EXPERIMENTAL AND NUMERICAL STUDY OF
HYDRODYNAMICS IN A CIRCULATING FLUIDIZED BED
SIRPA KALLIO+, JOHANNA AIRAKSINEN+, MATIAS GULDÉN*, ALF HERMANSON*,
JUHO PELTOLA‡, JOUNI RITVANEN#, MAIJU SEPPÄLÄ+, SRUJAL SHAH#,
VEIKKO TAIVASSALO+
+
VTT Technical Research Centre of Finland, P.O.Box 1000, FI-02044 VTT, Finland
Åbo Akademi University, Heat Engineering Laboratory, Piispankatu 8, FI-20500 Turku, Finland
‡
Tampere University of Technology, Department of Energy and Process Enginering, P.O.Box 527, FI33101 Tampere, Finland
#
Lappeenranta University of Technology, Department of Energy and Environmental Technology,
P.O.Box 20, FI-53851 Lappeenranta, Finland
*
sirpa.kallio@vtt.fi, +358 20 7224015
ABSTRACT
Fluid dynamics of circulating fluidized beds (CFB) can be computed with different
methods. Typically simulations are conducted as transient to fully describe the
complicated flow patterns of dense gas-solid suspensions. The computational mesh used
in these simulations must be reasonably fine, which in the case of large industrial
processes leads to unfeasibly long computations. To improve the modelling capabilities
it is necessary to develop faster simulation methods. The most attractive approach
seems to be time-averaged modelling facilitating steady-state simulation of a
fluidization process.
The model development and closure of the time-averaged equations requires transient
simulations. A prerequisite for utilisation of the transient results is that the simulations
are validated by measurements. For that purpose, a laboratory scale 2D CFB was built at
Åbo Akademi University and experiments were carried out. The apparatus and the
experiments are described in the present paper. The behaviour of the CFB was video
recorded and the videos were analysed to determine the average voidage distribution in
the CFB. The pressure distribution in the bed and the circulation rate of solids were also
measured. In addition, PIV measurement methods were tested to evaluate the
possibilities to analyse the velocities of individual particles and particle clusters.
Two experiments done in the laboratory scale 2D CFB were simulated as transient with
the Fluent software using the kinetic theory models of granular flow. The time-averaged
volume fractions and velocities of gas and particles were determined for the calculation
period in the studied cases. The simulation results were compared with measured flow
properties, observing a reasonably good agreement. The results were further analysed
with the aim to develop time-averaged CFD models. Since CFB simulations usually
require a coarse mesh, additional simulations were conducted to evaluate the mesh
dependence of the results with the aim to later develop closure relations that take the
mesh into account.
Keywords: CFB, CFD simulation, experimental, PIV
INTRODUCTION
Fluid dynamics of circulating fluidized beds (CFB) can be computed with different
methods. Typically the simulations are conducted with Eulerian-Eulerian models based
on the kinetic theory of granular flow. The simulations are run as transient to fully
describe the complicated flow patterns of dense gas-solid suspensions. These models
require a fine computational mesh, which in the case of large industrial processes leads
to unfeasibly long computations. To improve the modelling capabilities, it is necessary
to develop faster simulation methods. The most attractive approach seems to be timeaveraged modelling facilitating steady-state simulation of fluidization.
Several attempts to develop models for steady-state simulations, and for coarse-mesh
simulations that require similar equation closures, have been presented in the literature.
Agrawal et al. (2001) and Andrews et al. (2005) studied the average drag and stress
terms through simulations in small domains with periodic boundary conditions. Zhang
& VanderHeyden (2002) suggested an added-mass force closure for the correlation
between fluctuations of the pressure gradient of the continuous phase and fluctuations of
solids volume fraction. De Wilde (2007) analyzed the same term from simulations and
accounted also for the drag force in the derivation of a new closure. Zheng et al. (2006)
presented a two-scale Reynolds stress turbulence model for gas-particle flows.
Anisotropy and the influence of the real CFB geometry, both essential for successful
description of CFB hydrodynamics, were largely left out in the papers listed above.
These topics were, however, addressed in earlier work on steady state modelling done at
Lappeenranta University of Technology (LUT) and at Åbo Akademi University (ÅA) in
the 90’s but accurate equation closure was not possible at that time. New measurement
techniques and computational methods have now made a more elaborate equation
closure feasible by providing us detailed data on the fluctuations in velocities and
voidage. Transient simulations can now be utilized in development of equation closures.
With the objective to develop models for steady state simulation, work is now in
progress in a three-year joint project at the authors’research units. A prerequisite for
utilisation of the results from transient simulations is validation of the models by
measurements. For that purpose, a laboratory scale 2D CFB was built at ÅA and
experiments were carried out. The present paper deals with experimental and numerical
studies conducted in the joint project, which are related to this laboratory scale
circulating fluidized bed at ÅA.
EXPERIMENTAL
The 2D circulating fluidized bed
A 2D CFB unit was constructed at Åbo Akademi University (Guldén, 2008). The height
of the riser is 3 m and width 0.4 m. The distance between the riser walls is 0.015 m
which renders the bed two-dimensional. The CFB pilot is shown in Figure 1. The 0.4 m
wide riser walls, the side walls of the solids separator and the walls of the standpipe are
made of polycarbonate plates and the rest of plywood and metal plates. Air distributor
consists of 8 equally spaced air nozzles. Solids separation is done in a simple separation
box from which particles fall through the return leg into a loop seal consisting of two
fluidized 10 cm wide sections. The amount of solids in the loop seal and the solids
circulation rate are determined from videos taken of the loop seal region when the gas
flow to the loop seal is abruptly cut off. The fluidization air flow rates to the riser and
the loop seal are measured and controlled by Bronkhorst High-Tech B.V. Thermal Mass
Flow Controllers. Pressure was measured at several locations round the CFB with a
custom made manometer.
Figure 1. The 2D CFB system: a rough sketch, the 2D CFB, and its lower part with the wind box and the
loop seal.
The experiments
The bed material consisted of spherical glass particles with material density of 2480
kg/m3 and the Sauter mean diameter of 0.385 mm. In the ambient conditions of the
experiments, the terminal velocity of an average size particle is 2.9 m/s. Two
experiments with superficial gas velocities 3.1 m/s and 3.5 m/s were conducted. The
solids mass in the riser was in the two experiments 2.76 kg and 2.50 kg, respectively.
In the experiments, a dense vigorously fluctuating bottom region was observed with
highest particle concentration in the wall regions. Above the bottom zone, the
suspension travelled mainly upwards in form of clusters and more dilute suspension
between the clusters. At the side walls clusters were seen to fall down. Fig. 2 illustrates
the flow structure at the bed bottom and at 114-145 cm height. A denser wall region
with falling clusters is seen at both heights. The figures show long narrow clusters and
strands everywhere in the bed. The widths of the narrowest strands observed were about
2 mm. At the higher elevation, solids concentration inside the clusters was significantly
lower than in the clusters in the bottom region.
The behavior of the CFB at bed bottom and at 114-145 cm height was video recorded in
both experiments. From each case a 30 s section of the video was analyzed to estimate
the average volume fractions in the studied locations. The estimate was based on a
comparison between the local instantaneous grey scale values of the video image with
the reference values corresponding to an empty bed and to a packed bed. In the
interpretation of the concentration, Beer-Lambert law of absorbance of light was
utilized. The reference image corresponding to an empty bed at bed bottom and the
reference corresponding to a packed bed higher up were not available. Thus the
reference values had to be extrapolated from other regions. Since the lighting conditions
were not fully uniform, the poor reference values reduced the accuracy of the method.
Figure 2. Images from the experiments, from the left: at the bottom at U0=3.1 m/s, at the bottom at U0=3.5
m/s, at 1.14-1.45 m height at U0=3.1 m/s, and at 1.14-1.45 m height at U0=3.5 m/s.
Evaluation of the feasibility of PIV as a tool for analysis of solids velocities
To get more detailed data for model development and validation, local solids velocities
should be measured along with the local voidage. A good method for velocity analysis
is Particle Image Velocimetry (PIV) which was tested here to confirm the feasibility of
the method. Different imaging methods were tested. In PIV, images of a particle-laden
flow are recorded with a short time delay between consecutive frames. The images are
then divided into smaller interrogation areas, the intensity profiles of which are crosscorrelated between the consequent frames. The displacement of the particles can be
calculated from the correlation peak and translated to velocity. For backlit images the
local instantaneous void fraction can be estimated from the Beer-Lambert law on basis
of the average gray scale value of the interrogation area.
In this study a LaVision ImagerPro HS high speed camera was used with continuous
and pulsed lighting. The measurement setup for pulsed light is shown in Fig. 3. The
camera has a CMOS sensor with a resolution of 1280x1024 pixels. The maximum
recording frequency of the camera at the full resolution is 638 Hz for single-frame and
518 Hz for double-frame images. In the double-frame mode, the two frames to be
correlated are recorded with a very short time delay followed by a longer delay before
another double frame. In the single-frame mode, images are recorded with even time
intervals and each pair of consecutive frames is correlated. Before the cross-correlation
the intensities were locally normalized and inverted for the backlit images.
Figure 3. The measurement setup for the pulsed light measurements.
Pulsed light source
To control the exposure in the double-frame mode of the camera, a pulsed light source is
needed. Due to the limitations set by the CFB geometry, the light can only be directed
from the front or from the behind. Backlighting (light comes from behind) creates a
shadow image with the particles in the focus plane sharp and those outside of it blurred.
The cross-correlation algorithm weights the sharp particles as they create the highest
intensity peaks in the inverted image. However, the method can only be used if the light
can penetrate through the suspension. In the CFB this is not the case at the lowest void
fractions.
Figure 4 shows samples of the recorded images of a 45x34 mm2 window located in the
middle of the riser at 130 cm height, and the vector fields calculated from them.
Sections c and d in Figure 4 show examples of a situation in which the light doesn't
penetrate the clusters and the velocity calculation fails.
With frontlighting the light is
directed from the direction of
the camera. Intensity peaks
are generated by the particles
closest to the front wall. The
velocities of these particles
can be determined at any void
fraction but these measures
only represent the particles
right next to the wall. Another
drawback is that there is no
easy way to estimate the void
fraction. The frontlit method
can be used to study wall
Figure 4. Instantaneous velocity vector fields overlaid on the
and those regions
th effects
original images recorded using a diode-laser backlight. Every 16
of the calculated vectors is displayed. The measurement window is where the backlighting fails
marked with red in the schematic on the right.
due to low void fractions.
In this study a diode-laser was used for its portability. The power of the laser limits the
size of the usable measurement window to around 50-80 mm. With these image sizes
double-frame imaging has to be used to achieve a short enough time delay between
correlated frames. As the maximum pulse length of the laser also decreases as the
triggering frequency increases, no time resolution can be obtained and the calculated
fields have to be considered as discrete samples. The sampling can be spread over a long
period of time without collecting an unreasonable amount of data, making the
calculation of representative average fields more convenient. The spatial resolution of
the method is around 5 times the diameter of the particles.
Continuous light source
To get a larger measurement area than the available laser could provide, high frequency
fluorescent tubes were used to provide a backlight. Tests showed that with the single
frame imaging mode, a 600 Hz imaging frequency and a measurement window over the
whole width of the experimental device is a suitable combination. The individual
particles are not distinguishable, but the PIV correlation algorithm produced an
acceptable solids velocity field except in regions with the lowest voidage. The spatial
resolution was around 20 mm. The light source used proved to be slightly inadequate to
allow a short enough exposure time to eliminate the motion blur at the highest solids
velocities. A more powerful a light source is needed for future measurements. For good
average fields the data has to be collected over a long period of time, at least for 20
seconds. Figure 5 shows a sample of a velocity vector field measured over the whole
width of the riser just below the solids return inlet. The downward flow near the wall
can be seen on the left side, while the incoming solids disturb the wall layer on the right
side.
Figure 5. Velocity vectors overlaid on the original image, constant backlight. Every 4 th one of the
calculated vectors is displayed. The measurement window is marked with red in the schematic on the
right.
CFD SIMULATION OF THE 2D CFB AND VALIDATION
Models used
The two experiments conducted in the 2D CFB were simulated with the models based
on the kinetic theory of granular flow available in the Fluent 6.3.26 CFD software
(Fluent, 2006). The continuity and momentum equations used in the transient
simulations can be written for phase q (gas phase denoted by g and solid phase by s) as
follows:
∂α q ρ qm ∂α q ρ qmuq , k
+
=0
∂t
∂xk
(1)
∂α q ρ qm u q ,i
∂t
+
∂α q ρ qm u q ,k u q ,i
∂x k
∂p ∂α qτ q ,ik ∂α qτ
= −α q
+
+
∂xi
∂xk
∂x k
M
+ α q ρ qm g i + ( −1)
(δ qs +1)
q ,ik
−
∂p q
∂xi
K gs (u g ,i − u s ,i )
δ qs
(2)
where t is time, x is spatial coordinate, volume fraction, density, u velocity, p gas
phase pressure, ps solids pressure, g gravitational acceleration, K drag coefficient, qs
Kronecker delta, laminar stress, and M local scale turbulent stress.
The granular temperature was obtained from a partial differential equation using the
Syamlal et al. (1993) model for granular conductivity. The solid phase granular
viscosity was calculated from the model by Syamlal et al. (1993). The solids bulk
viscosity and solids pressure were calculated from the formulas by Lun et al. (1984).
The k- turbulence model producing the local scale turbulent stress was the version
modified for multiphase flows (“dispersed turbulence model”, Fluent (2006)). At the
walls, the partial slip model of Johnson and Jackson (1987) was used for the solids with
specularity coefficient equal to 0.001 and the free slip boundary condition was used for
the gas. For gas-particle interaction, a combination of the Wen & Yu (1966) (at the
voidage above 0.8) and Ergun (1952) equations was used. The frictional solids stresses
were calculated from the model of Schaeffer (1987). The first-order discretization for
time stepping and the second-order spatial discretization were employed. The 2D grid of
the simulation consisted of 31648 elements with the mesh spacing of 6.25 mm. The
time step in the simulation was 0.2 ms. Air inflow velocity at the bottom was described
by a function that reproduces the orifice locations.
Simulation results and comparisons with measurements
For comparison with measurements, the
simulations were first run till a steady
state and then for an extra 10 s time
period to obtain averages of the velocities
and void fractions. Fig. 6 illustrates the
typical flow patterns obtained in the
simulations and the corresponding time
averages at the two fluidization velocities.
The flow patterns are similar to the ones
observed in the experiments. Due to the
coarseness of the computational mesh,
however, the simulated clusters are wider,
the thinnest ones being 1.5 cm wide.
Otherwise the flow structure is correct
with a dense bottom bed and dense wall
regions with downflow of solids.
Figure. 6. Instantaneous and average solids
volume fractions at U0=3.1 m/s (left) and at
U0=3.5 m/s.
A comparison between measured and simulated solids circulation rates was also done.
Both in the experiments and in the simulations, an increase in the fluidization velocity
increased the circulation rate. Moreover, solids circulation rates obtained in the
simulations were of the same order as measured in the laboratory unit.
Gas phase static pressure was measured at several elevations at the wall opposite to the
solids return. Figure 7a shows a comparison between the measured and simulated
average pressure profiles at the wall. The simulated pressure profile is close to the
measured one at the lower fluidization velocity but a clear discrepancy is seen at the
higher velocity. From the measured pressure profile, the vertical voidage profile could
be estimated. The results are depicted in Fig. 7b together with the average solids volume
fraction profile obtained from the simulations. In addition, the average values obtained
from the analysis of the videos, taken at two elevations during the experiments, are
marked in Figure 7b. In general, the match is reasonably good considering the
inaccuracies in both of the experimental methods.
a)
b)
Figure. 7. Comparison of measured and simulated vertical pressure (a) and solid volume fraction (b)
profiles in the experiments with fluidization velocities 3.1m/s and 3.5 m/s.
From the videos, the average lateral solid volume fraction profiles were determined at
20 cm and 130 cm heights. Figure 8 shows a comparison between the measured and the
simulated lateral solids volume fraction profiles. In the profiles determined from the
videos there is clear asymmetry. This is very likely caused by the uneven lighting in the
experiments combined with the lack of good reference images for fixing the
concentration scale. Otherwise the measured and simulated profiles are in good
agreement. The thin dense wall region observed in the simulations could also be
detected in the experimental results.
Figure 8. Comparison of simulated and measured lateral solid volume fraction profiles at 20 cm and 130
cm heights at fluidization velocities 3.1 m/s and 3.5 m/s.
From
the
simulation
results, the average lateral
velocity profiles could be
calculated. Figure 9 shows
the profiles obtained in the
two simulations at 130 cm
height. A downflow region
is seen at the walls. This
corresponds to the visual
observations
and
the
velocity fields determined
by PIV. No average
velocity profiles were
measured at this stage.
Figure 9. Lateral profiles of the solids vertical velocity
component at fluidization velocities 3.1 m/s and 3.5 m/s
determined from the simulations 130 cm height.
Simulations of particle mixtures
In fluidized bed combustors the solid
phase consists of particles with different
sizes, densities and compositions. Thus
any model used to describe CFB
hydrodynamics
should
also
be
applicable to mixtures of particles.
Experiments are planned to be
conducted in the 2D CFB using
mixtures of particles to serve as
validation of multi-particle models. At
this stage, preliminary simulations were
done to test the capability of the Fluent
software to simulate mixtures. In the
first simulation, a single solid phase was
divided into two and Fluent was proved
to correctly treat this division. Next,
different mixtures of two types of
particles were simulated. An example of
resulting segregation of the two particle
sizes is shown in Fig. 10. The results
seem reasonable but validation through
measurements is required and will be
done in the future.
a)
b)
c)
d)
Figure 10. Simulation results for a particle mixture
with 10 mass% of particles with dp=0.650 mm and
90 mass% with dp=0.270 mm. Fluidization velocity
U0 = 2 m/s. a) volume fraction of particles with
dp=0.650 mm, b) volume fraction of particles with
dp=0.270 mm, c) 10 s average volume fraction of
particles with dp=0.650 mm, d) 10 s average volume
fraction of particles with dp=0.270 mm.
TRANSIENT CFD MODELING USING COARSER MESHES
As long as time-averaged models are not available, the most feasible alternative to
simulate gas-solid flow in a large scale CFB is to use a coarse mesh in a transient
simulation and to time-average the results. In coarse mesh simulations information on
the local flow structures is lost when the flow gets filtered by the mesh. The lost
information must be brought back into the system through closure models to predict the
hydrodynamics correctly. This corresponds to what has to be done in time-averaged
modelling where all information on time variations of the flow needs to be fed to the
model through equation closures.
In this study, the effect of mesh size was evaluated by simulating the experiment
conducted in the 2D CFB at fluidization velocity 3.1 m/s in three different meshes with
spacings 0.625 cm, 2.5 cm and 5 cm. The time step in all simulations was 0.2 ms. The
models were the same as used in the validation simulations of the previous section. The
only difference is in the description of the gas inlet. With the coarse meshes used here, it
is not possible to describe accurately the gas flow through the separate orifices. Thus the
entire bottom of the riser was defined as an inlet with a constant gas inflow velocity.
In each of the simulations, computations start from a situation where solids are
uniformly distributed throughout the riser. After 10-second simulation, a stable solution
is reached, with solids leaving the riser from the outlet and entering the bed through the
return leg. The mass flow rate of solids entering the system through the return channel is
adjusted in such a way that, at any instant of time the mass in the riser remains the same
as in the experiment. Averaging is then performed for another 20 seconds. Timeaveraged contours of solid volume fraction and the Favre-averaged gas y velocity for
different mesh sizes are shown in Figure 11.
Figure 11. Simulations with mesh spacings 0.625 cm, 2.5 cm and 5 cm: a) Time-averaged solid volume
fraction and b) Favre-averaged gas y velocity.
As seen from Fig. 11a, with the smaller mesh spacing of 0.625 cm, the time averaged
solid volume fraction contours shows presence of small scale clusters. When the mesh
spacing is increased, the time-averaged solids volume fraction shows a much more
uniform distribution and the mesoscale structures of the flow get filtered. This is also
seen in Fig. 11b. When the mesh spacing is increased, channelling in the middle of the
riser and wider wall layers are observed in the averaged velocity profiles. Thus the
simulations show that the results are mesh dependent. To obtain the same results in
different meshes, closure equations that take the mesh into account need to be
developed. A fine mesh is computationally expensive. For the above gas-solid flow
calculations of 30 s real time, the CPU time consumed by a single processor was around
8 hours with the 5 cm mesh spacing, with 2.5 cm mesh spacing it was 14 hours and with
0.625 cm mesh spacing 129 hours.
TOWARDS TIME-AVERAGED MODELING
A transient simulation in a coarse mesh is one alternative for simulating large CFBs.
Long simulation times are required to obtain a representative average flow field through
averaging of the transient simulation results. A faster alternative would be to directly
compute the average flow field from steady state models. Unfortunately, such models
are not available to date. For that purpose, models for CFBs are developed in the current
research project.
As an initial step in the development, the instantaneous continuity and momentum
equations, Equations (1) and (2), are averaged over time. First we need to define the
average quantities. A time average, also called Reynolds average, of a variable φ is
defined as
φ=
1 t + ∆t
φ dt
∆t ∫t
(3)
The instantaneous values can now be written as φ = φ + φ ' . This average is used for the
volume fraction q and pressure p. Thus, e.g., α q = α q + α q ' and p = p + p ' , and,
consequently, α q ' = 0 and p ' = 0 . A Favre average or a phase-weighted average is
defined as follows
⟨φ ⟩ =
α qφ
αq
(4)
Favre averaging is applied on velocities and we denote the average velocity by
U q , i ≡ ⟨u q ,i ⟩ . For the instantaneous velocity we have then u q ,i = U q ,i + u q ,i " . Note that
⟨u" q , i ⟩ = 0 but practically always u" q ,i ≠ 0 .
We obtain now the time-averaged continuity equation for phase q (
∂α q ρ qm ∂ρ qm α qU q , k
+
=0
∂t
∂xk
and time-averaged momentum equation for a phase q
qm
is a constant):
(5)
∂α q ρ qmU q,i
∂t
+
∂α q ρ qmU q, k U q,i
∂x k
= α q ρ qm g i − α q
∂p ′ ∂α qτ q ,ik
∂p
+
− α q′
∂x i
∂x k
∂xi
(6)
+
∂α qτ
M
∂x k
q ,ik
+ (−1)
(δ qs +1)
K gs (u gi − u si ) −
∂ pq
∂xi
δ qs −
∂ρ qm α q u q′′, k u q′′,i
∂x k
The terms on the right hand side are the gravitation term, pressure term, pressure
fluctuation term, laminar stress, turbulent stress, drag force, solid pressure term, and the
Reynolds stress term. The gravitation and pressure terms can be calculated from the
basic average flow properties but the rest of the terms need to be modelled. To evaluate
the importance of the different terms, a long (about 15 min) simulation of the
experiment at U0=3.5 m/s was conducted and the computation results were timeaveraged. Fig. 12 shows the terms in the time-averaged balance equation for the solid
phase vertical velocity component. The time derivative in the time-averaged equation
can be discarded in this case. In order to simplify the comparison of the terms, the
convection term in Equation (6) is moved to the right hand side.
∂α s ρ smU s , kU s, y
1: −
∂xk
2 : α s ρ sm g y
3: −αs
∂p
∂x y
4 : − α s′
∂p′
∂x y
5:
6:
∂α sτ s , yk
∂xk
∂α sτ M q , yk
∂ xk
7 : K gs (u gy − usy )
8: −
9: −
∂ ps
∂x y
∂ρ sm α s us′′, k us′′, y
∂xk
Figure 12. The different terms in the time-averaged balance equation of the solids vertical velocity
component plotted on the riser centre line as a function of height.
Although the simulated time period was very long, the average velocities are still not
perfectly smooth functions of the spatial coordinates. Consequently, the velocity
derivatives are even more restless which is seen in the large spatial variations in the
convection term. However, a comparison of the magnitudes of the different terms can
still be made. In the upper part of the riser, the largest terms are the drag term and the
gravitation term. The Reynolds stress term and the term originating from the correlation
between fluctuations in gas phase pressure and solids volume fraction are significant in
the bottom region. The turbulent stress term and solid pressure term are small over the
whole riser length on the centre line. However, a wider analysis of the terms in the
bottom bed showed that solids pressure can be significant in the dense wall regions
whereas the turbulent stresses are insignificant everywhere in the CFB. The largest
terms to be modelled are the drag forces and the Reynolds stresses. Modelling of the
pressure fluctuation term was considered in De Wilde (2007) and a similar approach
could be considered also here. The average drag forces have typically been described by
means of a correction to the drag coefficient as a function of suspension density, see e.g.
Kallio et al. (2008). Analysis of the average drag forces in the transient simulation could
be used to modify the earlier drag correction models.
In Kallio et al. (2008), the different components of the Reynolds stress tensors were
analysed from another transient simulation. An order of magnitude difference was
observed between the different components. Thus no assumption of isotropy, as often
made in modelling of single phase turbulence, can be made in modelling of turbulence
in dense gas-solid flows. The different components need to be modelled separately and
preferably through separate transport equations for each Reynolds stress component.
Transport equations for the Reynolds stresses of a phase q can be obtained from the
averaged momentum equations and they can be written as follows:
∂ρ qm α q u′qi′u′qj′
∂t
+
∂ρ qmU qk α q u′qi′u′qj′
∂xk
= Pq ,ij + Π q ,ij + Dq ,ij + M q ,ij − ε q ,ij + Gg ,ij + Fq ,ij + δ qs S q ,ij
(7)
where Pq,ij is the production of the Reynolds stresses
∂U q , j
∂U q ,i

Pq ,ij = −α q ρ q  u"q ,k u"q ,i
+ u"q ,k u"q , j
∂x k
∂x k

q,ij




(8)
is the pressure-strain covariance (or redistribution) term
 ∂α u"
∂α u" 
Π q ,ij = p'  q q , j + q q ,i 
 ∂x
∂x j 
i

(9)
Dq,ij represents the turbulent, pressure and molecular diffusion
Dq ,ij = −
(
∂
α q ρ q u"q,i u"q , j u"q , k + α q p ' u"q ,iδ jk + α q p ' u"q, jδ ik − α qτ q, jk (u" )u"q ,i − α qτ q ,ik (u" )u"q , j
∂xk
)
(10)
q,ij
is the dissipation term
ε q ,ij = α qτ q , jk (u" )
∂u"q ,i
∂xk
+ α qτ q ,ik (u" )
∂u"q , j
∂xk
(11)
Fq,ij is the turbulent mass flux
Fq ,ij = u"q ,i
∂α qτ q , jk (U )
∂xk
+ u"q , j
∂α qτ q ,ik (U )
∂xk
(12)
and Mq,ij arises from the modelling the local scale turbulence
M q ,ij = u"q ,i
∂α qτ qM, jk
∂x k
+ u" q , j
∂α qτ qM,ik
(13)
∂x k
The phase interaction term Gq,ij reads
Gq ,ij = K gs (u ′g′,i u ′s′, j + u ′g′, j u ′s′,i − 2u ′q′,i u ′q′, j ) + (−1)
+ (−1)
(δ qs +1)
(δ qs +1)
K gs u ′q′,i (U g , j - U s , j )
(14)
K gs u ′q′, j (U g ,i - U s ,i )
The two-phase term Ss,ij is
S s ,ij = −u"s ,i
∂ps
∂p
− u"s , j s
∂x j
∂xi
(15)
These terms in the balance equation for the vertical normal component of the solid
phase Reynolds stress are plotted in Fig. 13 on the centreline of the riser. Here again we
see that the simulation is too short to produce smooth averages. As shown in Fig. 13, the
largest term is the phase interaction term Gq,ij and a couple of other terms like the
production and dissipation terms are also significant. In a time-averaged CFD
simulation, only the stress production terms Pq ,ij needs no modelling. All the other terms
need to be modelled. Performance and applicability of various modelling alternatives
for the Reynolds stresses will next be examined and tested.
5000
1
2
4000
3
4
3000
5
6
2000
7
8
1000
9
10
0
∂U s, y
∂xk
1 : Ps, yy = −2α s ρ s u"s, k u"s, y
2 : Π s , yy = 2 p'
∂α s u"s , y
4 : Dsp, yy
∂x y
(
∂
α s ρ s u"s, y u"s, y u"s, k
∂x k
∂
= −2
α s p' u"s, y
∂x y
3 : DsT, yy = −
5 : Dsm, yy = 2
(
)
(
∂
α sτ s , yk (u" )u"s , y
∂xk
6 : Fs, yy = 2u"s, y
)
)
∂α sτ s , yk (U )
∂x k
7 : ε s , yy = 2α sτ s , yk (u" )
∂u" s , y
∂x k
8 : Gs , yy = 2K gs (u g′′, y u s′′, y − u s′′, y u′s′, y ) + 2 K gs us′′, y (U g , y - U s , y )
-1000
9 : S s, yy = −2u"s , y
-2000
0
0.5
1
1.5
y [m]
2
2.5
3
10 : M s , yy = 2u" s, y
∂p s
∂x y
∂α sτ sM, yk
∂x k
Figure 13. The terms in the balance equation for the vertical normal component of the solid phase
Reynolds stress [kg/ms3].
CONCLUSIONS
Fluid dynamics of circulating fluidized beds (CFB) can be computed with a variety of
methods. Typically the simulations are conducted with Eulerian-Eulerian models based
on the kinetic theory of granular flow. These models require a fine computational mesh
and, in addition, that the simulations are run as transient, which in the case of large
industrial processes leads to unfeasibly long computations. To improve the modelling
capabilities it is necessary to develop faster simulation methods. The most attractive
approach seems to be time-averaged modelling facilitating steady-state simulation of
fluidization. An order of magnitude slower but still feasible alternative would be to use
a coarse mesh and special mesh-dependent closure equations in a transient simulation.
A prerequisite for utilisation of transient simulations in steady-state model development
is validation by measurements. Thus, a 0.4 m wide and 3 m high laboratory scale 2D
CFB was built at Åbo Akademi University. Experiments were carried out for validation
of the CFD simulations. The behaviour of the process was video recorded and the
videos were analysed to determine the average voidage distribution in the CFB. The
pressure distribution in the bed and the circulation rate of solids were also measured.
The measured flow properties were compared with simulation results, observing a
reasonably good agreement. In addition, PIV measurements were conducted to evaluate
the possibilities to determine the velocities of individual particles and particle clusters.
The high speed imaging, together with time resolved velocity fields and void fraction
estimates based on the gray scale value, proved to provide an excellent visualization of
the velocity of the clusters in the CFB.
Results from a long simulation were analysed to evaluate the requirements for the
closure of the time-averaged equations. The largest terms to be modelled are the gassolid interaction and the Reynolds stress. In addition, the correlation between pressure
fluctuations and voidage can be significant in dense bottom bed conditions and solids
pressure in the wall layers at riser bottom. Modelling of the momentum equation terms
was discussed in the paper.
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ACKNOWLEDGEMENT
The financial support of Tekes, VTT Technical Research Centre of Finland, Fortum
Oyj, Foster Wheeler Energia Oy, Neste Oil Oyj and Metso Power Oy is gratefully
acknowledged.