Chapter 12 - Professional Reference Shelf

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R12.3 Fluidized Bed Reactors ^*

"When a man blames others for his failures, it's a good idea to credit others with his successes."

--Howard W. Newton

The fluidized-bed reactor has the ability to process large volumes of fluid. For the catalytic cracking of petroleum naphthas to form gasoline blends, as an example, the virtues of the fluidized-bed reactor drove its competitors from the market.

Fluidization occurs when small solid particles are suspended in an upward-flowing stream of fluid, as shown in Figure CD12-2. The fluid velocity is sufficient to suspend the particles, but not large enough to carry them out of the vessel. The solid particles swirl around the bed rapidly, creating excellent mixing among them. The material fluidized is almost always a solid and the fluidizing medium is either a liquid or a gas. The characteristics and behavior of a fluidized bed are strongly dependent on both the solid and liquid or gas properties. Nearly all of the significant commercial applications of fluidized-bed technology concern gas-solid systems, so these are treated in this section. The material that follows is based on what is seemingly the best model of the fluidized-bed reactor developed thus far--the bubbling-bed model of Kunii and Levenspiel.

Figure CD12-2
From D. Kunii and O. Levenspiel, Fluidization Engineering (Melbourne, Fla.: Robert E. Krieger Publishing Co., 1969).
Reprinted with permission of the publishers.

CD12.3-A Descriptive Behavior of the Kunii-Levenspiel Bubbling Bed Model

We are going to use the Kunii-Levenspiel bubbling-bed model to describe reactions in fluidized beds. In this model the reactant gas enters the bottom of the bed and flows up the reactor in the form of bubbles. As the bubbles rise, mass transfer of the reactant gases takes place as they flow (diffuse) in and out of the bubble to contact the solid particles, where the reaction product is formed. The product then flows back into a bubble and finally exits the bed when the bubble reaches the top of the bed. The rate at which the reactants and products transfer in and out of the bubble affects the conversion, as does the time it takes for the bubble to pass through the bed. Consequently, we need to describe the velocity at which the bubbles move through the column and the rate of transport of gases in and out of the bubbles. To calculate these parameters we need to determine a number of fluid-mechanics parameters associated with the fluidization process. Specifically, to determine the velocity of the bubble through the bed we first need to calculate:

The algorithm

Porosity at minimum fluidization,
Minimum fluidization velocity,
Bubble size, d _b

To calculate the mass transport coefficient we first must calculate:

Porosity at minimum fluidization,
Minimum fluidization velocity,
Velocity of bubble rise, U _b
Bubble size, d_b

To determine the reaction-rate parameters in the bed we first need to calculate:

Fraction of the total bed occupied by bubbles,
Fraction of the bed consisting of wakes,
Volume of catalyst in the bubbles, clouds, and emulsion, _b , _c and _e ,

It is evident that before we begin to study fluidized-bed reactors we must obtain an understanding of the fluid mechanics of fluidization. In Section CD12.3.2 equations are developed to calculate all the fluid mechanic parameters (e.g., d_b, umf

) necessary to obtain the mass transfer and reaction parameters.In Section CD12.3.3, equations for the mass transfer parameters are developed. In Section CD12.3.4 the reaction-rate parameters are presented, and the mole balance equations are applied to the bed to predict conversion in Section CD12.3.5.

CD12.3-B Mechanics of Fluidized Beds

In this section we describe the regions of fluidization and calculate the minimum and maximum fluidization velocities. Next, the Kunii-Levenspiel bubbling-bed model is described in detail. ⁹ Finally, equations to calculate the fraction of the bed comprising bubbles, the bubble size, the velocity of bubble rise, and the fractional volume of bubbles, clouds, and wakes are derived.

CD12.3.2A Description of the Phenomena

We consider a vertical bed of solid particles supported by a porous or perforated distributor plate, as in Figure CD12-3.1a. The direction of gas flow is upward through this bed. There is a drag exerted on the solid particles by the flowing gas, and at low gas velocities the pressure drop resulting from this drag will follow the Ergun equation, Equation (4-22), just as for any other type of packed bed. When the gas velocity is increased to a certain value however, the total drag on the particles will equal the weight of the bed, and the particles will begin to lift and barely fluidize. If rho

_c is the density of the solid catalyst particles, A _c the cross-sectional area, h _s the height of the bed settled before the particles start to lift, h the height of the bed at any time, and

_s and

the corresponding porosities of the settled and expanded bed, respectively, then the mass of solids in the bed, W _s, is

(CD12-3.1)

Figure CD12-3.1
Various kinds of contacting of a batch of solids by fluid.
Adapted from D. Kunii and O. Levenspiel, Fluidization Engineering (Melbourne, Fla.: Robert E. Krieger Publishing Co., 1977).

(Note nomenclature change: In the text and lecture, = porosity, while in this section,

= porosity.) This relationship is a consequence of the fact that the mass of the bed occupied solely by the solid particles is the same no matter what the porosity of the bed. When the drag force exceeds the gravitational force, the particles begin to lift and the bed expands (i.e., the height increases), thus increasing the bed porosity, as described by Equation (CD12-3.1). This increase in bed porosity decreases the overall drag until it is again balanced by the total gravitational force exerted on the solid particles (Figure CD12-3.1b).

If the gas velocity is increased still further, expansion of the bed will continue to occur; the solid particles will become somewhat separated from each other and begin to jostle each other and move around in a restless manner. Increasing the velocity just a slight amount causes further instabilities and some of the gas starts bypassing the rest of the bed in the form of bubbles (Figure CD12-3.1c). These bubbles grow in size as they rise up the column. Coincidentally with this, the solids in the bed begin moving upward, downward, and around in a highly agitated fashion, appearing as a boiling frothing mixture. With part of the gas bubbling through the bed and the solids being moved around as though they were part of the fluid, the bed of particles is said to be fluidized. It is in a state of aggregative, nonparticulate, or bubbling fluidization.

A further increase in gas velocity will result in slug flow (Figure CD12-3.1d) and unstable chaotic operation of the bed. Finally, at extremely high velocities the particles are blown or transported out of the bed (Figure CD12-3.1e).

The range of velocities over which the Ergun equation applies can be fairly large. On the other hand, the difference between the velocity at which the bed starts to expand and the velocity at which the bubbles start to appear can be extremely small and sometimes nonexistent. This observation means that if one steadily increases the gas flow rate, the first evidence of bed expansion may be the appearance of gas bubbles in the bed and the movement of solids. At low gas velocities in the range of fluidization, the rising bubbles contain very few solid particles. The remainder of the bed has a much higher concentration of solids in it and is known as the emulsion phase of the fluidized bed. The bubbles are shown as the bubble phase. The cloud phase is an intermediate phase between the bubble and emulsion phases.

Once the drag exerted on the particles equals the net gravitational force exerted on the particles, that is,

(CD12-3.2)

the pressure drop will not increase with an increase in velocity beyond this point (see Figure CD12-3.2). From the point at which the bubbles begin to appear in the bed, the gas velocity can be increased steadily over quite an appreciable range without changing the pressure drop across the bed or flowing the particles out of the bed. The bubbles become more frequent and the bed more highly agitated as the gas velocity is increased (Figure CD12-3.1c), but the particles remain in the bed. This region is bubbling fluidization. Depending on the physical characteristics of the gas, the solid particles, the distributor plate, and internals (such as heat exchanger tubes) within the bed, the region of bubbling fluidization can extend over more than an order of magnitude of gas velocities (e.g., 4 to 50 cm/s in Figure CD12-3.2). In other situations, gas velocities in the region of bubbling fluidization may be limited; the point at which the solids begin to be carried out of the bed by the rising gas may be a factor of only three or four times the velocity at incipient fluidization.

Eventually, if the gas velocity is increased continuously, it will eventually become sufficiently rapid to carry the solid particles upward, out of the bed. When this begins to happen, the bubbling and agitation of the solids are still present, and this is known as the region of fast fluidization, and the bed is a fast-fluidized bed. At velocities beyond this region, the particles are well apart, and the particles are merely carried along with the gas stream. Under these conditions, the reactor is usually referred to as a straight-through transport reactor (STTR) (Figure CD12-3.1e).

The various regions described above display the behavior illustrated in Figure CD12-3.2. This figure presents the pressure drop across a bed of solid particles as a function of gas velocity. The region covered by the Ergun equation is the rising portion of the plot (section I: image 12eq68.gif

.The section of the figure where the pressure drop remains essentially constant over a wide range of velocities is the region of bubbling fluidization (section II: image 12eq69.gif

cm/s). Beyond this are the regions of fast fluidization and of purely entrained flow.

Figure CD12-3.2
From D. Kunii and O. Levenspiel, Fluidization Engineering
(Melbourne, Fla.: Robert E. Krieger Publishing Co., 1977). Reprinted with permission of the publishers.

CD12.3.2B Minimum Fluidization Velocity

Fluidization will be considered to begin at the gas velocity at which the weight of the solids gravitational force exerted on the particles equals the drag on the particles from the rising gas. The gravitational force is given by Equation (CD12-3.1) and the drag force by the Ergun equation. All parameters at the point where these two forces are equal will be characterized by the subscript mf, to denote that this is the value of a particular term when the bed is just beginning to become fluidized. The combination g(^_c-^_g) occurs very frequently, as in Equation (CD12-3.1), and this grouping is termed :

(CD12-3.2)

The Ergun equation, Equation (4-22), can be written in the form

(CD12-3.3)

where the shape factor of catalyst particle, sometimes called the sphericity.

At the point of minimum fluidization the weight of the bed just equals the pressure drop across the bed:

(CD12-3.4)

For (Re _p = _g d _p U/

) we can solve Equation (CD12-3.4) for the minimum fluidization velocity, to give

Calculate

(CD12-3.5)

Reynolds numbers below 10 represent the usual situation, in which fine particles are fluidized by a gas. Sometimes, higher values of the Reynolds number do exist at the point of incipient fluidization, and then the quadratic equation (CD12-3.4) must be used.

Two dimensionless parameters in these two equations for deserve comment. The first is , the sphericity, which is a measure of a particle's nonideality in both shape and roughness. It is calculated by visualizing a sphere whose volume is equal to the particle's and dividing the surface area of this sphere by the actually measured surface area of the particle. Since the volume of a spherical particle in

and its surface area is

Calculate

(CD12-3.6)

Measured values of this parameter range from 0.5 to 1, with 0.6 being a normal value for a typical granular solid.

The second parameter of special interest is the void fraction at the point of minimum fluidization, image 12eq11.gif

. It appears in many of the equations describing fluidized-bed characteristics. A correlation exists which apparently gives quite accurate predictions of measured values of image 12eq11.gif

(within 10%) when the particles in the fluidized bed are fairly small: ¹⁰

Calculate

image 12eq77.gif

(CD12-3.7)

Another correlation commonly used is that of Wen and Yu ¹¹ :

(CD12-3.8)

and/or

(CD12-3.9)

When the particles are large, the predicted image 12eq11.gif

can be much too small. If a value of image 12eq11.gif

below 0.40 is predicted, it should be considered suspect. Kunii and Levenspiel ¹² state that image 12eq11.gif

is an easily measurable value. However, if it is not convenient to do so, Equation (CD12-3.7) should suffice. Values of image 12eq11.gif

around 0.5 are typical. If the distribution of sizes of the particles covers too large a range, the equation will not apply because smaller particles can fill the interstices between larger particles. When a distribution of particle sizes exists, an equation for calculating the mean diameter is

(CD12-3.10)

where

is the fraction of particles with diameter d _pi.

CD12.3.2C Maximum Fluidization

If the gas velocity is increased to a sufficiently high value, however, the drag on an individual particle will surpass the gravitational force on the particle, and the particle will be entrained in a gas and carried out of the bed. The point at which the drag on an individual particle is about to exceed the gravitational force exerted on it is called the maximum fluidization velocity.

Maximum
velocity through
the bed u_t

When the upward velocity of the gas exceeds the free-fall terminal velocity of the particle, u _t , the particle will be carried upward with the gas stream. For fine particles, the Reynolds numbers will be small, and two relationships presented by Kunii and Levenspiel ¹³ are:

image 12eq82.gif

(CD12-3.11)

We now have the maximum and minimum superficial velocities at which we may operate the bed. The entering superficial velocity, u ₀ , must be above the minimum fluidization velocity but below the slugging, u_ms and terminal, u _t , velocities.

Both of these conditions must be satisfied for proper bed operation.

CD12.3.2D Descriptive Behavior of a Fluidized Bed: The Model of Kunii and Levenspiel

At gas flow rates above the point of minimum fluidization, a fluidized bed appears much like a vigorously boiling liquid; bubbles of gas rise rapidly and burst on the surface, and the emulsion phase is thoroughly agitated. The bubbles form very near the bottom of the bed, very close to the distributor plate, and as a result the design of the distributor plate has a significant effect on fluidized-bed characteristics.

Literally hundreds of investigators have contributed to what is now regarded as a fairly practical description of the behavior of a fluidized bed; chief among these is to be the work of Davidson. ¹⁴ Early investigators saw that the fluidized bed had to be treated as a two-phase system: an emulsion phase and a bubble phase (often called the dense and lean phases). The bubbles contain very small amounts of solids. They are not spherical; rather, they have an approximately hemispherical top and a pushed-in bottom. Each bubble of gas has a wake which contains a significant amount of solids. These characteristics are illustrated in Figure CD12-5, which were obtained from x-rays of the wake and emulsion, the darkened portion being the bubble phase.

As the bubble rises, it pulls up the wake with its solids behind it. The net flow of the solids in the emulsion phase must therefore be downward. The gas within a particular bubble remains largely within that bubble, penetrating only a short distance into the surrounding emulsion phase. The region penetrated by gas from a rising bubble is called the cloud.

Davidson found that he could relate the velocity of bubble rise and the cloud thickness to the size of bubble. Kunii and Levenspiel ¹⁵ combined these observations with some simplifying assumptions to provide a practical, usable model of fluidized-bed behavior. Their assumptions are presented in Table CD12-3.1. Several of these assumptions had been used by earlier investigators, particularly Davidson and Harrison. ¹⁶ With the possible exception of assumption 3, all of these assumptions are of questionable validity, and rather obvious deviations from them are observed routinely. Nevertheless, the deviations apparently do not affect the mechanical or reaction behavior of fluidized beds sufficiently to diminish their usefulness.

Figure CD12-3.3
Schematic of bubble, cloud, and wake.

R12.3-13E Bubble Velocity and Cloud Size

From experiments with single bubbles, Davidson found that the velocity of rise of a single bubble could be related to the bubble size by

Single bubble

(CD12-3.13)

When many bubbles are present, this velocity would be affected by other factors. The more bubbles that are present, the less drag there would be on an individual bubble; the bubbles would carry each other up through the bed. The greater number of bubbles would result from larger amounts of gas passing through the bed (i.e., a larger value of u ₀ ). Therefore, the larger the value of u ₀ , the faster should be the velocity of a gas bubble as it rises through the bed.

Other factors that should affect this term are the viscosity of the gas and the size and density of the solid particles that make up the bed. Both of these terms also affect the minimum fluidization velocity, so this term might well appear in any relationship for the velocity of bubble rise; the higher the minimum fluidization velocity, the lower the velocity of the rising bubble.

Adopting an expression used in gas-liquid systems, Davidson proposed that the rate of bubble rise in a fluidized bed could be represented simply by adding and subtracting these terms:

Velocity of bubble
rise u _b

(CD12-3.14)

Bubble Size. The equations for the velocity of bubble rise, Equations (CD12-3.13) and (CD12-3.14), are functions of the bubble diameter, an elusive value to obtain. As might be expected, it has been found to depend on such factors as bed diameter, height above the distributor plate, gas velocity, and the components that affect the fluidization characteristics of the particles. Unfortunately, for predictability, the bubble diameter also depends significantly on the type and number of baffles, heat exchangers tubes, and so on, within the fluidized bed (sometimes called "internals"). The design of the distributor plate, which disperses the inlet gas over the bottom of the bed, can also has a pronounced effect on the bubble diameter.

Studies of bubble diameter carried out thus far have concentrated on fluidized beds with no internals and have involved rather small beds. Under these conditions the bubbles grow as they rise through the bed. The best relationship between bubble diameter and height in the column at this writing seems to be that of Mori and Wen, ¹⁷ who correlated the data of studies covering bed diameters of 7 to 130 cm, minimum fluidization velocities of 0.5 to 20 cm /s, and solid particle sizes of 0.006 to 0.045 cm. Their principal equation was

d _b

(CD12-3.15)

In this equation, d _b is the bubble diameter in a bed of diameter D _t observed at a height h above the distributor plate, d _b0 is the diameter of the bubble formed initially just above the distributor plate, and d _bm is the maximum bubble diameter attained if all the bubbles in any horizontal plane coalesce to form a single bubble (as they will do if the bed is high enough).

The maximum bubble diameter, d _bm , has been observed to follow the relationship

d _maximum

(CD12-3.16)

for all beds, while the initial bubble diameter depends on the type of distributor plate. For porous plates the relationship

(CD12-3.17)

is observed, and for perforated plates the relationship

d _minimum

(CD12-3.18)

appears to be valid, in which n _d is the number of perforations. For beds with diameters between 30 and 130 cm, these relations appear to predict bubble diameters with an accuracy of about 50%; for beds with diameters between 7 and 30 cm, the accuracy of prediction appears to be approximately +100%, -60% of the observed values.
Werther developed the following correlation based on a statistical coalescence model: ¹⁸

image 12eq91.gif

(CD12-3.19)

The bubble size predicted by this model is close to that predicted by Mori and Wen ¹⁹ for large-diameter beds (2 m) and smaller than that suggested by Mori and Wen for small-diameter beds (0.1 m).

CD12.2.2F Fraction of Bed in the Bubble Phase

Using the Kunii-Levenspiel model, the fraction of the bed occupied by the bubbles and wakes can be estimated by material balances on the solid particles and the gas flows. The parameter is the fraction of the total bed occupied by the part of the bubbles that does not include the wake, and

is the volume of wake per volume of bubble. The bed fraction in the wakes is therefore

. The bed fraction in the emulsion phase (which includes the clouds) is (1 - -

). Letting A _c and _c represent the cross-sectional area of the bed and the density of the solid particles, respectively, a material balance on the solids (Figure CD12-3.4) gives

Figure CD12-3.4
Wake angle w and wake fraction of three-dimensional bubbles at ambient conditions; evaluated from x-ray photographs by Rowe and Partridge.
Adapted from D. Kuknii and O. Levenspiel,
Fluidization Engineering, 2nd. Ed., (Stoneham, Mass.; Butterworth-Heinemann 1991).

Velocity of solids u _s

(CD12-3.20)

A material balance on the gas flows gives

image 12eq94.gif

(CD12-3.21)

The velocity of rise of gas in the emulsion phase is

Velocity of
gas in
emulsion u _e

(CD12-3.22)

(In the fluidization literature, u _s is almost always taken as positive in the downward direction.) Factoring the cross-sectional area from Equation (CD12-3.21) and then combining Equations (CD12-3.21) and (CD12-3.22), we obtain an expression for the fraction image greekd.gif

of the bed occupied by bubbles:

Volume fraction
bubbles image greekd.gif

CD12-3.23)

The wake parameter, image greeka.gif

, is a function of the particle size in Figure CD12-3.4. The value of a has been observed experimentally to vary between 0.25 and 1.0, with typical values close to 0.4. Kunii and Levenspiel assume that Equation (CD12-46) can be simplified to

(CD12-3.24)

which is valid for (e.g., image 12eq98.gif

Example CD12-3 Maximum Solids Holdup

^*This material was developed from notes by Dr. Lee F. Brown and H. Scott Fogler.
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Professional Reference Shelf

R12.3 Fluidized Bed Reactors*

CD12.3-A Descriptive Behavior of the Kunii-Levenspiel Bubbling Bed Model

CD12.3-B Mechanics of Fluidized Beds

R12.3 Fluidized Bed Reactors ^*