point is.

Along the first section of the graph (area A) the pressure drop can be related to the velocity using

the Ergun Equation (equation 1 below). This equation can only be used in this section because of
the small pressure drop and velocity where it is considered a packed bed (Bird, Stewart,

Lightfoot 191).




                                                                                 (1)


Where:

P0 = initial pressure [Pa]

PL = pressure at the end of the column [Pa]

G0 = mass flux [kg/m2*s]

Dp = particle diameter [m]

L = height of the bed [m]

ε = void fraction [dimensionless]

ρ = density of the particle [kg/m3]

μ = viscosity [kg/s*m]




However as the velocity and pressure drop increase the Ergun Equation (1) cannot be used in the

determination of fluidized bed values. Figure 2 below also shows this relation but with the

overall bed height verses the superficial velocity in the column.
                    Figure 2: Bed Height vs. Superficial Velocity (McCabe 3)


Point C on both figures is the point of minimum fluidization velocity, Vf, and can be caluculated

by the following equations. (Note: The small frictional force exerted on the wall was ignored).

First the determination of the upward force by the gas on the bed can be calculated using

equation 2 below.


                                                                                        (2)

Where:

ΔP = pressure drop across the bed [Pa]

A = cross-sectional area of the bed [m2]




Next the volume of the particles can be solved for using equation 3 below.


                                                                                            (3)

Where:
ε = void fraction [dimensionless]

A = cross-sectional area of the bed [m2]

L = height of the bed [m]




From here we can determine the net weight of the particles in the column by using equation 4

below.


                                                                                          (4)


Where:

A = cross-sectional area of the bed [m2]

L = height of the bed [m]

ε = void fraction [dimensionless]

   = density of the particle [kg/m3]

  = density of the fluid (or gas) [kg/m3]

g = acceleration due to gravity [m/s2]




The theoretical pressure drop can be determined by using equation 5 below.


                                                                                          (5)

Where:

L = height of the bed [m]

ε = void fraction [dimensionless]
   = density of the particle [kg/m3]


Typically for a bed with small particles (Dp < 0.1 mm), the flow conditions are such that the

Reynolds number (Re) is relatively small (Re < 10) that the Kozeny-Carmen Equation can be

used to find the velocity of fluidization (Vf) (McCabe 5). This Kozeny-Carmen Equation can be

seen as equation 6 below.



                                                                                           (6)


Where:

Vf = fluid velocity [m/s]

ε = void fraction [dimensionless]

   = density of the particle [kg/m3]

   = density of the fluid (or gas) [kg/m3]

g = acceleration due to gravity [m/s2]

Dp = particle diameter [m]

μ = viscosity [kg/s*m]




When the superficial velocity (Vs) is equal to the fluidized velocity (Vf) this state is known as

incipient fluidization (McCabe 5). Next the settling velocity can be determined by restricting the

size of the particle to be small like before so that Stokes Law can be used to calculate this

velocity. This can be seen in equation 7 below.



                                                                                           (7)
Where:

Vsettling = settling velocity [m/s]

   = density of the particle [kg/m3]

   = density of the fluid (or gas) [kg/m3]

g = acceleration due to gravity [m/s2]

Dp = particle diameter [m]

μ = viscosity [kg/s*m]




Once the velocity of settling and the fluidization velocity are determined a ratio of the two can be

formed relating the void fraction back to both the velocities, which can be seen in equation 8

below.


                                                                                            (8)



Where:

Vf = superficial velocity [m/s]

Vsettling = settling velocity [m/s]

ε = void fraction [dimensionless]




In cases where the particles a very small it is likely that they may be carried out of the bed

system, therefore filters or cyclones must be emplaced to recover these particles at high

superficial velocities. Bubbling fluidization will occur in this experiment because this is strictly a

gas-fluidized bed which will be seen as large pockets of gas moving through the free particles.
References:


Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena. New York:

J. Wiley, 2007. Print.


Sinnott, Ray, and Gavin Towler. Chemical Engineering Design. Amsterdam: Elsevier, 2009.

Print.


W.E. McCabe, J.C. Smith, and P. Harriott 2001. Unit Operations of Chemical Engineering,

McGraw Hill, New York.

								
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