Introduction
In this fluidized bed processing scenario, molten steel is poured from a tundish into a cylindrical
chamber (diameter = 0.5 m). The molten steel flows through high velocity inert gas jets, creating small
droplets of molten metal. The droplets have a diameter of 1 mm and solidify during free fall in the
chamber. For this example, the chamber has a possibility of four different inert gas atmospheres
(Helium, Neon, Argon, and Krypton) at 27°C. However, in an attempt to save space, the droplets are only
allowed to cool to 1000°C, well into the Austenite phase field for any range of steel compositions, by the
time they reach the end of the chamber. It is assumed that the steel shot has cooled enough that it does
not deform when it hits the bottom of the chamber. At the end of the chamber the steel shot rolls onto
a conveyor belt which passes over a set of fans blowing room temperature air up through the bed of
steel shot, fluidizing the bed. This is where the rest of the cooling occurs for the steel shot. The conveyor
is 0.5 m wide, moves at 5 cm/sec, and is to have the shortest length possible to conserve space. Since
the conveyor moves at such a low speed, it is assumed that all the cooling is due to the fluidization of
the bed, not from the movement of the conveyor. Figure 1 gives a visual representation for the entire
process, which is expected to produce 1 million pounds of steel shot per day.
Fig. 1: A visual depiction of the entire fluidized bed processing scenario.
Approach and Calculations
The first step to solving this processing example was to determine the height of the chamber
required to cool the droplets from the tundish temperature (1727°C) to 1000°C. The height of the
chamber depended upon how much time is required to cool the shot and the velocity at which the shot
is falling. The shot is small enough that it will reach its terminal velocity very quickly, so the assumption
is made that throughout freefall the shot is traveling at terminal velocity. The terminal velocity of
spheres in a fluidized bed is given in Equation 1.
Eq. 1
where d is the sphere diameter, ρs and ρf are the densities of the sphere and fluid, respectively, g is the
acceleration due to gravity, and f is the friction factor. While the diameter, densities, and acceleration
due to gravity are all known, the friction factor (found in Table 1) depends on the Reynolds number (Re,
Equation 2), which gives an indication of how the fluid is flowing (i.e., laminar, semi-turbulent, or fully
turbulent). The fluid viscosity is denoted ηf.
Eq. 2
Table 1: Values for the friction factor, f, based on the Reynolds number
Friction Factor, f
24/Re
18.5/Re3/5
0.44
Reynolds Number, Re
Re < 1
1 < Re < 500
500 < Re < 20000
Since Re cannot be found without the viscosity, the assumption was made that Re would fall
into one of the three ranges listed in Table 1 and the appropriate friction factor was substituted into
Equation 1 as a function of the terminal velocity. The resulting equation looked as follows
Rearranging and combining terms allowed a solution for vt to be found. This value was
substituted into Equation 2 and if the resulting Re fell into the range that was assumed, the terminal
velocity found was correct. However, if Re fell outside of the assumed range, calculations had to be
done again assuming a different friction factor. This was done for the 4 different atmospheric gases in
the chamber.
The next step to this process was to determine the heat transfer coefficient for each inert gas.
The heat transfer coefficient, h, gives an indication, in this case, of how well heat transfers between the
solidifying drop and inert atmosphere and was found using the Nusselt number. The Nusselt number
(Nu) is a dimensionless number which relates the heat transfer coefficient, the sphere diameter, and the
thermal conductivity of the fluid (kf). This relationship can be found in Equation 3. The Nusselt number
also has a relationship between the Reynolds number and the Prandlt number (Pr), another
dimensionless number, which can be found in Equation 4. The Prandlt number is related to the fluid’s
viscosity, specific heat capacity at a constant pressure (Cpf), and thermal conductivity by Equation 5.
Eq. 3
Eq. 4
Eq. 5
Substitution of Equation 3 into Equation 4 and rearranging the variables yielded Equation 6, one
which gives h in terms of known variables.
Eq. 6
Heat transfer coefficients were found for each inert gas and the Biot number (Bi) for each gas
was found. The Biot number, another dimensionless number, indicates if there are any significant
temperature gradients across the solidifying solid. If the Biot number (found using Equation 7) is below
0.1, no significant temperature gradients are found across the solid, making the calculations for cooling
time relatively simple. If Bi > 0.1, the calculations for cooling can become pretty complicated.
Eq. 7
In Equation 6, L is the characteristic length associated with the solidifying solid and is found as
the ratio of volume/surface area and ks is the thermal conductivity of the solid. In the case of a sphere,
the characteristic length is L = d/6. Since each gas has a different heat transfer coefficient, each gas will
also have a different Biot number; therefore, Bi values were calculated for each atmospheric gas to
ensure that the following equations for heat removal were valid.
Determining the time to solidify and cool the steel droplets was the next step. The following
calculations were made using equations which are based on there being no significant temperature
gradients across the droplets (Bi >0.1). The droplets are at 1727°C (superheated) when they first fall
from the tundish. Before they can solidify the droplets must cool to the melting temperature for the
steel, which was given as 1515°C. Equation 8 gives the equation that was used in determining the time
required to remove the superheat from the molten droplets, where t is the time required to go from
temperature T0 to temperature T1 in a fluid at temperature Tf and Cps is the specific heat capacity at a
constant pressure for the solidifying material. All temperatures entered into the equation were in units
of degrees Kelvin.
Eq. 8
Even after the superheat is removed the molten drops will not solidify until the heat of fusion
(characteristic of the material) is removed. The time required to remove the heat of fusion was found
using equation 9. In the equation Vs and As are the volume and surface area of the solid, respectively,
ΔHs is the latent heat of fusion for the solid, and Tmelt is the melting temperature of the solidifying
material.
Eq. 9
Once the heat of fusion is removed from the droplets, they are finally solid and can continue
cooling. The cooling is now exactly the same as outlined in cooling from the tundish temperature to the
melting temperature and therefore Equation 8 was used again. The calculations performed for this
report were done assuming that Cps and ρs stayed constant between the liquid and solid phases. The
total cooling time required to reach 1000°C in the chamber is simply the sum of the three times
mentioned above: time to remove superheat, time to remove the heat of fusion, and time to cool from
the melting temperature to 1000°C.
Knowing the velocity of the falling droplets and the amount of time required for them to cool
allowed for the determination of the required chamber height. Velocity is a measure of distance/time
and multiplying it by a time gives a distance. The terminal velocity for each gas was multiplied by the
total cooling time for that gas and that returned the distance the droplets would fall in the given
atmosphere. This distance is the required height of the chamber.
It was known that this process was expected to produce one million pounds of steel shot per
day. In order to satisfy the principles of the conservation of energy, the volume flow rate entering the
chamber had to be the same as that exiting the chamber. Volume flow rates are calculated by
multiplying the velocity of an object by the cross section area that it passes through. The volume flow
rate at the top was the production rate (1,000,000 lbs/day = 5.261 kg/s) divided by the density of the
material
However, at the exit of the chamber, the steel shot will not form one solid mass of metal.
Instead there will be voids as can be seen in Figure 2. To determine the area of the voids the size of the
square, characterized by the side of length a, had to be determined. This was done using the
Pythagorean Theorem with the hypotenuse of the triangle being four times the radius of the shot.
Knowing a it was easy to get the area of the square. The area of the circles inside the square was also
easily determined and the area of the voids was found by subtracting the area of the circles from the
area of the square. The void area fraction, ω, was determined using Equation 10.
Eq. 10
Knowing that the cooling bed was 0.5 m wide, moved at 5 cm/sec, and that the volume flow
rate at the exit was the same as at the entrance, a bed thickness could be determined. Since the spheres
only occupied a fraction of 1 – ω of the bed, the cross sectional area which it traveled through had to be
multiplied by 1 – ω as seen in Equation 11.
Eq. 11
In Equation 10 vbed is the velocity of the conveyor belt, w is the bed width, and z is the bed
thickness. Inputting the known variables allowed for a solution for z to be determined. The bed
thickness does not factor into any of the remaining equations, but was determined as a sort of reality
check to make sure that the cooling bed would not be unreasonably thick.
Figure 2: Schematic diagram that was used in determining the void area
fraction of the shot on the conveyor belt.
The last hurdle to this processing example was to determine how long the cooling conveyor belt
needed to be. Cooling of the shot on the conveyor belt is done by fluidizing the bed of shot with room
temperature air. In order to fluidize the bed, the air velocity must be greater than or equal to the
minimum fluidization velocity for the shot. Minimum fluidization is outlined through Equations 12 and
13.
Eq. 12
Eq. 13
The Galileo number, Ga, was first determined because it relies only on physical constants.
Inserting the calculated Galileo number into Equation 12 gave the Reynolds number required for
minimum fluidization. The minimum fluidization velocity was then determined using Equation 2. Now
that the minimum velocity to fluidize the bed was determined, the terminal velocity for the shot had to
be determined because when the air velocity reaches the terminal velocity of the shot, the shot would
be elutriated, or blown out of the bed. The terminal velocity for the shot being fluidized by air was found
in the same fashion as the terminal velocity of the shot in the chamber, using Equation 1 and the correct
friction factor.
The velocity of the air has an effect on how fast the shot will cool in the fluidized bed, so
determining the heat transfer coefficient as a function of the air velocity was done next. Equation 6 was
used to determine the heat transfer coefficient, with the Reynolds number defined as a function of air
velocity, in turn defining the heat transfer coefficient as a function of air velocity. With the heat transfer
coefficient defined relative to the air velocity, the Biot number was defined as a function of the air
velocity as well. Plots of the heat transfer coefficient and the Biot number versus air velocity can be seen
in Figure 3.
500
0.002
400

0.0015


hair vair 300

Bi air vair
0.001
200
100
5 10
0
5
10
15
4
0
5
10
vair
vair
Figure 3: Plots of the heat transfer coefficient (left) and Biot number
(right) for cooling in the fluidized bed. Note: Bi > 0.1 for all velocities,
meaning there are never any significant temperature gradients across
the shot.
35
30


25
tcoolair vair
20
15
10
0
5
10
15
vair
Figure 4: A plot of how the cooling time in the fluidized bed is affected
by the air velocity. Note: Larger air velocities result in lower cooling
times.
15
Determining the time for the shot to cool in the fluidized bed was the last step before a
conveyor belt length could be determined. Using Equation 8 the time to cool the shot from 1000°C to
room temperature as a function of air velocity was found and a plot of the cooling time can be found in
Figure 4. The temperature of the air was assumed to be 25°C and the shot was assumed to cool to 27°C
because if the shot were to cool to 25°C the equation would be invalid as ln(0) is undefined. Similar to
finding the height required for the chamber, the length of the conveyor belt was found by multiplying
the conveyor velocity by the time required to cool the steel shot. The conveyor belt length plotted
against the velocity of the air can be found in Figure 5. For the complete set of calculations and notes
that accompany them, refer to Appendix A.
2
1.5


ConveyorLength vair
1
0.5
0
5
10
15
vair
Figure 5: Conveyor belt length plotted against air velocity. Note: Since
higher velocities lower the cooling times, it makes sense that the
conveyor belt length would decrease with increasing velocity.