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). When particles in the bed become suspended on
the upward following gas the system becomes known as a fluidized bed. The point of fluidization
is where the particles and the gas find a point of equilibrium. 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.
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).
(
(𝑃0 −𝑃𝐿 )∗𝜌
𝐺02
𝐷𝑝
𝜀3
1−𝜀
) ∗ ( 𝐿 ) ∗ (1−𝜀) = 150 ∗ (𝐷𝑝 𝐺0 ) +
⁄𝜇
7
4
(1)
Where: P0 is the initial pressure [Pa]
PL is the pressure at the end of the column [Pa]
G0 is the mass flux [kg/m2*s]
Dp is particle diameter [m]
L is the height of the bed [m]
ε is the void fraction [dimensionless]
ρ is the density of the particle [kg/m3]
μ is the 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.
𝑈𝑝𝑤𝑎𝑟𝑑 𝑓𝑜𝑟𝑐𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑏𝑒𝑑 = ∆𝑃 ∗ 𝐴
(2)
Where: ΔP is the pressure drop across the bed [Pa]
A is the cross-sectional area of the bed [m2]
Next the volume of the particles can be solved for using equation 3 below.
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 = (1 − 𝜀) ∗ 𝐴 ∗ 𝐿
(3)
Where: ε is the void fraction [dimensionless]
A is the cross-sectional area of the bed [m2]
L is the height of the bed [m]
From here we can determine the net weight of the particles in the column by using equation 4
below.
𝑁𝑒𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 = (1 − 𝜀) ∗ (𝜌𝑝 − 𝜌𝑓 ) ∗ 𝐴 ∗ 𝐿 ∗ 𝑔
Where: A is the cross-sectional area of the bed [m2]
L is the height of the bed [m]
ε is the void fraction [dimensionless]
𝜌𝑝 is the density of the particle [kg/m3]
𝜌𝑓 is the density of the fluid (or gas) [kg/m3]
g is the acceleration due to gravity [m/s2]
The theoretical pressure drop can be determined by using equation 5 below.
(4)
∆𝑃 = (1 − 𝜀) ∗ (𝜌𝑝 − 𝜌𝑓 ) ∗ 𝐿 ∗ 𝑔
(5)
Where: L is the height of the bed [m]
ε is the void fraction [dimensionless]
𝜌𝑝 is the 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.
𝑉𝑓 =
(𝜌𝑝 − 𝜌𝑓 )∗𝑔∗𝐷𝑝2
150∗𝜇
𝜀3
∗ 1−𝜀
(6)
Where: Vf is the fluid velocity [m/s]
ε is the void fraction [dimensionless]
𝜌𝑝 is the density of the particle [kg/m3]
𝜌𝑓 is the density of the fluid (or gas) [kg/m3]
g is the acceleration due to gravity [m/s2]
Dp is particle diameter [m]
μ is the 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.
𝑉𝑆𝑒𝑡𝑡𝑙𝑖𝑛𝑔 =
(𝜌𝑝 − 𝜌𝑓 )∗𝑔∗𝐷𝑝2
18∗𝜇
Where: Vsettling is the settling velocity [m/s]
𝜌𝑝 is the density of the particle [kg/m3]
𝜌𝑓 is the density of the fluid (or gas) [kg/m3]
g is the acceleration due to gravity [m/s2]
Dp is particle diameter [m]
(7)
μ is the 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.
𝑉𝑠𝑒𝑡𝑡𝑙𝑖𝑛𝑔
𝑉𝑓
=
25
3
∗
1− 𝜀
𝜀3
(8)
Where: Vf is the superficial velocity [m/s]
Vsettling is the settling velocity [m/s]
ε is the 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.