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

#### AIChE Annual Meeting

#### 2013

#### 2013 AIChE Annual Meeting

#### Particle Technology Forum

#### Dynamics and Modeling of Particulate Systems III

#### Monday, November 4, 2013 - 3:33pm to 3:51pm

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. An additional issue

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:

*ε*

_{cell}= 1 ? (Σ

*V*)/

_{particles}*V*where

_{cell}*ε*

_{cell}is the voidage of the cell, Σ

*V*

_{particles}is the total volume of all the particles in the cell and

*V*

_{cell}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-U*= 0.1 m/s. These results match those

_{mf}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])