(172b) Novel Fluid Grid and Voidage Calculation Techniques for a Discrete Element Model of a 3D Cylindrical Fluidized Bed | AIChE

# (172b) Novel Fluid Grid and Voidage Calculation Techniques for a Discrete Element Model of a 3D Cylindrical Fluidized Bed

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## Authors

University of Cambridge

Fluidized beds are used
widely in industry, but despite this, the fundamental physics underlying
fluidized beds is still not well understood. Computational modelling of
fluidized beds has grown significantly recently, due in part to the drastic
increase in computational power. An accurate model is advantageous because it
can provide detailed time-averaged and instantaneous information on essentially
all aspects of fluid and particle motion in fluidized beds. This paper
describes the development and validation of a 3D cylindrical model
incorporating novel fluid grid and voidage
calculation methods to ensure accurate and stable simulations of fluidized
beds.

Discrete element
modelling (DEM) to simulate individual particle motion combined with
computational fluid dynamics (CFD) to model fluid motion has proven an effective
way to model laboratory size fluidized beds. Since it model particles
individually, governed by contact and Newtonian mechanics, DEM-CFD provides a
more accurate and insightful description of the particles in fluidized beds
than do two-fluid models (TFMs) which treat the particles as a continuous
phase. The main imperfection in DEM-CFD is that it requires a drag law to be
used to describe the force of interaction between fluid and particles; however,
more detailed models which do not require a drag law, such as direct numerical
simulation (DNS), are too computationally expensive to model laboratory-scale
fluidized beds.

Recently, researchers
have been able to develop DEM-CFD models which go beyond simulating 2D
rectangular beds and now model fully 3D cylindrical and more complex
geometries. These models have generally used one of two techniques to simulate
more complex geometries: (1) using a rectangular fluid grid with the immersed
boundary method [1] or (2) using an unstructured fluid grid generated by a
commercial CFD package, e.g. [2]. One
difficulty associated with both methods is that it is very difficult to ensure
that the fluid cells retain relatively equal volumes as well as reasonable
shapes. In order to satisfy the assumptions made in deriving the
volume-averaged fluid equations [3] used in DEM-CFD, it is necessary to ensure
that the fluid cells are significantly larger in volume than the particles and
do not have oblong shapes. If these criteria or not satisfied, the equations
can break down, corrupting the solution.

raised by using unstructured grids in DEM-CFD stems from trying to measure voidage in arbitrarily shaped CFD cells. Voidage is the volumetric fraction of a CFD cell free of
particles, ideally defined as:

εcell
= 1 ? (ΣVparticles)/Vcell

where εcell
is the voidage of the cell, ΣVparticles
is the total volume of all the particles in the cell and Vcell
is the volume of the cell. This calculation needs to be accurate and stable in
DEM-CFD because terms in the fluid equations and the drag law depend heavily on
voidage. The main difficulty in calculating voidage stems from the fact that individual particles often
lie in multiple fluid cells, and their volume must be divided between those
cells. A crude approximation, here forward referred to as the direct method,
would be to assume that the entire volume of a particle lies in a fluid cell if
its centre lies in the cell. A method originally proposed for rectangular grids
[4], here forward referred to as the grouping method, involves calculating voidage for each cell via the direct method and then
reassessing the voidage for each cell by averaging
its voidage with the voidage
in all of its surrounding cells. This methodology was developed to add
stability to the voidage calculation by spatially
smearing the results and can be directly translated for use in arbitrarily
shaped fluid cells. Another methodology for calculating voidage
in rectangular grids [5], here forward known as the cuboid approximation,
treats particles as being encapsulated by a cube, with the fraction of the
volume of a particle divided into fluid cells being equal to the fraction of
the volume of the cube located in the cells. This methodology provides
stability and accuracy to voidage calculation in
rectangular grids, yet it is not readily applicable to arbitrarily shaped fluid
cells, since it is computationally expensive to divide the volume of a cube
amongst arbitrarily shaped cells.

Thus a 3D cylindrical
DEM-CFD model was developed here, using novel methods to address the issues of
fluid cell volumes and voidage calculation associated
with complex geometry models. A structured CFD grid in cylindrical coordinates was
used, in which the length of cells in the radial direction remained constant,
angle subtended by cells decreased with distance from the centre of the
cylinder. This CFD grid conformation allowed the fluid cells to be kept with
constant volume and reasonable shapes, to ensure the volume averaged fluid
equations would be satisfied, while still fitting a 3D cylindrical geometry.
Additionally, a novel voidage calculation technique,
referred to as the ?square grid? technique, was developed to stably and accurately
calculate voidage in the cylindrical fluid cells. In
this technique, voidage was first calculated on a
square grid, using the cuboid approximation [5] to divide particle volume
between square cells. Then, based on the fraction of each cylindrical fluid
cell which lay in each square cell, calculated during initialisation, the voidage in each square cell was translated to calculate voidage in each cylindrical fluid cell. Figure 1 shows the
cylindrical fluid grid setup with the square grid for calculating voidage overlaid.

These novel techniques
were built on top of the rectangular DEM-CFD code described in Müller et al.[6].
Particles were still modelled in rectangular coordinates, yet contained by
cylindrical walls. Fluid particle interaction was modelled using the Beetstra et al.
[7] drag law. Fluid velocities were converted into rectangular coordinates at
the position of the particle of interest, in order to calculate the relative
velocity between the fluid and the particle for calculating drag on the
particles. The drag force was converted back into cylindrical coordinates to
calculate the cumulative drag on the fluid.

Figure 2 shows instantaneous voidage
and vertical fluid velocity maps of steady state flow through a packed bed
using the square grid, grouping and direct voidage
calculation methods. While the voidage calculations
are very similar, instabilities in the direct method caused even by minor
movement in particles between cells in a packed bed, make its simulated fluid
velocity unphysical.

Figure 3 shows instantaneous and time-averaged voidage and vertical fluid velocity maps in a short
bubbling fluidized bed using the square grid and grouping methods for
calculating voidage. While both have similar voidage results and time-averaged fluid velocity results,
only the square grid method shows the stability to physically simulate
instantaneous fluid velocity.

The 3D cylindrical DEM-CFD model was validated by comparison
to MR experiments. Figure 4 shows a comparison of time-averaged particle
velocity in a short bubbling bed; this comparison is discussed further
elsewhere [8]. Additionally, DEM-CFD results for slug rise velocity in a tall
slugging bed were compared to MR measurements [9] and theoretical values.
Simulated slugs had rise velocities of 0.23 m/s and 0.25 m/s at locations 85 mm
and 50 mm above the distributor in a 24 mm diameter bed with a settled height of
350 mm with an excess gas velocity U-Umf = 0.1 m/s. These results match those
from MR experiments of the same bed [9], which determined that at these heights
in this range of excess gas velocity, slug rise velocities should lie in between the theoretical values for wall and
axisymmetric slugs of 0.19 m/s and 0.30 m/s respectively.

The validation of the 3D cylindrical DEM-CFD model by
experimental results in vastly different fluidized bed systems lends confidence
to its use in predicting behaviour in a wide variety of fluidized bed systems.

References:

[1]
Guo, Y., Wu, C.Y., Thornton, C. (2012). A.I.Ch.E. J., 59, 1075-1087.

[2]
Chu, K.W., Yu, A.B. (2008a). Powder Technology, 179, 104-114.

[3]
Anderson, T. B., & Jackson, R. (1967). Industrial & Engineering
Chemistry Fundamentals, 6, 527?539.

[4]
Yu, A.B. (2006), A.I.Ch.E. J., 52, 496-509.

[5]
Khawaja, H.A., Scott, S.A., Virk,
M.S., & Moatamedi, M. (2012). The Journal of
Computational Multiphase Flows, 4, 183-192.

[6]
Müller, C.R., Scott, S.A., Holland, D.J., Clarke, B.C., Sederman,
A.J., Dennis, J.S., Gladden, L.F. (2009). Particuology,
7, 297-306.

[7]
Beetstra, R., van der Hoef,
M. A., & Kuipers, J. A. M. (2007). AIChE J., 53, 489?501.

[8]
Boyce, C.M., Holland, D.J., Scott, S.A., Dennis, J.S. (submitted 2013). Phys Rev E.

[9]
Müller, C.R., Davidson, J.F., Dennis, J.S., Fennell, P.S., Gladden, L.F, Hayhurst, A.N., Mantle, M.D., Rees, A.C., Sederman, A.J. (2007). Chemical Engineering Science, 62,
82-93.

Figures:

Figure 1 Novel cylindrical CFD
grid, each colour representing a different fluid cell, with square grid for voidage calculation overlaid.

Figure 2.
Comparison of voidage calculation methods for packed
bed flow.

Figure 3.
Comparison of voidage calculation
methods for instantaneous (left) and time-averaged (right) fluidized bed flow.

Figure 4 Comparison of time-average axial
particle velocity images from: (a) experimental MR imaging and (b) DEM-CFD
simulation of a bubbling fluidized bed. A difference map is shown in (c) (Reproduced
from Boyce et al. [8])