A fluidized bed reactor, in principle, is not an ideal reactor. Hence, it is not easy to achieve ideal conditions corresponding to a plug flow reactor. If plug flow behavior is achieved, the moisture content of the particle can be easily predicted from knowl-edge of the residence time. However, plug flow is influenced by several parameters, such as particle properties, operating conditions, and bed geometry. Plug flow is characterized by a dimensionless parameter known as the axial dispersion coefficient (αH), which is the ratio of the product of particle diffusivity (D) and its residence time to the square of the bed height (H) (i.e., αH = Dtm/H2). This parameter is used in the equation that relates the change in concentration rate (¹C/¹t) of the particle to the axial concentration gradient (¹C/¹x):
(2.4) Taking θ = tm, ϕ = x/H, the dimensionless form of Equation 2.4 is
(2.5)
Equation 2.4 has to be modified by subtracting from the right-hand side the term
¹C/¹x if the particles have a net flow velocity u. For the case of plug flow, αH = 0, and for perfect mixing αH → ×. The value of αH is close to 0.1 for most plug flow fluid bed dryers. The mean standard deviation3 of the residence time distribution (i.e., σθ = σ/tm = Ð2αH) is Ð0.2. A deviation from plug flow is overcome by several perfectly mixed reactors connected in series. In deeper beds, where particle circu-lation cells may be formed, αH values will tend to increase, thereby indicating a departure from plug flow characteristics. In other words, shallow beds can approach the plug-flow-type reactor. It has been well documented in the literature that a fluidized bed of particles with magnetic properties can be operated without bubbles over a wide range of gas flow rates by the use of a magnetic field. The fluidlike behavior of bubbleless fluidization under the influence of an external magnetic field is often termed a magnetically stabilized fluidized bed (MSFB).14 These beds have less turbulence even at relatively high velocities, thereby allowing operation of the bed without particle attrition and elutriation. Furthermore, the MSFB can behave
Xm=
∫
0∞X t f t dt( ) ( )
like a plug flow reactor, as axial and radial dispersion can be suppressed. Geuzens and Thoenes15 reported that radial mixing in an MSFB is comparable to a packed bed and axial mixing is comparable to something intermediate between a packed and fluidized bed. Thus, an MSFB reactor has the advantages of a packed as well as a fluidized bed reactor. Literature on drying in this class of reactors is scant.
3. Models
a. Definition of Models
There are several models to assess the performance of a fluidized bed for a chemical reaction. The models and their assumptions as applied to a fluidized bed reactor are presented in Chapter 5. The models of a fluidized bed as applied to drying are different. In fact, the process of drying has usually been dealt with in terms of thermal balance or mass transport of moisture from a wet solid to a dry gas. For a well-fluidized bed, it is widely accepted that the fluidizing gas finds it way through the reactor by dividing itself in the form of bubbles and as interstitial gas through the solid particle-rich dense or emulsion phase. The bubble which rises along with the bed is assumed to have either a boundary layer due to a gas film or a cloud due to a mixture of dense gas–solid phase. A mathematical model for fluidized bed drying was proposed by Alebregtse.16 The model is based on hydro-dynamics and mass transfer for the powders which can be fluidized well. The model assumes three phases: (1) the solid phase which contains the moisture, (2) the dense phase which is a mixture of gas and solid (also known as the emulsion phase), and (3) the bubble phase (also known as the lean phase). Figure 2.5 illustrates the model definition usually considered in the mass transport of moisture in a three-phase model.
b. Mass (Moisture) Transport
The moisture from the wet solid particles diffuses to the surface, and the moisture thus emerged is found in the emulsion-phase gas of a fluidized bed dryer. During the diffusion of moisture from the solid, the internal diffusion resistance may be neglected in some situations; in others, it must be considered. During constant rate drying, the diffusional resistance for moisture transport can be neglected. Palanez17 proposed a three-phase model that neglects the heat and mass transport inside the particle as the limiting factor. However, he emphasized the need for a refined approach if any transport resistance for moisture within the particle is encountered.
Hoebink and Rietema18 proposed a three-phase model for drying of a particle that has internal diffusion as the limiting factor. Hence, particle drying is assumed to be a slow process. Once diffused to the surface, the moisture is assumed to have negligible resistance for transport into the emulsion gas. This is because of the large surface area of the solid particle available for transport of moisture to the emulsion gas, thereby bringing an equilibrium quickly. The moisture content of the emulsion gas then can be assumed to be transferred to the bubble-phase gas. During such a transfer, the resistance can be assumed to be offered either by the boundary layer
of gas or by a cloud of gas–solid mixture that exists around a rising bubble. In a model developed for continuous drying of solids, Verkooijen19 proposed a simplified approach for drying smaller particles (i.e., dp < 120 µm). The transport of moisture is assumed to take place through a boundary layer. The boundary theory of Chiba and Kobayashi20 can be used in this case. The following is a model proposed by Verkooijen19 for a fluidized bed where the bubbles are not surrounded by a cloud of gas–solid mixture.
c. Moisture at the Surface of the Particle (CsR)
For a wet solid particle, the moisture distribution inside the particle and the surface concentration can in principle be obtained from solution of the differential equation
(2.6)
Solution of Equation 2.6 for the boundary conditions at t = 0 (i.e., Cs = CsO for 0 <
r < Rp) and t > 0 (i.e., r = 0, ¹Cs/¹r = 0 and r = R, Cs = CsR) will result in:
Figure 2.5 Schematic diagram for model definition: (a) fluid bed with various phases and (b) mass transport in three-phase model.
∂
∂ ∂
∂
∂
∂ C
t D
r r r C r
s = ⎛ s
⎝⎜ ⎞
⎠⎟
⎡
⎣⎢ ⎤
⎦⎥
2 2
(2.7)
For a perfectly mixed bed, the average moisture concentration (CsM) that is the same as the exit concentration (Cse) can be obtained from the age distribution:
(2.8)
Using Equation 2.7 in Equation 2.8,
(2.9)
where αp = Dtm/Rp2.
d. Mass Balance Across a Gas Bubble For a bubble of radius Rb and volume Vb,
(2.10)
where a is the surface area available for mass transport. Since
(2.11) Equation 2.10, after integration for the boundary condition h = 0, Cb = Cgo, results in:
(2.12)
From Equation 2.12, one obtains:
(Cge – Cgo) = (1 – β) (CgR – Cgo) (2.13) e. Overall Mass Balance of a Fluid Bed Dryer
If Qs is the volumetric feed rate of solid and Qg is the volumetric flow rate of gas, moisture lost by solid = moisture gained by gas gives:
Qs (CsO – Cse) = Qg (Cge – Cgo) (2.14)
Using Equation 2.13 in Equation 2.14, Cse can be predicted.
f. Model Testing
If the equilibrium moisture content of the solid is Cseq, then the ratio (CsO – Cse)/(CsO – Cseq) is defined as the drying efficiency. The exponential term in Equation 2.12 is a function of the bubbling characteristics of a fluidized bed and has to be evaluated from basic hydrodynamic data. Verkooijen19 tested his model for drying in a fluidized bed by experimentally measuring the drying efficiency and comparing it with his model prediction. His experiments were conducted on a 30-cm-ID fluid-ized bed provided with a distributor with 197 holes 2 mm in diameter. Silica gel with an initial moisture content CsO = 88–145 g water per kilogram dry solid was dried using air at 50°C at velocities ranging from 5.5 to 11 cm/s. Bed height was 16 and 30 cm. His model prediction showed that for particles <120 µm, cloud resistance caused the drying efficiency to fall; for large particles, the drying effi-ciency predicted in the presence of cloud and by the hydrodynamic boundary layer model were the same. In other words, the model suggests that for particles of dp
<120 µm, mass transfer in the cloud phase must be considered. The model developed by Alebregtse16 for drying in a fluidized bed is based on mass balance for solid phase, dense phase, and bubble phase. Simple two-phase theory is assumed to be obeyed. According to his model and experimental findings on the constant drying rate of wet salt (dp = 40 µm) by air (Tinlet = 473 K), the following results were obtained:
1. The bed height above a certain limit has an inverse effect on the drying rate.
2. If the superficial gas velocity is maintained at low magnitudes, the drying gas at the exit can be well saturated.
3. The drying rate can be increased by distributing the gas uniformly and maintaining a small bubble size. Hence, a distributor with a low catchment area is recommended for this purpose.
4. The fluidized bed diameter has an insignificant effect at a high operating gas velocity.
g. Constraints
The above models do not take into account the volume increase of the gas inside the reactor due to the evaporation of moisture. Most basic data used for prediction of the drying rate in fluidized bed models were developed for dry particles that can be well fluidized. The fluidizing characteristics of wet particles are different from those of dry particles. Hence, the models should take this aspect into account. Furthermore, some of the data, such as sorption isotherm and diffu-sion coefficient for moisture inside the drying particle, are to be experimentally determined for use in model predictions. As previously mentioned, drying is a phenomenon that has complex heat and mass transport, and modeling this using a fluidized bed reactor increases the complexity further. Hence, there is room for further research on this tropic.