Pore scale study of fluid flow and drag force in randomly packed beds

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Thermodynamics, Transport, and Fluid Mechanics

Pore scale study of fluid flow and drag force in randomly packed beds of different porosities Yongli Wu, Qinfu Hou, and Aibing Yu Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b06418 • Publication Date (Web): 05 Mar 2019 Downloaded from http://pubs.acs.org on March 6, 2019

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Industrial & Engineering Chemistry Research

Pore scale study of fluid flow and drag force in randomly packed beds of different porosities Yongli Wu,1 Qinfu Hou,1 and Aibing Yu1,2* 1ARC

2Centre

Research Hub for Computational Particle Technology, Department of Chemical Engineering, Monash University, VIC 3800, Australia

for Simulation and Modelling of Particulate Systems, Southeast University - Monash University Joint Research Institute, Suzhou 215123, PR China

Abstract Understanding the relationship between a fluid flow and the structural properties of a bed is important for various industrial applications and a proper description of particle-fluid flow. This work presents a pore scale study of fluid flow through randomly packed beds by a network model. The model incorporates the inertial effect, thus can be applied to flows of high Reynolds numbers. The results show the structure heterogeneity associated with the connection of pores is important in determining the distribution of fluid flow. The statistical distribution of normalized drag forces on individual particles is Gaussian, related to the statistical distribution of pore properties. The mean drag force of a bed by the pore scale model agrees well with that from a previous study by the lattice-Boltzmann method. These findings support the application of the pore network model for simulating particle-fluid flows under a wider range of Reynolds numbers.

Keywords: packed bed, pore scale study, pore network model, drag force, particle-fluid flow

Corresponding authors. Emails: [email protected] (QFH) and [email protected] (ABY)

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Graphical abstract

A packed bed

Drag force

Delaunay tessellation

Pore pressure and fluid flow


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1. Introduction Fluid flow through packed beds or porous media has been a very important and attractive phenomenon for decades.1-3 On the one hand, understanding this phenomenon is beneficial to various industrial applications, such as oil and gas production,4 heat and mass transfer in packed bed reactors,5-7 heap leaching,8 blast furnace ironmaking,9-11 and the electrochemical characteristics of solid oxide fuel cells.12 On the other hand, the relationship between structural properties and fluid flow in packed beds is of fundamental importance in describing particlefluid flows.13-20 Due to the complexity of the inner structure of packed beds or porous media, experimental and theoretical studies often assume the bed as a homogeneous system, and mainly focus on bulk properties.21 With the development of some advanced experimental techniques,3,22-24 the local features of packed beds and fluid flow can be depicted, whereas it is still expensive and laborious to investigate key variables comprehensively through experiments. By comparison, numerical simulations are promising, as they are cost-effective and can offer the details of fluid flows within the bed. In the past decades, there are many numerical simulations in this area, and they can be generally classified into three groups according to the computational approaches adopted. The first group is based on the traditional mesh-based approaches using computational fluid dynamics (CFD).25-31 These Eulerian methods are advantageous in solving the Navier–Stokes equations, and the flow dynamics even at a subparticle/sub-pore level can be obtained with fine meshes.16,32 However, the meshing of the complicated pore space can be very challenging, and fine meshing can lead to a high computational cost. The second group is based on direct numerical simulations (DNS), using latticeBoltzmann method (LBM),5,15,17,33-38 smoothed particle hydrodynamics,39-41 immersed boundary method,6 etc. The governing equations of mass and momentum balances are solved at a sub-particle or sub-pore scale, much finer than the particle or pore space. Due to the fine 3 ACS Paragon Plus Environment


scale, the detailed and accurate fluid flow can be obtained. For instance, the fluid velocity distribution, as well as the effects of bed structures and flow conditions on fluid flows can be readily examined.17 However, this group of methods is computationally expensive, and upscaling is often required to sum or average the sub-particle/sub-pore scale information to a particle/pore scale for the information of general interest. The third group is based on the pore network model.2-4,42-44 For this method, the void space of a packed bed or a porous medium can be represented by a network of nodes and bonds. The nodes in the network correspond to pore bodies, and the bonds connecting the nodes correspond to pore throats. Such a topological network represents essential features of the porous media. The heterogeneity and interconnectivity of the media can be well reflected. The information offered by this method at the pore scale is of general interest. Also, it is computationally more efficient than DNS. A lot of progress has been made for the pore network model since it was introduced by Fatt.45 Generally, the description of the pore geometry becomes more realistic, and the method has also been applied to understand various problems, as reviewed by Blunt et al.4 Among those studies, many continuous efforts were dedicated to the relationship between pore structures and flow properties of a bed.1,23,46-48 Although the large scale flow dynamics such as the permeability and flow dispersion are well understood, there are controversies about the flow dynamics at the scale of individual pores. For example, the distribution of flow velocity inside a bed has either a single peak or two peaks, but there is limited understanding of these different observations.47,49,50 It is believed that different packed bed structures may give different flow distributions. However, the fluid flow within loosely packed beds of spheres is rarely studied because the bed porosity higher than 0.5 is hard to be achieved for the random packing of monosized coarse spheres.51 Recently, with the variation of particle cohesion and packing conditions, packing structures of spheres with a wide range of porosities can be generated numerically.5254

It is known that those packings of cohesive particles actually have different microscopic 4 ACS Paragon Plus Environment

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properties compared with those of coarse particles.52,54 Nonetheless, it is not clear how those packing structures affect the fluid flows. In addition, the pore network model which solves the fluid flow at a pore scale is at the equivalent scale to the discrete element method (DEM)55 which solves the solid dynamics at a particle scale. Hence, it is promising to combine the pore network model and the particle scale DEM to simulate particle-fluid flows, as demonstrated in several recent studies.56-60 But these studies were mainly limited to low Re conditions, due to the use of a linear Hagen-Poiseuille equation or the neglect of the inertial effect in the momentum equation. Although there are some studies which considered the inertial effect in the pore network model for the application to the so-called non-Darcy flows under high Re conditions,61,62 the effect of such fluid flows (under high Re) on particles has not been investigated. In fact, the fluid induced forces like the drag force on particles can be critical for the coupling of DEM and the pore network model. It is thus essential to examine the drag force calculated by the pore network model under different flow conditions (with varying Re) for the application to particle-fluid flows in a wide range of Re. In this work, a pore scale network model is developed to study the fluid flow through randomly packed beds. The model incorporates an inertial term into the Hagen-Poiseuille equation for the application to high Re conditions. The packed beds of a wide range of porosity are generated by DEM simulations with different material properties and packing conditions. Thus, the effects of Re and bed structures on fluid flow and drag force are explicitly investigated. The paper is organized as follows. First, the pore scale network model and simulation conditions are introduced in Sections 2 and 3. Then, after the model validation, the relationships between the pore scale structural properties and fluid flow/drag force are discussed in Section 4. At last, a summary of the work is given in Section 5.



2. Model description In the pore network model, the mass and momentum balances of fluid flow are established on the pore units of the packed bed. Each pore unit corresponds to a tetrahedron obtained by Delaunay tessellation. As shown in Figure 1, the bed space is tessellated into a series of tetrahedra based on the locations of particle centers. Note that these tetrahedron meshes are generated with periodic boundary conditions (Figure 1b) as used by Thompson.63 The final periodic meshes are composed of tetrahedra, each of which has at least one vertice (particle) in the primary domain (the red domain in Figure 1b).

(b)

(a)

(c)

Figure 1. Delaunay tessellation: (a) a packed bed of particles, (b) a 2D diagram for the periodic boundary conditions, where the primary domain is in red and the adjacent domains are in blue (there are eight adjacent domains in 2D and 26 adjacent domains in 3D) and (c) tetrahedron meshes. It is noted that these periodic meshes own the space-filling feature, e.g., the meshes at the left boundaries can be well connected with those at the right boundaries with common faces. 6 ACS Paragon Plus Environment

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This feature allows the realization of periodic fluid flows. Both the mass and momentum transfers of fluid flows are related to the connection between a pore and its neighbors. Therefore, the neighbors of each tetrahedron are critical for the periodic fluid flow. Here, the neighbor of a tetrahedron is defined as the one that shares a common face composed of three common particles with it. Note that the identity of a particle used for determining the neighbors is its original identity in the primary domain. Hence, even two tetrahedra at different sides of the domain can still be the neighbor to each other. The well-constructed neighbor list ensures the following periodic pore scale mass and momentum balances, thus the periodic fluid flow is realized. Figure 2 illustrates a pore unit and the connection of two adjacent pores. The four vertices of a Delaunay tetrahedron are at the centers of four spheres (Figure 2a). The void space located in the center of the four spheres is the pore space. Besides, at each face of the tetrahedron, there is a free area (

) that is not taken up by the particles (Figure 2b). The connection or the fluid

flow between two neighboring pores is through such free areas, as shown in Figure 2c. The socalled pore throat is the tube connecting the centers of the two neighboring pores. It is assumed to be cylindrical with the effective throat diameter

, given by: (1)

= Another important parameter is the throat length

which is the distance between the centers

of two neighboring pores. With these parameters, the relationship between the fluid flow and the pressure of the two pores can be established based on the Hagen-Poiseuille equation, given as: =

(2)

2

Note that the real geometry of a pore throat is not cylindrical, but with a converging-diverging shape.43 Hence, the effective throat diameter

is modified as

by a shape factor

(

=

) to describe the pore throat as a hyperbolic tube. In addition, Equation (2) only considers 7 ACS Paragon Plus Environment


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the viscous effect and is restricted to low Re. To consider the kinetic energy loss or the inertial effect, a nonlinear term should be added as introduced by Ergun and Orning,64 given as: =

2

+2

2

(3)

Pore throat

Pore i Pore j (a)

(c)

(b)

Figure 2. Diagram of a pore unit (a), the effective throat diameter

based on the void area

of each face (b), and the pore connection (c). Moreover, under the steady flow condition, the mass balance of fluid flow in and out of each pore requires: 4

(4)

=0

=1

where the sum runs over the four throats connecting pore i and adjacent pore j (j=1 to 4). Based on Equations (3) and (4), a set of equations of fluid velocity and pressure can be established. Theoretically, these nonlinear equations can be solved by Newton’s method. However, due to the large set of equations, it is difficult to obtain a converged solution. Here, we use an iterative procedure for the solution. To begin with, Equation (3) is re-written as: =

where

1

=

2

,

2

=

2

2

and

(

2

+

2

2

)

=(

1

+

2)

(5)

. k1 is just related to fluid viscosity and the

=

geometry of the pore throats, thus can be calculated explicitly. However, k2 is related to fluid velocity (Re), thus it is implicit in the computation. To solve this, the procedure in Figure 3 is 8 ACS Paragon Plus Environment

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used. First, k2 for each equation is obtained by assuming the initial fluid velocity as zero or using the fluid velocity from the previous iteration. Then, the set of equations is linear at each iteration and can be solved by algorithms like the Gauss-Seidel method or using some efficient solvers like the PARDISO of the Intel® Math Kernal Library (we used the PARDISO solver in this work). Finally, the pressure and fluid velocity are obtained when the iterative calculation is converged (the converging factor in Figure 3 is set to be 1.0×10-3 in this work). Start

Calculate k1, and set k2 = 0

Solve the linear equations and obtain initial velocity U0

Use U0 to obtain k2

Set U0 = U

Solve the linear equations based on k1 and k2, and obtain the updated velocity U

Converged ?

No

)

( Yes

End and output U

Figure 3. The procedure of solving the nonlinear equations.

3. Simulation conditions In this study, packed beds of dimensions 8d×8d×8d (d is particle diameter, and its default value is set as 0.01m) are extracted from the packings obtained by DEM simulations.52,53,65,66 By varying material properties (e.g., particle cohesion52) and packing conditions (e.g., 9 ACS Paragon Plus Environment


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mechanical vibrations66), the packings with porosity from 0.260 to 0.761 and mean particle contact from 2.4 to 12.052,54,66 are achieved, thus enabling the investigation of the effect of bed structure on flow properties. It is noted that the spheres in these packed beds are static without any movements in the simulations.

Pressure (Pa) Pressure (Pa)

(a)

(b)

Figure 4. Solved pressure fields of a packed bed ( = 0.761) under different pressure gradients: (a) !" = 0.0047 Pa/m, (b) !" = 2.73 Pa/m. The points represent the centers of pores. The simulation is performed through giving different pressures at the top (low pressure) and the bottom (high pressure) of a packed bed, hence a pressure gradient is introduced along the vertical (Z) direction of the packed bed. Under such a pressure difference, the fluid (density #

= 1.25 kg m-3, viscosity

= 1.8 × 10

5

kg m-1 s-1) can flow into the bed from the bottom.

Figure 4 gives two instances of the solved pressure fields for a packed bed under different pressure gradients. Note that the points in Figure 4 represent the pore centers, instead of spheres in the packed bed. The solved pressure varies linearly along the Z direction for both instances. This indicates that the pressure loss is linear along the Z direction of the uniform packed bed even under different the pressure gradients. After the pressure at each pore center is obtained, the fluid flux or velocity can be obtained based on Equation (5). In addition, desired fluid velocities and Reynolds (Re) numbers of the 10 ACS Paragon Plus Environment

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bed can be achieved by adjusting the pressure gradients. For example, the first instance in Figure 4 corresponds to the Re = 0.5, and the second one corresponds to Re = 87.

4. Results and discussion 4.1. Model validation Since the present model computes the pressure drop and fluid flow, it can be naturally validated via the comparison between the computed data and the experimental data in terms of the permeability and the frictional factor. 106

Permeability, k (Darcy)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


105

Ergun and Orning (1949) Beavers et al.(1973) Chu and Ng (1989) Present model

104 103 102 101 10-1

100

101

Particle diameter, d (mm)

Figure 5. Comparison of the calculated permeabilities by the present model with those measured permeabilities by experiments (adapted in part with permission from Ergun and Orning.64 Copyright 1949 American Chemical Society; Beavers et al.67 Copyright 1973 American Society of Mechanical Engineers ASME; and Chu and Ng.68 Copyright 1989 John Wiley and Sons) for random packings of equal spheres. The permeability of a bed gives the basic relationship between the fluid flow and structural properties. There were experimental data for the relationship between permeability and particle diameter for dense packed beds of equal spheres.42 In this work, the permeability k of the bed can be calculated according to Darcy’s law:



=

(6)

(

where U is the superficial fluid velocity corresponding to the pressure gradient ( of the bed (equivalent to

/L). Note that the Re is smaller than 1.0 by using a small ( in the calculation,

thus the Darcy’s law is applicable. Then, the calculated permeabilities for dense beds of different particle diameters are compared with experimental data. Note that the porosity of dense beds used here is about 0.360 which is similar to the porosity of Finney’s packing as used by Bryant et al.42 Figure 5 shows the calculated permeabilities agree well with those measured from experiments. 106

Rumpf and Gupte, = 0.41 Linear model, = 0.41 Present model, = 0.41 Rumpf and Gupte, = 0.64 Linear model, = 0.64 Present model, = 0.64

5

10

Frictional factor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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104 103 102 101 100 10-1 10-2

10-1

100

101

102

Re

Figure 6. Relationship between the frictional factor (

/

2L)

and Re for different bed

porosities (Experimental data were adapted in part with permission from Rumpf and Gupte.69 Copyright 1971 John Wiley and Sons). In addition, the relationship between the frictional factor (defined as

/

2L,

see

Gibilaro et al.70) and Re is concerned in particular in understanding the drag correlations for particle-fluid interactions.70 Figure 6 shows the results calculated by the present model are consistent with the experimental data by Rumpf and Gupte69 even for high Re (~ 100). However, if only the linear term is considered for the momentum equation (neglecting the nonlinear inertial term in Equation (3)), the frictional factor obtained is smaller than the experimental data 12 ACS Paragon Plus Environment

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when Re > 10. This illustrates that the inertial effect is important and should be incorporated into the momentum equation, as done in the present model (Equation (3)). Overall, the comparable results in terms of the permeability and the frictional factor validate the present model, giving the basis for the following examination of fluid flows and drag forces of different bed structures. 4.2. Pore structure Since the simulated fluid flow in this study is directly related to the bed structure, the networks of pore throats are characterized, as shown in Figure 7. Here, each stick represents the connection of two pores and its thickness represents the effective throat diameter. The pore connections and pore throats change noticeably with the bed porosity. Generally, a bed with a higher porosity tends to possess more pore throats of large sizes, and the pore structure becomes increasingly uniform with the decrease of bed porosity. This can be clearly observed from the probability distribution of the effective throat diameter (Figure 8). There are two peaks with the first located in Di,j/d < 0.5 and the second located in 0.5 < Di,j/d < 1. The first peak corresponds to the small throat formed in the dense packing structure where particles are in close contact, whereas the second one corresponds to the large throat in the loose packing structure where particles may not necessarily contact each other. With the decrease of the bed porosity, the second peak becomes weaker, and the first peak becomes higher. Also, the distribution varies from a wide Gaussian to a narrow exponential distribution with the decrease of the bed porosity. The increasingly narrower distribution indicates the throat size becomes more uniform. Such statistical features are consistent with those observations in Figure 7, both indicating that the pore structure is closely related to the bed porosity.



(a)

(b)

(c)

(d)

Figure 7. Networks of pore throats for packed beds of: (a) = 0.370, (b) = 0.469, (c) = 0.557, (d) = 0.761. 6 = 0.370 = 0.469 = 0.557 = 0.761

5

P(Di,j /d)

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4 3 2 1 0 0

0.5

1

1.5

2

D /d i,j

Figure 8. Distribution of effective throat diameter for different bed porosities. 14 ACS Paragon Plus Environment

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As stated, the packings used in this study were generated using different packing methods and material properties. However, it was found that the structures of static packings of monosized spheres at a same porosity are largely similar, despite different packing methods.54,71,72 To further confirm this observation, the effect of packing methods on pore properties of packings at a same porosity is examined below. Table 1. Cases used for structural comparison. (kg/m3)

Bo

Packing method

0.45

20

Packing in air

-

0.45

3

Settling in liquids

2300

0.58

105

Packing in air

-

0.58

3

Settling in liquids

2400

0.72

422

Packing in air

-

0.72

86

Settling in liquids

2400

f

As shown in Table 1, there are three pairs of cases, where each pair of cases have the same porosity achieved by using different particle cohesion (characterized by bond number Bo, i.e., )* =

+,-. /0

) and different packing methods (like packing in air52 or settling in liquids54). The

distribution curves of pore properties of these packings are compared in Figure 9. Only slight differences can be observed for the pairs with a high porosity (e.g., R 0.58), i.e., the curve of the one from settling of less cohesive particles has a higher peak or slightly shifts to the left, compared with the other one of more cohesive particles. However, the distribution curves of each pair are very similar with the main features kept the same. This illustrates the effect of different packing methods at a same porosity is not significant for our system with static monosized spheres. Since those structures are largely similar for a given porosity, one packing structure is chosen for each porosity point in the following discussions.



5 = 0.45 - #1 = 0.45 - #2 = 0.58 - #1 = 0.58 - #2 = 0.72 - #1 = 0.72 - #2

4

P(Di,j/d)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3 2 1 0 0

0.5

1

1.5

2

D /d i,j

Figure 9. Comparison of pore properties for the beds in Table 1 obtained under different packing conditions. 4.3. Fluid flow and bed structure With the pore scale model, fluid flows around individual particles can be determined by the fluid flows at those pore throats adjacent to them. Here, a group of particles at the bed center and their surrounding flow networks are illustrated in Figure 10. The flow networks are geometrically the same as the networks of pore throats (Figure 7), and the color represents the magnitude of fluid velocity. The fluid velocity tends to be larger at those throats aligned along the vertical pressure gradient. For those horizontal throats, the velocities are very small regardless of their sizes.


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|V|/|Vmax| (a)

(b)

(c)

(d)

1.000 0.857 0.714 0.571 0.429 0.286 0.143 0

Figure 10. Flow network for a group of particles in a packed bed of: (a) = 0.370, (b) = 0.469, (c) = 0.557, (d) = 0.761. The stick thickness represents the throat size and its color represents the fluid velocity. The red sphere is the particle at the bed center, and those white spheres are the Delaunay neighbors of the center one. Note that the diameter of the particle is reduced in the figure for a better visualization.



80

(a)

(b)

60 40 V/

20

(degree)

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2

0 80

(c)

1.5

(d)

60

1

40

0.5

20

0

0 0

0.5

1

1.5

2

2.5 0

0.5

1

1.5

2

2.5

Di,j / Figure 11. Distribution of fluid velocity in terms of the normalized throat diameter and throat orientation angle

1

2

for a packed bed of: (a) = 0.370, (b) = 0.469, (c) = 0.557, (d) = 0.761.

Since the throat orientation is so important, it is quantified by an angle , defined as the angle between a vertically upward vector and the vector along the pore throat; 90°, and a larger

is from 0° to

corresponds to a more horizontal throat orientation. The throat orientation

is related to the pore-to-pore connection, reflecting the structure heterogeneity. Then, the distribution of fluid velocity in terms of the normalized throat diameter orientation angle

1

2 and the throat

is given in Figure 11. A large throat diameter does not always correspond

to a large fluid velocity. As shown in Figures 11c-d, a large fluid velocity is likely to associate with a large

1

2 and a small . In particular, when the porosity is low (Figure 11a), a large

fluid velocity could be observed for the combination of a small

and a small throat diameter.

This is because throat diameters tend to be uniform with a small deviation when the porosity is low (Figures 7 and 8); the fluid flow is thus mainly affected by the throat orientation. Moreover, the throat diameter is closely related to the local porosity, i.e., a large throat diameter 18 ACS Paragon Plus Environment

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corresponds to a high local porosity around the neighboring particles. Hence, from Figure 11, it is revealed that in addition to local porosity (related to throat diameter), the structure heterogeneity is also critical in determining the fluid flows around particles. 0.8

(a)

= 0.370 = 0.469 = 0.557 = 0.761

P(Vz /)

0.6

0.4

0.2

0 -1

0

1

2

3

4

Vz / 1 Re = 0.01 Re = 10 Re = 40 Re = 320

(b) 0.8

P(Vz /)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 0.4 0.2 0 -1

0

1

2

3

4

5

Vz / Figure 12. Probability distribution of normalized velocity

(34 1342) for: (a) different bed

porosities (Re = 1); (b) different Reynolds numbers Re ( = 0.761). The statistical distribution of fluid flow is further analyzed here. Figure 12a gives the probability distributions of normalized fluid velocity along Z direction

(34 1342)

with

different bed porosities . On the one hand, the distributions share an exponential decay trend, similar to those observed in previous experimental22 and LBM17 studies. On the other hand, the 19 ACS Paragon Plus Environment


distributions of different bed porosities are almost identical. This is different from the previous observations of LBM simulations17 where the distribution varies from a wide bi-mode distribution to a narrow single-mode distribution. Such a difference is attributed to the different computational scales of the two methods. LBM is at a sub-pore scale, and the detailed fluid flows within the inner space of pores are depicted;17 the effect of different pore structures (bed porosities) on

(34 1342) can thus be well captured. By contrast, the fluid flow by the pore

network model is depicted at a pore level. There is just one velocity vector corresponding to each pore throat regardless of the throat size (Figure 10). Therefore, the effect of different pore structures on

(34 1342) is not captured by the by pore network model.

In addition,

(34 1342) is affected by different Re. As shown in Figure 12b, the peak of

(34 1342) decreases with the increase of Re. Generally, a higher peak of (34 1342) indicates a more uniform distribution of fluid flow. This is consistent with previous findings.17 Nevertheless, the effect of Re on effect of Re on

(34 1342) is not very strong. Further understanding to the

(34 1342) at the pore scale is still necessary in future work. 0.8 = 0.370 = 0.469 = 0.557 = 0.761

0.6

P(Q/)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4

0.2

0 -1

0

1

2

3

4

Q/ Figure 13. Probability distribution of normalized fluid flux porosities . 20 ACS Paragon Plus Environment

( 1 2) for different bed

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In fact, for the pore scale model, the relationship between flow dynamics and pore structures can be more clearly indicated by the distribution of fluid flux Q. The reason is that Q represents the collective information of structure (throat area) and fluid velocity at each pore throat. Here, the distribution of normalized fluid flux different bed porosities.

( 1 2) is given in Figure 13 for

( 1 2) varies from a two-peak distribution to a one-peak distribution

with the decrease of the bed porosity. Such an observation is consistent with the recent work by Alim et al.,50 who performed both simulation and mathematical analysis of 2D porous media; and the transition from a two-peak to a one-peak distribution was observed when the packing varies from an ordered to a disordered state. The radial distribution function g(r), extensively used in the past to characterize the interparticle correlations and local configurations in a packing,52,54,73-75 is employed to further understand the two-peak to one-peak transition of

( 1 2) in a 3D structure. It is defined as

the probability of finding a particle center at a given distance from a reference one, given by g (r) =

78 9 92:9

, where

(r) is the number of particle centers situated at a distance between r and

0

rVWr from the center of a given particle and

0 is the average number of particle per unit volume

in the packing. As shown in Figure 14a, g(r) has the peak at r = 1.0 regardless of the bed porosity; this peak represents the structure of two particles in contact, which generally exists in a packing. However, g(r) for

= 0.370 has a splitting second peak at r X 1.7 and r = 2.0, respectively.

According to Clarke and Jonsson,73 the first subpeak (r X 1.7) is resulted from two local structures denoted by the 211 configuration (edge-sharing in-plane equilateral triangles) and the 333 configuration (face sharing tetrahedra), as shown in Figure 13b. The second subpeak (r = 2.0) is mainly from the 100 configuration (three particles along a line). Such short-ranged ordered features increase the structural heterogeneity of a random packing. Especially, the structures correspond to the first subpeak (211 and 333) can form pore units of regular 21 ACS Paragon Plus Environment


tetrahedral shape; their existence in a random packing may account for the two-peak

( 1 2) for = 0.370. Besides, with the increase of the bed porosity, the

distribution of

splitting second peak disappears gradually, and the first peak becomes narrower. These structural variations indicate the interparticle correlations become weak and the bed structure becomes more random and uniform, which explains why

( 1 2) becomes more uniform

with a single peak when the bed porosity increases. Hence, the transition from a two-peak to a one-peak distribution of the fluid flux (Figure 13) with increasing porosity is resulted from the decreasing heterogeneity (indicated by the splitting second peak of g(r)) of the bed structure. 2

= 0.370 1

100

0 2

= 0.400 1

Radial distribution function, g(r)

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211

0 2

= 0.469 1 0 2

= 0.557 1

333

0 2

= 0.761 1 0 0

1

2

3

4

5

Radial distance, r (particle diameter)

(a)

(b)

Figure 14. Radial distribution function g(r): (a) for different bed porosities; (b) three typical local structures lead to the second peak in g(r),73 where the red particles represent the common neighbors in the structures. 22 ACS Paragon Plus Environment

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4.4. Drag force Fluids experience pressure losses when they flow through a packed bed of particles. At the pore scale, the pressure loss :

of fluids through a pore throat ij (connecting pores i and j) is

mainly resulted from two particle-fluid interaction forces,57 i.e., ;"

induced by pressure

gradient and ;- due to local frictional drag, expressed as: :

= (;" + ;- )

Pore scale study of fluid flow and drag force in randomly packed beds

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