258
A.C. Hoffmann
Dept. of Chemical Engineering, University of Groningen, Nijenborgh
4, 9747 AG Groningen, The Netherlands.
H.G. Dehling
Dept. of Mathematics, University of Groningen, Blauwborgje 3,
9747 AC Groningen, The Netherlands.
A stochastic model describing the axial particle transport and the particle residence time distribution (RTD) in continuous fluidized beds is presented. This model has the advantage of clarity and intuitive appeal. Moreover, the method of solution of the model is far simpler and more powerful than for the traditional modelling approach based on conservation equations. After presentation of the model and the method of solution, predictions for the transient axial distribution of tracer particles in a fluidized bed are shown graphically. Predicted RTD curves are also shown graphically and compared with experimental data from the literature. Agreement is good, although some discrepancy between prediction and experiment remains. A discussion is provided.KEY WORDS: continuous fluidized bed; stochastic model; particle; residence time distribution.
The range of application of fluidized beds in the processing industry is, in spite of some difficulties with design and scale-up, increasing rapidly. At the same time there is a general trend towards process miniaturization and intensification. Continuously operated gas fluidized beds with a good gas-solid contacting, that is with only a small fraction of the gas bypassing the bed in fluidization bubbles, are therefore attractive.
In many continuous fluidized bed processes the particles are being treated in some way (e.g. fluidized bed dryers, catalyst regeneration, particle coating). In such processes, the quality and uniformity of the particulate product depends on the particle RTD in the bed.
A number of research articles are dedicated to the modelling of the particle RTD in continuous fluidized beds. In most of this work, the traditional tools of RTD theory have been invoked 1-8. However, it was found that these methods could not fully describe the experimental RTD data. In a few studies phenomenological models are formulated 9-11, although there is no agreement in these articles as to which phenomena actually govern the particle RTD.
Partridge and Rowe 12 first proposed the particle motion in batch fluidized bed to be governed by the following phenomena (see also Figure 1):
Hoffmann and Paarhuis 13 showed that these processes also could account for the particle RTD in continuous beds.
The model presented in this article is based on these transport processes. The mathematical formulation of the model is based on modelling the axial position of the particle by means of a Markov chain: the probability distribution of the axial position of the particle at a given time-step n depends only on its position at time-step n-1 and a set of transition probabilities. The transition probabilities are quantified in accordance with the particle transport processes, which, in turn, are quantified from empirical models in the literature.
Figure 1. Sketch of a bubbling fluidized bed
We divide the fluidized bed in a number of discrete cells as shown in Figure 2. The model calculates the probability distribution of the axial position of one particle as a function of time. Time is also discretized, and therefore denoted by n, the number of time steps since t=0. If the particle is in cell i at a given time-step, it is assigned probabilities for the following time-step of: i) remaining where it is, ii) moving forward to the next cell, iii) moving back to the previous cell or iv) being returned back to the origin. These probabilities are, respectively (ai+bi+di=1):
qi,i=ai(1-li); qi,i+1=bi(1-li); qi,i-1=di(1-li); qi,1=li (1)
At the boundaries we assume reflection at the origin (the surface of the bed) and absorption at cell N+1, which is under the distributor plate. Thus:
qi,1=1-bi(1-l1);
qi,2=b1(1-l1);
qN+1,N+1=1 (2)
Figure 2. The discretized fluidized bed
The transfer probabilities thus form a matrix, Q, of dimension (N+1,N+1) with the elements qi,j, and the probability distribution of the particle position at time t is a vector of dimension (N+1), p(t). For discretized time the position of the particle at the n'th time step is denoted by p(n), with the elements p(n,i). Knowing p(n-1), p(n) can be found recursively:
or in matrix notation: 
(3)
Iterating this, we obtain the formula for the probability distribution of the position of the particle at time n in terms of its initial probability distribution:
(4)
p(n) with p(0)= (1,0,......,0) represents the axial distribution of marked particles after n time steps, if they were added to the top of the bed at time t=0. The particle residence time is equal to the first exit time T, also a random variable, and the RTD of the particles is equal to the distribution of T. The RTD can be described as a cumulative function, F(n)=Prob.(T<=n), or by its probability function: E(n)=Prob.(T=n). The latter is a discrete 'exit age distribution'. F(n) it given by p(n,N+1).
Figure 3 shows two simulations of movements of a particle, evaluated with the model using constant parameters (i.e. not varying with position in the bed) in the transition matrix. In the left figure, the return probability is set to zero, which renders a process of convection with superimposed dispersion. In the right figure, the return probability is set to 0.01.
Figure 3. Simulations of the movement of a particle with constant parameters throughout the bed. In the left figure N=20, di=0.2, ai=0.5, bi=0.3 and li=0. In the right figure the parameters are the same, except the return probability, which is set to: li=0.01
The model described above is a discrete one. Although the actual process under consideration is also, in a sense, discrete (fluidization bubbles are discrete events), the model parameters will be related to the physics by describing the particle transport in terms of continuous processes. We denote the vertical position in the bed by x, and the convective axial velocity due to in-and outflow and circulation by v(x). The dispersion due to disturbance by fluidization bubbles we describe by a dispersion coefficient, D(x). The rate of returns to the origin is denoted by l(x).
Denoting the length of time-step in the model by e and the cell width by D, the parameters in the transition matrix are defined as follows to obtain a limit for e, D-> 0:
; 
; 
;
(5)
With these choices for the model parameters, the mean displacement per unit time becomes v(i D), while the mean square displacement becomes D(i D) under condition that the return probability, l i is zero. We let e depend on D via:
(6)
where Do is the maximum value of the diffusivity in the calculation domain. Equation (6) ensures that the calculated probabilities are positive for sufficiently small D. Inserting Equations (5) and (6) in (1) and (2) gives the transfer probability matrix.
If the particle starts at the origin, the RTD can be calculated from Equation (4) as the probability of it having reached the absorbing cell (N+1) as a function of time. Moreover, the distribution at a given time, t, of marked particles added to the top of the bed at t=0 can be found as the probability distribution of the position of the particle over the cells at timestep n=t/e. We expect the results of the discrete model to converge to those of the continuous process as e->0.
The residence time distribution in the continuous model depends crucially on the model parameters v, D and l. Figure 4 shows RTD curves for 10 different values of the parameters, obtained by all possible combinations of speed (v=0.01), diffusion constant (D1=0.01 and D2=0.001) and return rate (l 1=0, l 2=0.005, l 3=0.01, l 4=0.02, l 5=0.04).
The effects of variation of the parameters are clearly visible.
Increasing the diffusion coefficient results in a larger spread
of the residence time distribution. This effect is strongest in
the top rows with small distortion due to the return rate. The
variation in return rates manifests itself mostly in the tail
of the distribution. Already for small return rates the tail becomes
almost exponential, i.e. of the type F(t)=1-e-at.
Intuitively this shows that towards the end of the return time
the reactor becomes almost an ideal mixer. For larger values of
l the returns to the origin dominate
the entire picture, resulting in an exponential shape of the RTD
curve over the whole range.
Figure 4 RTD curves for the continuous model with different parameters:
v=0 in all cases, D=0.01 in the left column, and
D2=0.001 in the right one, l=0,
0.005, 0.01, 0.02 and 0.04 from top to bottom row.
It is possible to quantify the parameters v and D from empirical relationships in the literature. The return probability is due to formation of bubbles with wakes in the bottom of the bed on the one hand, and increase of the total flow in the wake phase with height (due to an increase in the volume of individual bubbles through coalescence and a consequent increase in the fraction of the bubble/wake sphere filled with wake material) on the other. The convective particle velocity is due to the in- and outflow and the circulation caused by the wake transport and therefore increases with increasing height in the bed. A full account of the quantification of the model parameters is given in 14. The essence of it is that using empirical formulae for:
(7)
(8)
(9)
the parameters v and D can be calculated approximately, leaving no adjustable parameters in the model. Werther 17 and others have shown that the two-phase theory generally overestimates the gas flow in the bubble phase. Values of QB were reduced in accordance with his work.
Figure 5 shows the probability distribution of the particle after 0.167 of Danckwert's residence times for two systems. The operational conditions are shown in Table 1.
In case A, the value of U-Umf, and therefore
the bubbling intensity, is relatively low compared to the rates
of in-and outflow. Consequently, the circulation in bubble wakes
and the dispersion due to disturbance of the bulk are low, with
marked particles added to the top of the bed travelling for some
time as a wave down the bed. This wave is still recognizable in
the figure, although asymmetrical due to the return of particles
to the surface, and only little marked particles have exited the
bed. In case B, the bubbling intensity is higher, resulting in
better axial mixing and a more marked particles having exited
the bed.
Table 1 Operational conditions for cases shown in Figures 4 and 5
Figure 5. Probability distribution of particle position after
0.167 Danckwert's residence times for cases A (black columns)
and B in Table 1.
Experimental grade-efficiency curves exist for these cases 18. In Figure 6, the RTD curves predicted by the model are compared with experiment. It can be seen that agreement is good, in spite of the fact that no adjustable parameters are incorporated.
Comparing in more detail the time at which the first wave of marked
particles arrive at the exit of the bed is clearly predicted well.
In case A, however, the experimental F-curve rises more sharply
than the model predicts. This sort of discrepancy was seen consistently
in the cases with low bubbling intensity. The explanation for
this may be found in the presence of dead or less active regions
in the bed at low bubbling intensities. La Rivière et al.
18 found that when the bubbling intensity was low,
the first reduced moments of their experimental RTD data were
below unity, indicating dead space in the bed. These are probably
located close to the distributor plate in between the discharge
orifices.
Figure 6. F-curves for cases A and B in Table 1. Points are experimental 18 and lines model predictions. Time is made dimensionless by dividing with Danckwert's mean residence time.
The agreement seen in Figure 6 shows that the particle transport processes upon which the model is based will explain experimental RTD data. On the other hand, the agreement is not enough in itself to demonstrate the correctness of the modelling assumptions. Both transport in bubble wakes without dispersion and dispersion without transport in wakes will go a long way in matching experimental curves and, as mentioned in the Introduction other workers have managed to match experimental RTD curves envisaging other transport processes. We stress, however, that the particle transport processes modelled here are in agreement with extensive research on the axial particle transport in batch fluidized beds, and that no adjustable parameters are included in our model.
The convection with axial dispersion and return probability gives rise to F-curves with a fairly steep initial rise. The model shows that 'intensity curves' result which in some cases exhibit a local maximum. This shape of intensity curve has been noticed in experimental intensity curves by other workers 19 as characteristic to continuous fluidized beds.
The stochastic modelling approach has proved to be a powerful and intuitively appealing modelling approach. Potentially it can provide information not only about RTD curves, but also about the transient particle behaviour in fluidized beds. Other possibilities are modelling of particles differing in size and density from the bulk bed particles and modelling of the particle transport in beds with internals and beds with in-and outflow via standpipes. Other systems, such as bubble columns, may also be described using this approach.
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A cross-sectional area of bed
D dispersion coefficient
Do maximal dispersion coefficient
DB diameter of bubble/wake sphere
fW fraction of bubble/wake sphere filled with wake material
g gravitational acceleration
h height in bed from the distributor plate
i,j indices denoting the number of cell
N number of cells internal to the discretized bed
n index denoting number of time step
p probability vector
p elements of p
Qin volumetric inflow of particles
QB volumetric flow of gas in the bubble phase
Q transition probability matrix
qi,i elements of Q
t time
T first exit time
U superficial velocity
Umf U at conditions of minimum fluidization
v convective particle velocity
x vertical coordinate with origin in the bed surface
Greek:
a,b,d,l parameters in the transition probabilities
e length of time interval
D cell width
qW angle subtended
from the centre of the fluidization bubble by the wake