University of Groningen
Stochastic models for transport in a fluidized bed
Dehling, H.G; Hoffmann, A.C; Stuut, H.W.
Published in:
Siam Journal on Applied Mathematics
DOI:
10.1137/S0036139996306316
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Publication date:
1999
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Citation for published version (APA):
Dehling, H. G., Hoffmann, A. C., & Stuut, H. W. (1999). Stochastic models for transport in a fluidized bed.
Siam Journal on Applied Mathematics
,
60
(1), 337 - 358. https://doi.org/10.1137/S0036139996306316
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STOCHASTIC MODELS FOR TRANSPORT IN A FLUIDIZED BED
∗
H. G. DEHLING
†
, A. C. HOFFMANN
‡
, AND H. W. STUUT
†
SIAM J. A
PPL. MAT H .
c
°
1999 Society for Industrial and Applied Mathematics
Vol. 60, No. 1, pp. 337–358
Abstract. In this paper we study stochastic models for the transport of particles in a fluidized
bed reactor and compute the associated residence time distribution (RTD). Our main model is
basically a diffusion process in [0,A] with reflecting/absorbing boundary conditions, modified by
allowing jumps to the origin as a result of transport of particles in the wake of rising fluidization
bubbles. We study discrete time birth-death Markov chains as approximations to our diffusion model.
For these we can compute the particle distribution inside the reactor as well as the RTD by simple
and fast matrix calculations. It turns out that discretization of the reactor into a moderate number
of segments already gives excellent numerical approximations to the continuous model. From the
forward equation for the particle distribution in the discrete model we obtain in the diffusion limit
a partial differential equation for the particle density p(t, x)
∂
∂t
p(t, x)=
1
2
∂
2
∂x
2
[D(x)p(t, x)] −
∂
∂x
[v(x)p(t, x)] − λ(x)p(t, x)
with boundary conditions p(t, 1) = 0 and
1
2
∂
∂x
[D(x)p(t, x)]
|x=0
− v(0)p(t, 0) −
Z
1
0
λ(x)p(t, x)dx =0.
Here v(x) and D(x) are the velocity and the diffusion coefficients and λ(x) gives the rate of jumps
to the origin.
We also study a model allowing a discrete probability of jumps to the origin from the distributor
plate, thus incorporating the experimentally observed fact that a fixed percentage of particles gets
caught in the wake of gas bubbles during their formation at the bottom of the reactor. It turns out
that this effect contributes to an extra term to the boundary condition at x =0.
Finally, we model the particle flow in the wakes of rising fluidization bubbles and derive λ(x)as
well as v(x). We compare our results with experimental data.
Key words. Markov processes, diffusion approximation, stochastic modeling, transport phe-
nomena, residence time distribution, chemical reactors, fluidized beds, partial differential equations
AMS subject classifications. Primary, 60J25; Secondary, 60J70, 92E20
PII. S0036139996306316
1. Introduction. In this paper we study the transport of particles in a certain
type of chemical reactor, with special emphasis on computing residence time distri-
butions. The reactor we will consider is a fluidized bed reactor. A fluidized bed is
obtained by forcing gas upward through a bed of powder. This is done through a
distributor plate permeable to the fluidizing gas but not to the particles. If the gas
velocity is high enough, the bed will be supported in the gas stream, the particles
entering a more or less floating state. The bed will then exhibit liquid-like behavior.
If the gas velocity is increased further, fluidization bubbles will start to form and rise
through the bed much as in a boiling liquid. A further feature of the bed considered
here is that it is “continuous,” meaning that particles are added to (and removed
from) the bed continuously. The process is shown schematically in Figure 1.
∗
Received by the editors July 8, 1996; accepted for publication (in revised form) December 22,
1998; published electronically December 21, 1999.
http://www.siam.org/journals/siap/60-1/30631.html
†
Department of Mathematics, University of Groningen, Blauwborgje 3, 9747 AC Groningen, The
Netherlands (dehling@math.rug.nl).
‡
Department of Chemical Engineering, University of Groningen, Nijenborgh 4, 9747 AG Gronin-
gen, The Netherlands (a.c.hoffmann@chem.rug.nl).
337
Downloaded 12/18/18 to 129.125.148.19. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
338 H. G. DEHLING, A. C. HOFFMANN, AND H. W. STUUT
Fig. 1. Fluidized bed reactor. Fluidizing air is introduced through the porous support plate.
The fluidization bubbles increase in size with height in the bed due to coalescence, as does the bubble
wake fraction and, therefore, the material transport in the wake phase.
A number of studies in the scientific literature are dedicated to predicting the
particle residence time distribution (RTD) in such continuous fluidized beds. In most
of this work attempts have been made to formulate semiempirical models using the
traditional tools of RTD theory. Klose and Herschel [10] and Pudel, Str¨umke, and
S¨undermann [15] used series of ideal mixers. They fitted the number of mixers to
match predictions to experimental results. Another modeling approach has been
based on plug flow with superimposed dispersion, where the dispersion coefficient was
adjusted to fit experimental data [13], [20], [18].
However, a discrepancy between the model predictions and the actual behavior of
continuous fluidized beds remained when using these simple approaches. Whittmann
et al. [22] tried series of ideal mixers with reverse flow, and Heertjes, de Nie, and
Verloop [7] tried a model incorporating combinations of ideal mixers in series and in
parallel. Krishnaiah, Pydisetty, and Varma [11] proposed a model with a combination
of a mixed section with a stagnant one and short circuiting.
A few studies have attempted to take the actual physical phenomena occurring in
a continuous fluidized bed into consideration. Berrutti, Liden, and Scott [1] proposed
a model based on a series of compartments in the fluidized bed each with gross solid
circulation. Morris, Gubbins, and Watkins [14] presented a model based on the pro-
posed existence of a velocity profile in the vertically moving fluidized solids, similar to
the profile in a viscous fluid flowing in a pipe. Haines, King, and Woodburn [5] used
the plug flow with axial dispersion approach, but they augmented their dispersion
coefficient (which they assumed was caused mostly by the random collisional move-
ment of individual particles) with a term accounting for the extra dispersion caused
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STOCHASTIC MODELS FOR TRANSPORT IN A FLUIDIZED BED 339
by rising fluidization bubbles.
Rowe and Partridge [17] first proposed physical processes governing the vertical
particle transport processes in batch fluidized beds. These are (see also Figure 1)
• transport upward in bubble wakes and deposition on top of the bed,
• transport down in the bulk due to the removal of material low in the bed in
bubble wakes (“circulation”), and
• dispersion due to the disturbance of the bulk material by fluidization bubbles.
Hoffmann and Paarhuis [8] showed by means of a computer simulation that these
processes could account also for the RTD of particles in continuous fluidized beds.
In this article a mathematical model based on these principles—and starting with
a stochastic model for the particle motion in the bed—is formulated, solved, and
compared with experimental results. We believe that stochastic models for the mo-
tion of an individual particle should be the basis for an analysis of the evolution of
the particle density over time. Such models are easily made—at least for a discrete
approximation—and have a stronger intuitive appeal than traditional macroscopic
models. The macroscopic description in terms of PDEs with boundary conditions
can, of course, be derived from the stochastic model.
Our mathematical analysis is based on a Markovian model for the motion of
individual particles through the reactor. Denoting the vertical distance of the particle
from the top of the reactor at time t by X
t
, we model the motion by a stochastic
process (X
t
). We first study a discrete model, obtained by dividing the reactor into
N horizontal cells of equal width and modeling the particle’s location at integer times
only. The cells are numbered from top to bottom, with an extra cell with index N +1
denoting the lower exit of the reactor. Particles that have entered state N + 1 cannot
return to the interior of the reactor. The dynamics of our process is described by a
Markov chain (X
n
)
n≥0
with state space {1, 2,...,N+1}and an absorbing boundary
at N + 1. Our Markov chain is basically a birth-death process, modified to allow
for instant jumps to the first cell—thus modeling the possibility that a particle gets
caught in the wake of a rising gas bubble (see Figure 2).
We show that this simple model gives rise to RTDs that capture the main features
of empirically observed RTDs in fluidized beds. We show that the long tails of the
RTD function are a consequence of the fact that the second-largest eigenvalue of our
Markov transition matrix is very close to 1.
In reality, the transport process occurs in continuous space and time, and thus
we should also study continuous models. We introduce such models as limits of
discrete Markov chains, obtained by letting space and time discretizations converge
to zero in an appropriate way. The limit process is an ordinary diffusion process with
reflecting/absorbing boundary conditions, modified to allow for instant jumps to the
origin. We derive a PDE for the particle density p(t, x) in the continuous model which
then also provides the RTD function via F (t)=1−
R
1
0
p(t, x)dx. We show numerically
that the particle density and the RTD function of the continuous model can be well
approximated by the corresponding functions in the discrete model, provided the
discretization is fine enough—for all practical purposes, N = 50 cells turned out to
be sufficient.
In the last section we take a model for the particle flow in the wakes of rising gas
bubbles as basis for modeling the parameters of the continuous jump-diffusion model.
For several parameter settings we compare the model RTD with experimental values.
It turns out that our theoretically obtained RTD functions capture the main features
of the experimental RTDs quite well.
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340 H. G. DEHLING, A. C. HOFFMANN, AND H. W. STUUT
6
?
f
6
λ
i
i
i +1
i−1
N+1
1
Fig. 2. Discrete Markov model for transport in a fluidized bed. The reactor is vertically
partitioned into N cells of equal width, with cell N +1 symbolizing the exterior of the reactor. The
possible transitions are that a particle can move one cell down, stay in the same cell, move one cell
up, or move to the top of the reactor.
2. Discrete Markov model. In this section we study discrete mathematical
models for particle transport in a fluidized bed. We discretize space by subdividing the
interior of the reactor into N horizontal cells of equal width, labeled i =1,2,...,N,
and identifying the lower exit of the reactor with the index i = N + 1. Time is
discretized by considering the particle position at integer times only. We denote the
index of the cell that the particle visits at time n by X
n
. We model the dynamics of
the process by assuming that (X
n
)
n≥0
forms a Markov chain. This Markov chain is
fully specified once we know
1. the probability vector p(0)=(p(0, 1),...,p(0,N + 1)) of the particle’s initial
position, where p(0,i)=P(X
0
=i), and
2. the transition matrix P =(p
ij
)
1≤i,j≤N +1
, where p
ij
gives the conditional
probability that the particle is in cell j at time n + 1, given it was in cell i at
time n, i.e., p
ij
= P (X
n+1
= j|X
n
= i).
Probabilities of arbitrary events concerning (X
n
)
n≥0
can then be computed. For
example, the probability that the particle starts at time n = 0 in cell i
0
, then visits cells
i
1
,...,i
n−1
, and finally at time n ends up in cell i
n
is given by p(0,i
0
)p
i
0
i
1
...p
i
n−1
i
n
.
Definition 1. We define the probability function of X
n
by
p(n, i)=P(X
n
=i).
The corresponding probability vector is denoted by p(n).
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