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This paper compares coarse grid simulations completed with various filtered models to computationally very expensive resolved simulations of fluidization in a 2D setup. It was shown that the original one-marker filtered model performed best. Surprisingly, more recently presented two-marker models showed substantial deviations from the resolved simulations. Unfortunately, one-marker models are not suitable for general large-scale fluidized bed simulations. Therefore, the performance of a new two-marker model is assessed, which respects certain limits when correcting the drag force and uses a normalized filtered slip velocity. While the initial results with this new model are encouraging, substantial uncertainty still exists due to the present lack of a dedicated filtered stress model.
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Verification of filtered Two-Fluid Models in different flow regimes
Jan Hendrik Cloete1, Schalk Cloete2, Stefan Radl3, Shahriar Amini1,2*
1 Department of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU),
NO-7491 Trondheim, Norway
2 Flow Technology Department, SINTEF Materials and Chemistry, NO-7465 Trondheim, Norway
3 Institute of Process and Particle Engineering, Graz University of Technology, Inffeldgasse 13/III, Graz,
AbstractThis paper compares coarse grid simulations completed with various filtered models
to computationally very expensive resolved simulations of fluidization in a 2D setup. It was
shown that the original one-marker filtered model performed best. Surprisingly, more recently
presented two-marker models showed substantial deviations from the resolved simulations.
Unfortunately, one-marker models are not suitable for general large-scale fluidized bed
simulations. Therefore, the performance of a new two-marker model is assessed, which respects
certain limits when correcting the drag force and uses a normalized filtered slip velocity. While
the initial results with this new model are encouraging, substantial uncertainty still exists due to
the present lack of a dedicated filtered stress model.
Computational fluid dynamic simulations using the Two-Fluid Model (TFM) closed by the Kinetic Theory
of Granular Flow (Gidaspow et al., 1992, Lun et al., 1984) have become a popular tool for the investigation
of fluidized beds. This method allows the performance of the reactor to be evaluated and new concepts to be
tested without costly experiments or plant trials. However, for accurate results from TFM simulations, it is
essential that the transient multiphase structures, i.e. bubbles and clusters, are resolved accurately in time and
space (Cloete et al., 2015, 2016b). Despite the rapid increase of computational power over the last few
decades, performing resolved simulations remains impractical for large, industrial-scale fluidized beds,
especially when small particles (i.e., < 100 µm) are considered.
For this reason filtered TFMs (fTFMs) were developed, with different formulations provided by groups from
Princeton (Milioli et al., 2013), INPT (Ozel et al., 2013) and JKU (Schneiderbauer and Pirker, 2014). In this
approach, the conservation equations are spatially averaged, with additional terms appearing for the effects
of the unresolved, meso-scale structures on the drag force and the solids stresses. Data from highly resolved
simulations are then analyzed by statistically averaging over differently sized regions (filters) to develop
correlations for these additional terms. Consequently, reasonable predictions of overall reactor behavior can
be obtained in coarse grid simulations at computational times that are several orders of magnitude smaller
than what would be required for resolved simulations. Most of the research in the field has focused on
developing a filtered drag model, since the contribution of the drag term is generally most important in the
filtered momentum equations (Ozel et al., 2013).
The current state-of-the-art filtered drag models are two-marker models. Here the corrections for sub-grid
effects are described (next to the averaging volume, i.e., filter size) as a function of (i) the filtered solids
volume fraction, and (ii) the filtered solids slip velocity. The filtered slip velocity will tend to increase in less
homogenous flows as the gas will tend to bypass dense regions, creating a larger slip between the phases
inside the filter region. The filtered slip velocity therefore acts as a measure of the non-homogeneity of the
flow and therefore serves as a good choice as independent variable in correlations for the sub-grid
corrections. One disadvantage of the slip velocity, however, is that it is highly correlated with the volume
fraction. For example, larger slip velocities will tend to occur in dilute regions, whereas the slip velocities
will tend to be smaller in dense regions. This leads to a situation where the occurrences are distributed
unevenly in the two-dimensional parameter space, making it more difficult to fit an accurate model to the
Our present contribution proposes the filtered slip velocity magnitude scaled by the steady state slip velocity
as a second marker for the drag correction correlation. Since the steady state slip velocity is a function of the
filtered solids volume fraction, it results in a much more even distribution of data in the two-dimensional
parameter space. Additionally, a simple dependency of the drag correction on the scaled filtered slip velocity
was revealed, which allows a correlation to be accurately fitted to the data. The resulting model is tested in
this study by comparing coarse grid simulation results with resolved simulation results, as well as to coarse
grid simulations from filtered drag models previously proposed in literature, for 3 cases operating in different
fluidization regimes. This will show the potential of this approach for future study.
Resolved simulations
Fine grid simulations are performed in this study for the evaluation of the filtered models in coarse grid
simulations using the Two Fluid Model closed by the Kinetic Theory of Granular Flow. A more detailed
description of the resolved simulation setup can be found in (Cloete et al.).
Filtered simulations
The coarse grid simulations are performed using the spatially averaged, or filtered, continuity and
momentum equations. The filtered continuity equations remain similar in form to the equations used in the
resolved simulations. However, the filtered momentum equations contain several additional terms that
require closure, as shown for the filtered solids momentum equation below. The filtered gas momentum
equation contains similar terms, but the gas phase meso-scale stresses are generally considered to be small
(Milioli et al., 2013) and are therefore neglected in this study.
( )
 
( )
( )
sss ssss s s ssss
s s s gs g s s
gK p
ραυ ραυυ α ραυυ
τ αρ υ υ α
+∇⋅ = ∇ −∇ −∇⋅ +
∇⋅ + +
  
Here, the third term on the right requires closure for the meso-scale solids stresses, and the two last terms
require closure for the meso-scale interphase momentum exchange. Three literature formulations of these
closures will be evaluated (Igci and Sundaresan, 2011, Milioli et al., 2013, Sarkar et al., 2016) in what
follows. In addition, this study will evaluate a new filtered drag model.
The closure for the filtered drag force is usually determined in the following form:
 
( )
gs g y s y s
gy sy
gs coarse
υυ α
,gs coarse
is evaluated at the filtered solids volume fraction and filtered slip velocity. Note, that only
the forces and slip velocities in the y-direction are considered when calculating
(i.e., the drag correction
factor) since the filtered drag force is most important in that direction.
The following equation is proposed for the filtered drag correction factor:
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
** ** 3
1 2 max
** **
3 45
arctan arctan
log 2
arctan log
f fine s f fine s
f fine f fine
Cx xx
α αα π
∆ −∆ ∆ −∆
∆ −∆ ∆ −∆
(the scaled filtered slip velocity magnitude) is the filtered slip velocity magnitude divided by the
steady state slip velocity at the filtered solids volume fraction and
is the dimensionless grid size in the
resolved simulations used to develop the model. The filter size is non-dimensionalized by
The following values of the coefficients are used: 1=32.3, 2=53.7, 3=16.0 , 4= 0.933, 5=
0.0327, 6= 0.746, 7= 0.870, 8= 0.962, and  = 0.55. The fine grid scaled filter size is set to
= 0.129.
This approach offers several improvements over the current state-of-the-art two-marker models (Milioli et
al., 2013, Sarkar et al., 2016). Firstly, using the scaled filtered slip velocity as the second marker leads to a
more even distribution of data in the two-dimensional parameter space when collecting data from resolved
simulations. This allows a more accurate model to be fitted to the data. Secondly, by fitting a correlation to
, more emphasis is placed on large corrections, therefore improving the model fit in regions where
the filtered drag correction is most important. Thirdly, we find that using the scaled filtered slip velocity
results in a linear dependency of
( )
log C
log slip
. This allows a simpler correlation to be chosen as the
basis of our fit.
Lastly, the equation form proposed obeys all the physical constraints for the filtered drag correction: the first
and the second
functions cause a zero correction for very dilute and very dense flows, when there are
no meso-scale structures. The third
function leads to a zero correction when the filter size is equal to
the grid size used in the resolved simulations, and causes the drag correction to reach an asymptotic limit at
very large filter sizes. Last, the correction C is ensured to be positive due to the nature of logarithmic
Verification cases
Three cases, operated in different fluidization regimes, are considered to thoroughly test the generality of the
filtered models derived in this study. The average superficial inlet gas velocity is chosen to be at the
geometric center of the bubbling fluidization and turbulent fluidization regimes, according to (Bi and Grace,
1995), and quarter way between the transitions to turbulent fluidization and homogenous dilute phase
transport for the fast fluidization case. A solids flux of 150
/kg m s
is specified, and the simulation was run
for 10s to reach a pseudo-steady state before time-averaging the simulation for 30s.
Additionally, the profile of the velocity and solids volume fraction at the inlet is specified to be non-uniform.
This forces a mean gradient in the system, allowing for rigorous testing of the filtered models for the meso-
scale solids stresses. The gas phase superficial velocity is chosen to be half the average superficial velocity at
the sides of the domain and increase linearly towards the center. The solids inlet velocity is set equal to the
gas inlet velocity. The solid volume fraction is set to a minimum at the centre, with a value equal to half the
solids volume fraction required to deliver the specified solids flux at the mean gas superficial velocity. The
solids volume fraction then increases linearly towards the sides.
The average superficial velocity for each case is given in Table 1, as well as the geometrical proportions.
Figure 2 to Figure 4 offer graphical examples of the three simulated geometries. The simulation domain
consists of a rectangular reactor region and a small outlet region. The aspect ratio of the reactor region is
increased with a factor of 2 as the fluidization velocity is increased to allow sufficient cluster formation
inside the fluidization region. The sides of the fluidization region are specified as periodic boundaries, since
the model evaluated in this study was derived for periodic flows not influenced by wall-effects. In the outlet
region, walls with a free-slip boundary condition for the solids slope at an angle of 45° towards the outlet.
The outlet has a width of 10 cm to prevent backflow, which would cause numerical instabilities. The average
solids volume fraction in the rest of the study is evaluated in the fluidization region only, to minimize the
influence of the walls in the outlet region on the results.
Table 1 - Description of the configuration for the three verification cases
region height
region width (m)
Average gas
velocity (m/s)
Fast fluidization
All simulations were performed in 2D to allow feasible computational times of the resolved simulations,
which were performed at a grid size equal to 11.8 times the particle diameter. Testing filtered models in 2D
should, however, still remain a valid approach, since filtered models derived from 2D and 3D simulations
have been shown to be qualitatively similar (Igci et al., 2008). The particle and fluid properties used in the
simulations are summarized in Table 2.
Table 2 - Summary of particle and fluid properties
Particle diameter
75×10-6 m
Particle density
1500 kg/m3
Gas density
1.3 kg/m3
Gas viscosity
1.8×10-5 kg/m s
Terminal settling velocity
0.2184 m/s
Model performance comparison
An overall view of model performance is shown in Figure 1. The bars represent the percentage deviation in
terms of overall solids hold-up from the fine-grid simulations for the three different fluidization regimes. A
positive deviation implies that too much solids is present in the domain, most likely caused by an under-
prediction of the drag force. A negative deviation indicates too little solids resulting from an over-predicted
drag force.
Figure 1: Deviation in the overall solids holdup from the resolved simulation for several different model setups. Coarse:
no filtered modelling; Cloete: Equation (3); Igci: (Igci and Sundaresan, 2011); Milioli: (Milioli et al., 2013); Sarkar:
(Sarkar et al., 2016). All simulations were carried out on a grid size of 20 mm (~23 times larger than the fine grid
The importance of a filtered model is clearly visible in the Coarse simulations for the bubbling and turbulent
regimes shown in Figure 1. In both these cases, a significant negative deviation (i.e., ca. 25%) in the overall
solids hold-up is observed, indicating that clustering is not sufficiently resolved to accurately predict the
momentum coupling between gas and solids. As a result, the momentum coupling term is over-predicted,
resulting in an under-prediction of the solids hold-up. This is not observed in the fast fluidization regime case
because clustering only took place in a relatively small region of the domain (see Figure 4).
Of the four different filtered models employed, the Igci model consistently showed the best performance.
This is the original filtered model, using only the filtered volume fraction as a marker for both the drag and
the stresses. Also, the Igci model has been derived from 2D simulations. Unfortunately, this model is not
suitable as a general solution for large-scale fluidized bed reactor modelling because of two important
limitations: (i) it requires specialized wall functions to give reliable predictions in wall-bound domains
(Cloete et al., 2013), and (ii) it cannot create sufficiently large drag corrections to predict flows in domains
with very large cell sizes (Cloete et al., 2016a).
The more complex two-marker models, which employ the filtered slip velocity as an additional marker for
the drag and the filtered scalar strain rate as an additional marker for the stresses, proved to be less reliable.
These models are required to overcome the fundamental limitations of the one-marker model outlined above,
but it is clear that additional work is required to improve performance to the point where these models can be
safely employed for large-scale fluidized bed reactor simulations.
Figure 2: Instantaneous solids volume fraction contours arranged from left to right for the (a) Resolved, (b) Coarse, (c)
Cloete, (d) Igci, (e) Milioli and (f) Sarkar models in the bubbling regime. The models are referenced in the caption of
Figure 1.
Figure 3: Instantaneous solids volume fraction contours arranged from left to right for the (a) Resolved, (b) Coarse, (c)
Cloete, (d) Igci, (e) Milioli and (f) Sarkar models in the turbulent regime. The models are referenced in the caption of
Figure 1.
Figure 4: Instantaneous solids volume fraction contours arranged from left to right for the (a) Resolved, (b) Coarse, (c)
Cloete, (d) Igci, (e) Milioli and (f) Sarkar models in the fast fluidization regime. The models are referenced in the
caption of Figure 1.
As shown in Figure 2, Figure 3 and Figure 4, both the Milioli and Sarkar models lead to large inaccuracies
when compared to the resolved simulation. Not only is the solids hold-up much too high, but the bottom
region where no clustering takes place is not captured correctly. Given that the Sarkar model was derived for
3D flows, it is understandable that the drag force is under-predicted. However, such a clear reason for the
discrepancy cannot be identified for the Milioli model. The Cloete model performs better, both in terms of
overall solids hold-up and qualitative flow behavior, but significant discrepancies are still evident.
It is important to note that the Cloete filtered drag model was implemented with the Sarkar filtered stresses,
which gave the most accurate results in all three cases. Figure 5 shows that the filtered stress model has a
significant impact on the solids hold-up and the general flow behavior in the filtered simulation. It is
therefore recommended that care should be taken when deriving filtered models for solids pressure and
viscosity since these models strongly impact the accuracy of the overall filtered simulation.
Figure 5: Instantaneous solids volume fraction contours for the Cloete filtered drag model with (a) no filtered stress
modelling, (b) Igci filtered stresses, (c) Milioli filtered stresses and (d) Sarkar filtered stresses in the bubbling regime.
The models are referenced in the caption of Figure 1.
Drag model comparison
In order to better understand the results presented thus far, the drag correction functions of the four different
filtered drag models investigated in this study are represented in Figure 6. Most important, the wide range of
drag corrections accessible by the two-marker models relative to the one marker model is clearly visible. In
addition, the large filtered drag corrections predicted by the Milioli and Sarkar models relative to the Cloete
and Igci models are also evident. This helps to explain the trends shown in Figure 1.
The four filtered drag models share similar characteristics. All the two-marker models increase the drag
correction with an increase in the slip velocity. In addition, all the models except for Sarkar respect the
physical limits that no drag correction is present at zero and very high filtered solids volume fractions (no
clusters can form under these conditions). The large drag correction predicted by the Sarkar model at high
solids volume fractions and scaled slip velocities could lead to unphysical results.
It is also noteworthy that the Milioli and Sarkar models encounter a maximum bound within the range of
scaled slip velocities investigated. Such a maximum bound could restrict the generality of the model,
especially when large cell sizes are used in large-scale filtered fluidized bed simulations. Another important
fundamental challenge with the Milioli and Sarkar formulations is that the drag correction sometimes
increases with the slip velocity to a power greater than unity. This characteristic causes the drag force in a
filtered simulation to decrease with increased slip velocity, creating an unphysical self-reinforcing feedback
mechanism where more slip leads to less drag and even more slip.
The Cloete model attempts to rectify these issues by respecting all physical limits (including the fact that the
drag force must increase with slip velocity) and not imposing a maximum correction. As stated in the
methodology, this correlation also ensures a relatively even distribution of data throughout the parameter
space, thereby maximizing the likelihood of a good correlation fit. It therefore appears to be a good
foundation on which to build an improved set of two-marker filtered models.
Figure 6: Graphical representation of the drag corrections of Cloete (top left), Igci (top right), Milioli (bottom left) and
Sarkar (bottom right) expressed as (). The different lines represent different scaled slip velocities ranging from
0.5 to 16. The models are referenced in the caption of Figure 1.
This verification study has illustrated that substantial research efforts are still required to develop filtered
models that are suitable for general application in large-scale fluidized bed reactor simulations. Verification
studies in 2D domains and over three fluidization regimes; bubbling, turbulent and fast fluidization, have
illustrated that the simple original one-marker filtered model results in accurate predictions, whereas more
recent two-marker models show substantial deviations from well-resolved verification simulations.
Given that the one-marker approach is not suitable as a general large scale fluidized bed modelling solution,
two-marker model development studies should continue. One such new two-marker drag model was tested
with improved results relative to existing models. However, evaluation of the new model involved
significant uncertainty because a dedicated solids stress model is not yet available. It was shown that the
implementation of different available solids stress models with the new drag model resulted in widely
different solutions.
The new proposed two-marker drag correlation aims to improve on existing models by evenly distributing
the data in bins within the 2D parameter space, respecting physical limits, avoiding a maximum bound on the
correction and devising a simple equation form that is easy to implement. Despite the uncertainty introduced
by the solids stress model, the results presented in this paper are encouraging. Future work will derive
dedicated models for solids pressure and viscosity in order to more conclusively verify the performance of
this new drag correlation.
density, kg/m3
volume fraction
pressure, Pa
velocity, m/s
gravitational constant, m/s2
Momentum exchange coefficient
Filtered drag correction factor
Dimensionless filter size
Particle diameter, m
Scaled filtered slip velocity magnitude, m/s
Gas viscosity, kg/m s
The authors would like to express their gratitude for the financial support from the European Commission
under the NanoSim grant (project number: 604656), as well as for the computational resources provided at
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Full-text available
Filtered modelling of dynamic gas-particle flows has been actively studied by various groups around the world for more than a decade. Even so, the great complexity of this field of study means that several important knowledge gaps still exist. This thesis represents a significant step forward by closing several of the most important knowledge gaps through the development and rigorous assessment of new closures via detailed a priori and a posteriori analyses. The resulting set of filtered closures clearly outperforms the current state of the art, resulting in several valuable conclusions and recommendations. The primary conclusion from the present work is related to the critical importance of accounting for anisotropy in the filtered closures for drag and solids mesoscale stresses. For the filtered drag force, it was found that conventional isotropic closures strongly underpredict the drag correction in the directions perpendicular to gravity. A new formulation based on the drift velocity concept was found to account for this anisotropic effect in an efficient and natural manner. For the solids mesoscale stresses, the present work confirmed that the conventional approach based on the Boussinesq approximation results in large errors. In fact, studies showed that coarse grid simulations completely neglecting the solids mesoscale stresses perform better than those relying on the Boussinesq-based approach. Based on this knowledge, a new closure formulation was devised, conveniently allowing the prediction of the anisotropic solids mesoscale stresses via a single expression. Findings from the present study also challenged other conventions in the field. Firstly, the use of the filtered slip velocity as a second marker in the filtered drag force closure was found to lead to poor model performance. Secondly, a filter size to grid size ratio of unity appears to be the fundamentally correct ratio instead of the commonly employed ratio of 2. And thirdly, the 2D models derived in this work outperformed a 3D model from the literature in a validation study, suggesting that domain size independence of resolved simulations is more important than performing simulations in 3D. For reactive flows, the present work showed that a relatively simple closure can accurately predict the filtered reaction rate. In addition, the closure for the mesoscale species dispersion rate used in the filtered species transport equation was shown to have only a minor effect on reactor performance predictions. However, coarse grid reactive simulations were sensitive to the accuracy of the hydrodynamic filtered closures employed. Good hydrodynamic modelling is therefore the most important prerequisite for accurate large scale reactor performance predictions. Despite the progress made in this thesis, some important knowledge gaps persist. Firstly, this study did not attempt to quantify the generality of the proposed closures to flow situations with different particle and fluid properties. Such studies are required before the newly proposed closures can be recommended for use in reactors with particle and fluid properties that are very different from the FCC-type system considered in the present work. Secondly, an important effect related to the ratio of the domain width to the length of macro-clusters resolved in coarse grid simulations was identified. This effect required the use of a larger filter size to grid size ratio in narrow domains and further studies are required to find a general solution to this challenge. However, informed application of the anisotropic closures proposed in this thesis to real fluidized bed reactor problems can already be recommended. Experience from such studies can further accelerate the development of closures for filtered models towards the goal of their ubiquitous deployment for design, optimization and scale-up of fluidized bed reactors in industry.
Over the past decade, filtered Two Fluid Models (fTFMs) have emerged as a promising approach for enabling fluidized bed simulations at industrially relevant scales. In these models, the filtered drag force is considered to be the most important quantity that requires closure. To date, such closures have typically relied on an isotropic interphase momentum exchange coefficient by applying a drag correction factor to the microscopic drag closures commonly used in resolved simulations. In the present study, both isotropic and anisotropic closures are developed for predicting the filtered interphase forces. The relative performance of these two approaches is then evaluated by means of an a priori assessment, considering data obtained from simulations in which all flow variables are resolved, which were also used for closure derivation. Also, an a posteriori assessment, which compares coarse grid simulation results to a benchmark resolved simulation of a bubbling fluidized bed, is presented. The primary conclusion from the present study is that it is essential to account for the anisotropy of the filtered momentum exchange coefficient. It is shown that this can be done by employing a drift velocity formulation of the filtered drag force and by considering a gravitational contribution that only acts in the vertical direction. Furthermore, it is found that for the large computational grid sizes that are typically required in industrial scale fluidized bed simulations, a closure for the meso-scale interphase force is essential. Finally, also for coarse grids, a non-linearity correction factor, which accounts for assumptions in deriving the drift velocity-based form of the filtered drag force, requires closure. The present study therefore highlights multiple avenues for improving drag closures used in fTFMs. Hence, these results may critically strengthen the predictive capabilities of fTFMs, as well as guide future modelling efforts.
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CFD simulations of fluidized bed reactors are generally limited to the laboratory scale because of the fine grid sizes that are required to resolve complex particle clustering phenomena. The filtered Two Fluid Model (fTFM) approach has recently emerged as a promising method for allowing reasonable predictions of large-scale fluidized beds. This paper presents a verification study of new two-marker fTFM closures. In general, the fTFMs matched well to the resolved simulations. It was shown that the two-marker models significantly increased the predicted degree of phase segregation (resolved in coarse grid simulations), and hence have superior capabilities compared to simpler one-marker models. Also, the two-marker model predicted a more dynamic transient flow behaviour. However, further work is recommended to extend the present study over a wider range of flow conditions.
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This paper compares the performance of two filtered two fluid model (fTFM) formulations and two dense discrete phase model (DDPM) formulations in two different geometries. Comparisons to lab-scale experimental data with a fine Geldart A powder showed the ability of both the fTFM and DDPM approaches to deliver reasonable predictions of fluidized bed behavior on relatively coarse grids. Large-scale simulations then clearly showed the advantage of recently developed two-marker fTFMs where the drag correction depends on both the volume fraction and the slip velocity. It was also shown that both DDPM formulations become inaccurate on very coarse grids, suggesting clear limits to this approach when simulating large-scale fluidized beds.
Full-text available
The filtered two fluid model (fTFM) is a promising approach for enabling large scale fluidized bed reactor simulations using multiphase flow modelling. A substantial amount of research has been conducted in this field to derive filtered closures for all applicable transport processes including drag, stresses, heat transfer, scalar transport and reactions. This work aims to investigate the effect of TFM closures used in the resolved simulations from which fTFM closures are derived in order to assess the degree of uncertainty stemming from this source. In addition, the filtering approach provides a good platform for building a detailed understanding of the effects of different TFM closures. Simulations showed that the drag model used in the resolved simulations had the most significant effect on the filtered quantities derived. Inclusion of a frictional pressure model also had a large influence in the dense regions of the domain. Selection of a more generic drag model and inclusion of a frictional pressure model is therefore recommended for future studies. Finally, results revealed that the effects of TFM closures on filtered quantities could be grouped into two categories: interphase transport (drag, heat transfer and reactions) and diffusive transport (stresses and scalar transport). This suggests a common methodology in terms of filtered model form for these two groups.
The accuracy of fluidized-bed CFD predictions using the two-fluid model can be improved significantly, even when using coarse grids, by replacing the microscopic kinetic-theory-based closures with coarse-grained constitutive models. These coarse-grained constitutive relationships, called filtered models, account for the unresolved gas-particle structures (clusters and bubbles) via sub-grid corrections. Following the previous 2-D approaches of Igci et al. [AIChE J., 54(6), 1431–1448, 2008] and Milioli et al. [AIChE J., 59(9), 3265–3275, 2013], new closures for the filtered inter-phase drag and stresses in the gas and particle phases are constructed from highly-resolved 3-D simulations of gas-particle flows. These new closure relations are then validated through the bubbling-fluidized-bed challenge problem presented by National Energy Technology Laboratory and Particulate Solids Research Inc.
This short communication builds on previous work on the grid independence behaviour of the Two Fluid Model in reactive bubbling fluidized bed simulations. Regarding hydrodynamic grid independence behaviour (the numerical accuracy with which phase segregation was resolved), the particle relaxation time was confirmed as being directly proportional to the cell size achieving sufficiently grid independent behaviour. This relationship held over different particle sizes, particle densities, gas densities, gas viscosities and drag laws, but the slope of the proportionality changed for particle relaxation times above 0.4. For reactive grid independence behaviour (the numerical accuracy with which reactor performance was resolved), the relationship between the particle relaxation time and the sufficiently grid independent cell size was more complex, depending not only on the resolution of phase segregation, but also on the kinetic rate implemented and on the permeability of the emulsion phase. Simple and practical rules of thumb were proposed for estimating the sufficiently grid independent cell size for hydrodynamic and reactive simulations. For most practical purposes, the simpler and more accurate hydrodynamic grid independent cell size correlation can safely be used to run sufficiently accurate bubbling fluidized bed reactor simulations.
Conference Paper
Gas-particle flows exhibit large fluctuations in velocities and local suspension density. In riser flows, these fluctuations are associated with the random motion of the individual particles and with the chaotic motion of particle clusters, which are repeatedly formed and broken apart. These clusters occur over a wide range of length scales and their dynamics span a broad range of time scales. We construct filtered hydrodynamic models for such systems by averaging over fluctuations occurring on short length and time scales, and determine the associated constitutive models by filtering results obtained from highly resolved simulations of a kinetic theory based model for fluidized suspensions. In this presentation, we will describe results obtained by simulating such filtered hydrodynamic models and compare them against simulations of the kinetic theory model which formed the starting point of the filtering analysis. Such a verification study exposes conditions when the filtered model is able to afford grid-size independent solutions at much coarser resolutions than the original model and when the macroscale character predicted by the filtered and kinetic theory models are in agreement.
New constitutive relations for filtered two‐fluid models (TFM) of gas‐particle flows are obtained by systematically filtering results generated through highly resolved simulations of a kinetic theory‐based TFM. It was found in our earlier studies that the residual correlations appearing in the filtered TFM equations depended principally on the filter size and filtered particle volume fraction. Closer inspection of a large amount of computational data gathered in this study reveals an additional, systematic dependence of the correction to the drag coefficient on the filtered slip velocity, which serves as a marker for the extent of subfilter‐scale inhomogeneity. Furthermore, the residual correlations for the momentum fluxes in the gas and particle phases arising from the subfilter‐scale fluctuations are found to be modeled nicely using constitutive relations of the form used in large‐eddy simulations of single‐phase turbulent flows. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3265–3275, 2013
Two different approaches to constitutive relations for filtered two-fluid models (TFM) of gas- solid flows are deduced. The first model (Model A) is derived using systematically filtered results obtained from a highly resolved simulation of a bubbling fluidized bed. The second model (Model B) stems from the assumption of the formation of sub-grid heterogeneities inside the suspension phase of fluidized beds. These approaches for the unresolved terms appearing in the filtered TFM are, then, substantiated by the corresponding filtered data. Furthermore, the presented models are verified in the case of the bubbling fluidized bed used to generate the fine grid data. The numerical results obtained on coarse grids demonstrate that the computed bed hydrodynamics is in fairly good agreement with the highly resolved simulation. The results further show that the contribution from the unresolved frictional stresses is required to correctly predict the bubble rise velocity using coarse grids.
Full 3D flow simulations of lab and industrial scale dense fluidized beds were carried out using a filtered Eulerian–Eulerian approach. Filtered closures for interphase momentum exchange, solids stresses and additional wall corrections were implemented in the standard equations of motion. These closures had a very large effect on the overall model performance when solved on the large cell sizes required for computationally affordable 3D fluidized bed simulations. Numerical experiments conducted under different fluidization conditions showed that the current model formulation performs well over a wide range of operating conditions. It was found that additional modelling accounting for flow non-uniformity is essential under certain fluidization conditions. The current method for dealing with flow non-uniformity by means of wall corrections yielded good results under vigorous fluidization, but caused a slight inaccuracy at low fluidization velocities. In general, comparisons to a wide range of experimental data showed good quantitative agreement, suggesting that the formulation of the filtered model is highly generic. The filtered approach was also successfully verified in a large scale bubbling fluidized bed reactor by comparisons with a highly computationally expensive, well resolved, non-filtered flow simulation.