A Coupled DEM and CFD Simulation of Flow Field ... - ACS Publications

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Ind. Eng. Chem. Res. 2009, 48, 4060–4074

A Coupled DEM and CFD Simulation of Flow Field and Pressure Drop in Fixed Bed Reactor with Randomly Packed Catalyst Particles Hua Bai,* Jo¨rg Theuerkauf, and Paul A. Gillis The Dow Chemical Company, 2301 N. Brazosport BlVd., Freeport, Texas 77541

Paul M. Witt The Dow Chemical Company, Midland, Michigan 48674

Packed bed unit operations are required for many commercial chemical processes. The ability to a priori predict void fraction and pressure drop in a packed bed would significantly improve reactor design as well as allow for optimization around catalyst performance, catalyst design, and the resulting process pressure drop. Traditionally, the packed bed reactor designs are based on a homogeneous model with averaged empirical correlations. These correlations are often inapplicable for low tube-to-particle diameter ratios (D/d < 4) in which tube wall and local phenomena dominate. In this work, the discrete element method (DEM) and computational fluid dynamics (CFD) are coupled to model a fixed bed reactor with low tube-to-particle diameter ratios (D/d < 4). DEM is used to generate a realistic random packing structure for the packed bed with spherical or cylindrical particles, which is then imported into the CFD preprocessor (Gambit) to generate the mesh for the CFD simulation. Two types of experiments were conducted: the laboratory-scale experiments with up to ∼150 particles to allow simulation of entire packed beds in CFD, including random packing and structured packing, and the plant-scale experiments conducted with up to ∼1500 randomly packed particles. The concept of a “porosity correction factor” was introduced to compensate for the effect of porosity deviation between actual packing and the CFD model, which can be accumulated during DEM simulation, and particle shrinkage for purposes of grid generation in the CFD model. The predicted pressure drops match well with the experimental measurements with errors less than the desired limit (10%) for industrial design of packed bed reactors. The pressure drops calculated by the empirical correlations confirmed the inconsistency and unreliability of the empirical correlations for the packed beds with low tube-to-particle diameter ratios (D/d < 4) as well as the advantage of the DEM/CFD approach. 1. Introduction Traditionally, packed bed reactors are designed by a trialand-error process. The packing void fraction and the pressure drop across the packed bed are two critical variables for the design, and they are usually predicted using empirical correlations, such as the Ergun equation for pressure drop1 and the Leva2 or Dixon3 equation for void fraction (packing porosity). A small error in the predicted void fraction can translate to a significant error in pressure drop prediction. Because of this inherent error in the pressure drop prediction, many packed beds are oversized with a capacity safety factor. An alternate method to handle the expected error is to conduct specific pilot plant tests with representative particles. Given the packing material and the correct tube diameter, the Ergun equation can be fitted to the test data by varying the Ergun equation constants as well as the effective particle diameter and void fraction. If the actual tube-to-particle diameter (D/d) combination is used in the test rig, the predicted pressure drop from the modified Ergun equation can be sufficient. However, these tests are expensive and require significant prework to investigate. Typical catalyst supports are alumina or silica which can break and degrade after pouring. To minimize the effects of broken particles, the tests need to be completed with fresh particles, which may require drums of catalyst to be tested. Because of this, there is a significant limitation on the variations of catalyst shapes and sizes that can be reviewed for optimal performance. * To whom correspondence should be addressed. Tel.: (979) 2387621. Fax: (979) 238-7463. E-mail: [email protected].

The empirical correlations are usually based on homogeneous assumptions with averaged characteristics; thus they often do not apply to the packed beds with low D/d ratios where the tube wall local phenomena dominate. The local changes in porosity can lead to large variations in the predicted velocity profile and therefore nonuniform head loss along the packed bed.4 Accurate prediction of local voidage is also important for predicting heat transfer in packed beds, which is critical for stability analysis5,6 and reactor control. In order to reflect the local effects, a spatially resolving three-dimensional (3D) flow simulation is needed. A first requirement for such a detailed simulation is an appropriate 3D representation of the geometric structure of the packing. The advent in high power computing technology has resulted in significant advancements in fundamental modeling which provide an alternative approach, by coupling DEM (discrete element method) and CFD (computational fluid dynamics) technologies to address this issue. DEM is used to generate a realistic packing structure for the packed bed. The simulated packing structure is then imported into the CFD preprocessor to generate a mesh for the CFD simulation. The flow through the space between particles in the packed bed is obtained by solving the transport equations. This yields a very detailed solution containing local values of all relevant variables such as pressure, velocity, shear stress, turbulence properties, and temperature. Such detailed information is of great importance in understanding the phenomena occurring in the packed bed. The ability to a priori predict void fraction and pressure drop in a packed bed would significantly improve reactor design as

10.1021/ie801548h CCC: $40.75  2009 American Chemical Society Published on Web 03/12/2009

Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4061

Figure 1. Photograph (a) and schematic (b) of the laboratory experiment setup.

well as allow for optimization around catalyst performance, catalyst design, and the resulting process pressure drop. In this work, the packing structure in cylindrical tubes randomly packed with spherical or cylindrical particles and the flow through the packing were modeled by coupling DEM and CFD. Two types of experiments were conducted: laboratory-scale experiments with less than 160 particles packed to allow modeling of entire packed beds, including random packing and structured packing, and plant-scale experiments conducted with up to ∼ 1500 randomly packed particles. The goal is to predict the packed bed pressure drop with an error of less than 10%. The work was focused on the packed beds with low tube-to-particle diameter ratio (D/d < 4) where empirical correlations often become unreliable.

years.10-15 In these published works, usually only a small number (a few to tens) of particles were modeled to reduce the requirements for computational resources, and the particles were packed in special patterns such as regular or periodic packing (also called “structured packing”) so that the packing structures can be directly reconstructed in CFD without DEM or other packing predictions. These works provided some insights into how the packing structure influences the transport characteristics. However, they are not applicable directly to industrial applications, where hundreds and thousands of particles are randomly packed typically. The lattice Boltzmann method has also been used to compute the flow through a packed bed of spheres.16-18 It has a potentially high efficiency which allows somewhat larger packing to be simulated, but it is still in its early stage and has some limitations such as difficult to handle energy balance.11

2. DEM and CFD DEM7 is an explicit numerical scheme which simulates the dynamic and static behavior of assemblies of particles based on contact mechanics. A soft particle model is used in the DEM to simulate the particle contact behavior with springs, dashpots, and frictional sliders. The motion of each particle is tracked, and interaction with other particles or boundaries is considered. Forces are calculated based on interaction of particles and the physical properties of the entities, including the hardness of particles, expressed with a spring, and the particle energy dissipation, expressed with a dampener or dashpot. The hardness of the particle is proportional to the Young’s modulus, while the dashpot is related to the coefficient of restitution. The friction between entities is defined with a Coulombic type of friction and implemented with a friction factor. Based on the physical properties the particle forces are calculated, which leads to new particle positions.7,8 The commercial DEM software package PFC3D by ITASCA9 was used in this work. Once the packing structure of the packed bed is predicted with DEM, it is then incorporated into CFD to simulate the flow through the bed. Using CFD to model the detailed flow in packed beds has gained increasing attention in the past few

3. Experiments Two types of experiments were conducted. One is the laboratory-scale experiment in which less than 160 particles were packed to allow modeling of the entire packed beds in CFD, including random packing and structured packing. With the structured packing simulation, particles are positioned in specific patterns and the CFD geometry can be directly built up without DEM simulation. This experiment is mainly for the purpose of validating the CFD model. The other type is the more realistic plant-scale experiment in which up to thousands of particles were randomly packed but only a segment of a entire packed bed could be included in the CFD simulations due to the limit in available computational resources. The plant-scale experiment provided data for investigation of pressure drop scale-up from a segment to the corresponding entire packed bed. 3.1. Experimental Setup. Figure 1 shows a photograph (a) and sketch (b) of the laboratory setup for the laboratory-scale experiment. The tubes for these experiments were made of acrylic so that the structure of the packing could be visualized. The air flow, directed downward through the packing, was controlled by a manual valve and measured by an Aquamatic

4062 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 Table 1. Experimental Cases and Measurements case label

packed bed tube diam (D), mm

D/d

LS-1

square duct, 20 × 20

2.00

LS-2

no. particles packed (N)

packing height (H), mm

flow rate (Q), kg/h

measd press. drop (∆P), Pa

16

40


32

80

LS-3


64

160

LU-1

26.67

2.67

153

312.7

LU-2

20.51

1.79

82

444.7

PU-1

21.41

2.14

799

1722.1

PU-2

26.12

2.61

916

2319.0

PU-3

32.28

3.23

1545

1742.4

10.41 17.69 20.81 26.43 10.41 17.69 20.81 26.43 10.41 17.69 20.81 26.43 10.41 17.69 20.81 26.43 10.41 16.65 20.81 3.30 6.59 9.89 3.65 7.31 10.96 5.75 11.49 17.24

862 2 482 3 516 5 723 1 986 4 964 6 895 11 238 3 123 9 515 14 548 24 270 343 977 1 398 2 285 5 594 16 138 26 242 1 686 6 461 14 178 884 3 491 7 593 1 581 5 965 12 945

AT21052 flow cell rotometer. A water manometer was used to measure pressure drop across the packing. The manometer measuring locations were more than 2 times the tube diameter from the packing to ensure stable manometer readings. The back pressure (or the pressure at the packing top) was also measured along with the pressure drop measurement. Two kinds of particles were used in the laboratory-scale experiment: 10 mm diameter steel balls for the packing with spherical particles and acrylic cylinders with a diameter of 10 mm and a length of 10 mm for the packing with cylindrical particles. The packing was supported by three pins that passed through the cross section of the column. This design was chosen to minimize pressure drop from the packing support. The plant-scale experiment had a setup similar to that in Figure 1 except that pressurized air (1.72 bar) was used and the packing tubes were constructed of steel; therefore, the packing structures were not visible. 3.2. Experimental Cases and Data. Table 1 summarizes the key parameters and measurements of both laboratory-scale and plant-scale experiments, including the tube diameter (D), particle diameter (d), and length of the cylindrical particles (h), the total number of particles in packing (N), the packing height (H), the flow rate (Q), and the measured pressure drop (∆P). Each case or setup was tested under three or four different flow rates. For each flow rate, the test was repeated at least three times to quantify experimental precision. The pressure drop shown in Table 1 is the average of the repeats. All experiments were conducted at ambient temperature (∼25 °C). The first three cases (LS-1, LS-2, and LS-3) in Table 1 are the laboratory-scale experiments using structured packing of spherical particles. The square duct, with internal dimensions of 20 mm by 20 mm, allows four 10 mm steel balls to be positioned in a simple cubic lattice structure, giving a tube-toparticle dimension ratio of 2.0. The three cases have 16, 32, and 64 particles, corresponding to 4, 8, and 16 packing layers. All layers have the same four-ball packing. This structured packing can be built in the CFD model without DEM simulation since all particles’ positions are known. These three structured packing cases were used to verify the CFD model since the

particles/packing/scale steel balls/structured/lab scale

steel balls/structured/lab scale

steel balls/structured/lab scale

steel balls/random/lab scale

cylinders/random/lab scale steel balls/random/plant scale steel balls/random/plant scale steel balls/random/plant scale

errors due to randomness of packing, DEM simulation, and use of a segment of entire packed bed were eliminated. The next case in Table 1 (LU-1) is a packed bed with 153 steel balls randomly packed in a 26.67 mm diameter tube, giving a tube-to-particle diameter ratio of 2.67. The relatively small number (153) of particles allowed the entire packed bed to be simulated in CFD so that the effect of using only a segment of entire bed could be eliminated. The last case (LU-2) of the laboratory-scale experiments is a packed bed with 82 cylindrical particles randomly packed in a tube with an inner diameter of 20.51 mm. Each cylindrical particle has 10 mm diameter and 10 mm length, and its effective particle diameter, dp, defined as the diameter of a sphere with the same volume as the particle, is 11.45 mm, giving a tube-to-particle diameter ratio of 1.79. The entire packed bed (LU-2) was included in the CFD simulation. The last three cases (PU-1, PU-2, and PU-3) in Table 1 are the plant-scale experiment with hundreds of (up to 1545) steel balls (10 mm diameter) randomly packed in three different steel tubes of diameters of 21.41, 26.12, and 32.28 mm, giving tube-to-particle diameter ratios of 2.14, 2.61, and 3.23, respectively. The large numbers (799, 916, and 1545) of particles, more realistically reflecting the plant operation conditions, were beyond the capability limit of the available computational resources. Therefore, only a segment of the entire packed bed with 125 particles was included in the corresponding CFD simulation. 3.3. Packed Bed and Packing Porosity. Figure 2 shows a few examples of packed beds investigated in this paper. Figure 2a shows a packed bed with structured packing of spherical particles. Figure 2b shows two packed beds with random packing of spherical particles,19,20 and Figure 2c shows two fixed beds packed with random packing of cylindrical particles.21 Marked in Figure 2 are the tube inner diameter (D), the height of the packed bed (H), the particle diameter (d), and the height of cylindrical particles (h). The packed bed porosity, ε, is defined as the fraction of the void volume within the packing height. For the packed beds with spherical particles, the packing porosity can be expressed as


ε)1-

3

2Nd 3D2H

(1)

For the packed beds with cylindrical particles, the packing porosity can be expressed as ε)1-

Nd2h D2H

(2)

where N is the total number of particles in the packing. 4. Models and Methodology The general procedure to model the flow field and pressure drop in a packed bed is briefly described below. First, the packing structure in a packed bed is simulated using PFC3D, a commercial DEM package.9 Then, the DEM-simulated packing is imported into Gambit,22 a commercial CFD preprocessor to create the geometry and mesh for CFD simulation. If a CFD simulation of the entire packed bed becomes unmanageable or beyond the limit of available computational resources (common for commercial-scale packed beds), a representative segment from the entire packed bed is selected for the mesh generation. Next, the flow field through the packing is solved using FLUENT,23 a commercial CFD package. Finally, the pressure drop across the packing is retrieved from the CFD simulation result. If only a segment of the entire packed bed is modeled, the CFD-predicted pressured drop needs to be scaled up for the entire packed bed.

4.1. Packed Bed Simulation with DEM. For the DEM simulation with spherical particles, the packing is generated by dropping spheres from a specified height into a tube. The launch position of the spheres at the specified height is randomly assigned in the tube to mimic the real filling procedure of the tube with particles. Figure 3a illustrates the filling process, while cross sections are shown in Figure 3b for D/d ) 2.6. Local differences in the packing can be observed. Further, the DEM packing (Figure 3c) is compared with the steel balls in a Plexiglas pipe (Figure 3d). Another example of particle packing is depicted for a D/d ) 2.2 in Figure 3e,f. The DEM result is a view from the top, while the experimental result is a side view. The packing structure is in very good agreement. For simulation with cylindrical particles, it is more complicated since the DEM code (PFC3D) allows only spheres as the basic particle geometry. If a shape other than a sphere is required, it has to be built by assembling spheres. With the spheres the cylinders are generated by only placing particles on the surface of the cylinder which is defined by the diameter and number of particles on the surface. The number of particles on the surface has an impact on the roughness. Since the spheres are staggered on the surface, “valleys” and “hills” can be found. However, the smoothness of the surface can be adjusted by the relative positions of the spheres and their diameters. A smoother surface can be achieved by more spheres with smaller diameters for the composition of one cylinder. This adds to the numerical complexity of the packing simulation. During the packing process of the cylinders in the tube, the surface structure can be important if the resolution of the particles is not high; e.g.,

Figure 2. Examples of packed beds with low tube-to-particle diameter ratios (D/d): (a) structured packing with spherical particles, (b) random packing with spherical particles, and (c) random packing with cylindrical particles.

4064 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009

Figure 3. Fixed beds with low tube-to-particle diameter ratios, randomly packed with spherical particles (DEM simulation and photos of experimental packing)

few spheres lead to a rough surface. If two cylinders are in contact and they interlock due to surface resolution, frictional effects will be masked. Further, the CFD mesh generation for such intersecting cylinders is difficult. A sensitivity study was carried out to determine how many single spheres are required to avoid influence on the packing. The number of particles used for each cylinder depends on the size of the particle that is to be represented in DEM without introducing numerical effects that influence the packing. For the cylinder packing case (LU2) in this work, each cylinder was created with 1000 spheres, which is computationally manageable but fine enough to not significantly influence the packing. A typical DEM cylindrical particle is shown in Figure 4a. By default the PFC3D code is a single phase simulation tool; thus it does not use drag or buoyancy forces on particles falling though the air. To minimize the computational time to fill a tube, cylindrical particles were not dropped from the very top of the tube but were generated at defined heights in the tube. Based on the vertical location of a cylinder, the velocity of the cylinder was calculated based on potential and kinetic energies. It was ensured that the terminal settling velocity of the particles was not exceeded. This procedure ensures a correct starting velocity which affects the interaction of the cylinder at impact on the packing. Depending on the packing, the impact of falling particles can affect the structure of the packing. To mimic the effect that the particles rotate while they are falling in the tube, the cylinders are rotated randomly before they fall in the tube. The initial rotation of the cylinders prevents a preferential orientation and packing structure of the cylinders. Figure 4b shows the filling of the tube. Once the cylinders fill the pipe, the next cylinder’s initial release location is moved upward. Figure 4c is a close-up of a section of the DEM-simulated packing. It can be seen that local packing structure varies based on the vertical position. The structure of the packing in the tube changes locally based on geometry and the way the cylinders interact during the filling process. In addition to the geometric effects, the friction parameters also influence the packing. For the DEM simulations the particle-particle friction as well as the particle-wall friction were considered. This effect was studied with an experimental design where the geometries of the pipe and cylinder were kept

constant but the friction factors were varied between 0 and 1. The results reveal that the wall friction has a higher impact on the porosity than the ball friction. Based on the observations that the selection of the simulation parameters affects the packing structure, here expressed with the overall packing porosity, experiments are required to calibrate the simulation parameters.24 4.2. Geometry and Mesh Generation. A DEM-simulated packing geometry is exported into a text file of multiple columns containing the information regarding dimensions and positions of all particles in the packing. Current computational power allows DEM simulation of a entire packed bed containing thousands of particles. However, the number of particles that can be modeled in CFD is significantly less and is limited by available computational resources. For a typical commercialscale packed bed such as the plant-scale experiment conducted in this work, only a small section of the entire packed bed is included in the CFD model to fit the available computational resources. The chosen section should have a packing porosity that is the closest to that in the corresponding entire packed bed. Particles contact each other, and some also contact the tube wall. At these points of contact, mesh cells become highly skewed, which results in a poor-quality mesh and often causes a convergence problem in CFD simulation. To avoid this problem, each particle is slightly shrunk in size but the particle position remains unchanged during the geometry buildup. The particle shrinkage used in this work was 0.5% for spherical particles and 1% for cylindrical particles. Correspondingly, the spherical particle diameter used in the CFD model is 99.5% of the actual ball diameter and the cylindrical particle diameter used in the CFD model is 99% of the actual particle diameter. The length of the cylindrical particle is also shrunk by 1%. Shrinking particles in the CFD model is a common approach adopted in published works10-15 using body-fitted mesh for the packed bed. Typically, 1% shrinkage was used in those publications. The particle shrinkage directly affects the packing porosity (voidage) and therefore the pressure drop, and this effect is further discussed later in this paper.


Figure 4. DEM simulation of a fixed bed with randomly packed cylindrical particles: (a) a cylindrical DEM particle assembled with balls, (b) packing of the tube at different stages in time, and (c) close-up of a section of the DEM-simulated packing.

Tetrahedral elements were used to generate unstructured mesh for the packed bed, and the mesh size can be characterized by the edge length of a tetrahedral element. Small mesh size is often required not only for accommodating complex packing geometry but also for avoiding grid generation problems, especially when the shrinkage is very small. For example, with 1% particle shrinkage, the mesh size can be as large as 1 mm (10% particle diameter) for 10 mm particles. However, if the particle shrinkage is reduced to 0.5%, the mesh size has to be 0.7 mm or less in order to avoid failure in the grid generation. Empty sections are added to each end of the packed bed, as shown in Figure 2. This is necessary in order to minimize the effect of boundary conditions specified the inlet and outlet of the tube in CFD. The mass flow rate specified at the inlet (top of the modeling domain) is internally converted into a uniform velocity at the inlet boundary. The flow velocities start to vary significantly within the packing. The pressure-outlet boundary specifies a uniform pressure at the outlet (bottom of the modeling domain). The empty sections allow flow to naturally develop as approaching or leaving the packing. Parametric study showed that the length of the empty sections should be at least one tube diameter (D). It was also found that there was no noticeable difference in the pressure drop predictions if longer empty sections were used. Scripts have been developed to automate the tedious process of generating the packing geometry and mesh described above for spherical particles and cylindrical particles separately. The

script reads in the data file exported from DEM simulation and converts the packing data into a Gambit journal file. Running the journal file in Gambit generates a mesh for CFD simulation. Mesh size and particle shrinkage can be adjusted either in the script or in the Gambit journal file if needed. 4.3. Flow Field Simulation with CFD. The three-dimensional steady-state flow field through the packed bed is simulated by solving the Reynolds averaged Navier-Stokes (RANS) equations and the mass conservation equation. The commercial CFD package FLUENT 6.1 was used in this work. For the interested gas flow range, flow in the packed bed channel is turbulent with the tube Reynolds number ranging from 2000 to 20 000. The renormalization group (RNG) k-ε model25 is chosen for the turbulence closure. Different turbulence models were examined to understand their effects on the accuracy of pressure drop prediction, including the standard k-ε model, the realizable k-ε model, RNG k-ε model, the SpalartAllmaras model, the standard and SST k-ω model, and the Reynolds stress model (RSM). These models are available and built in FLUENT 6.1. Detailed descriptions of each model can be found in the FLUENT 6.1 User’s Manual;23 thus they are not repeated here. The turbulence model study was conducted with the same packed bed and flow conditions so that the turbulence model was the only difference. The study shows that there is no significant difference among various k-ε and k-ω models with variations of pressure drop of less than 4%. The


Figure 5. Typical CFD simulation results: (a) flow pattern shown as flow path lines, (b) pressure contours on particles and tube wall, (c) velocity on central plane, and (d) pressure profile across the packed bed central plane.

Spalart-Allmaras model predicted the lowest pressure drop and the largest discrepancy from the measurement. The RSM model predicted the largest pressure drop that was the closest to the measurement. However, the simulation took significantly (2-3 times) more CPU time and the predicted pressure drop was less than 3% larger than the selected RNG k-ε models. The RNG k-ε model25 was developed for the flows featuring strong stream-line curvature, separation, vortices, and boundary layers under strong adverse pressure gradients; thus it was considered a proper choice for the packed bed applications. The turbulent flow field through a packed bed is dominated by fluid-surface interactions. These surfaces include both the particles and the tube walls which are prevalent throughout the bed. The k-ε models are primarily valid for turbulent core flows (i.e., the flow in the regions somewhat far from walls). Very close to the wall, viscous damping reduces the tangential velocity fluctuations, while kinematic blocking reduces the normal fluctuations. Toward the outer part of the near-wall region (boundary layer), however, the turbulence is rapidly augmented by the production of turbulence kinetic energy due to the large gradients in mean velocity. Directly resolving the near-wall velocity profile requires very high mesh resolution for the boundary layer with y+ close to 1 (y+ is the dimensionless normal distance from the wall-adjacent cell to the wall). This high resolution requirement can significantly increase the number of mesh elements for the CFD model to the level beyond the limit of available computational resources. Otherwise, the number of particles included in CFD has to be significantly reduced, such as less than a dozen,26,27 which would be far insufficient to represent the packing characteristics of a randomly packed bed for a pressure drop study. In this work, the standard wall functions28 were used for the near-wall modeling, which allows relatively larger near-wall mesh elements; thus sufficient particles were included in the CFD simulation. With the standard wall functions, the viscous sublayer and buffer layer are not resolved. Instead, semiempirical formulas called “wall functions”

are used to bridge the viscosity-affected region between the wall and the fully turbulent region. During the CFD simulations, y+ on the particles and tube wall were checked to make sure that y+ is within the applicable range: 30 < y+ < 250. Local grid refinements were conducted to refine boundary layer mesh for the surface region with y+ > 250. The second-order UPWIND discretization scheme23 was used for all equations. The air density is calculated by the ideal gas law at room temperature (25 °C). Details of model formulations can be found in the FLUENT 6 User’s Manual;23 thus they are not repeated here. Figure 5 shows a typical CFD simulation result for a segment of packed bed, including the flow field, represented by the flow path lines (Figure 5a), the pressure contours on particles as well as on the tube wall (Figure 5b), the velocity magnitudes on the central plane across the packed bed (Figure 5c), and the pressure profile on the same central plane (Figure 5d). Each point on the pressure profile plot corresponds to one spatial position in the packing. The two almost flat lines at both ends correspond to the empty sections of the flow channel, indicating little pressure change in the nonpacking zones (the empty sections). These local pressure variations correspond to the local velocity variations through the packing structure. 4.4. Pressure Drop across Packed Bed. The pressure drop is simply the difference of average pressures at the inlet, P1, and outlet, P2, as illustrated in Figure 5d. This is the pressure drop across the packed bed if the entire bed is simulated in CFD. However, if only a segment of the entire packed bed is included in CFD, the predicted pressure drop must be scaled for the entire packed bed. Two linear scaling approaches were examined: scaling by packed bed height (H) and scaling by the number of particles (N). Both approaches gave similar results, but the latter was more robust and consistent; it was thus used in this paper. Scaling by the particle number was also found more accurate when the number of packing layers (H/d) in a


Figure 6. Effect of number of particles included in selected segment on packing porosity deviation between the segment and the corresponding entire packed bed.

CFD simulation was small. The pressure drop for the entire packed bed, ∆P, can be expressed as follows: ∆P ) ∆PS

N NS

(3)

where N is the number of particles in the entire packed bed and NS is the total number of particles in the segment included in the CFD simulation. In chemical plants, an accurate number of particles in a packed bed is often hard to obtain but the packing height information can be available. In this case, scaling can be done using packed bed height. The fundamental reason for the linear scaling of pressure drop is explained from the typical pressure profile shown in Figure 5d. The average pressure drops almost linearly along the packing height even though local pressure can be noticeably higher or lower than the average. 4.5. Porosity Deviation between CFD Model and Packed Bed. Porosity for a packed bed can be calculated from eq 1 or 2 for an actual packing (experiment) or a DEM-simulated packing. In CFD, the packing porosity can be easily obtained from the volume of fluid zone and the tube volume. Since the pressure drop is very sensitive to packing porosity, the porosity deviation between the packing in the CFD model and actual packing can greatly affect the accuracy of model prediction. The sources of the deviation are (1) DEM simulation, (2) only a segment of the entire packed bed included in CFD, and (3) particle shrinkage in the CFD model. 4.5.1. DEM Simulation. The nature of packing randomness naturally reflects as the difference between an actual or experimental packing and corresponding DEM simulation. In addition, the parameters in the DEM simulation, such as the particle dropping position, the initial dropping velocity, and the friction factor, can also affect the simulation result.24 Overall, it has been found that DEM is capable of simulating the packed beds of spherical or cylindrical particles with porosity deviation less than 3%. DEM-simulated packing may have a larger or smaller porosity than the actual packing. For all experimental cases listed in Table 1, the DEM simulations were done for whole packed beds. 4.5.2. Selected Segment of Entire Packed Bed. In most industrial applications where thousands of particles are packed, only a segment of the entire packed bed can be afforded to be included in CFD due to the limit of available computational

resources. The packing porosity of a selected segment can be larger or smaller than the corresponding entire bed, depending on where the segment is selected and how many particles are included in the selected segment. Figure 6 illustrates how the porosity of the selected segments deviates from the entire packed bed when the numbers of particles in the segments vary. These data are from the plant-scale experiments with spherical particles and the corresponding DEM simulations. Generally speaking, the deviations decrease as the number of particles included in the selected segments increases. There are also small-scale (up and down) periodic variations in the porosity deviation as the number of particles changes, which can be more clearly seen in the zoomed-in plot in Figure 6b. These periodic variations correspond to the layer changes in the packing. The porosity deviation suffers a small jump every time an additional particle constitutes a new layer of packing, and then decreases as more particles are added to the same layer. The segments with zero porosity deviation from the entire packed bed have minimum 159, 800, and 1182 particles for three different D/d ratios 2.14, 2.61, and 3.23, respectively. As the particle number further increases, the segment porosity can be either larger or smaller than the entire bed, but the deviation is relatively small (less than 0.5%). The larger the D/d ratio is, the more particles are needed to reach this low level deviation of packing porosity. This is because sufficient layers are required to represent the entire packed bed and a packing with a larger D/d ratio contains more particles per layer. Obviously, the packing porosity of a segment also varies within the selected segment. A recommended practice is to include as many particles as possible in the segment and select the segment position to have minimum porosity deviation from the entire packed bed. In this paper, for the three plant-scale experimental packed beds where up to 1500+ spherical particles were packed, only 125 particles were included in selected segments in the CFD simulations. Compared to the minimum number of particles required to have close-to-zero deviation from the entire packed bed (Figure 6), this number (125) is close for the bed with the lowest D/d ratio (2.14) but insufficient for the other two beds with larger D/d ratios (2.61 and 3.23). 4.5.3. Particle Shrinkage. As discussed earlier, particle shrinkage is necessary to avoid a highly skewed CFD mesh which can cause failure in mesh generation or a serious


more accurate representation of the packing in the CFD model comes with increased cost of computational resources. 4.6. Compensation for Porosity Deviation. The packing porosity deviation between CFD and experiment is accumulated during the processes of DEM simulation, use of a selected segment of entire packed bed in CFD model, and particle shrinkage. Each of these processes is an inherent part of the DEM/CFD methodology, and the porosity deviation cannot be avoided in the DEM simulation and particle shrinkage. A particle shrinkage of 1% alone would cause 3% porosity deviation which may lead to ∼10% error in the pressure drop prediction. Therefore, it is necessary to compensate the effect of the porosity deviation in order to meet the prediction accuracy goal (error 10. Much research has been conducted to identify more accurate Ergun constants a and b for different particle geometries and particle-to-tube diameter ratios. A widely used set of Ergun constants is from Handley and Heggs:29 a ) 1.248 and b ) 368 a ) 1.28 and b ) 458

for spherical particles (9) for cylindrical particles (10)

For the void fraction of a packed bed, the most widely used correlations are probably Dixon’s correlations: for spherical particles (eqs 11a-11c) ε ) 0.4 + 0.05(d/D) + 0.412(d/D)2 if d/D e 0.5

(11a)

ε ) 0.528 + 2.464(d/D - 0.5) if 0.5 e d/D e 0.536 (11b) ε ) 1 - 0.667(d/D)3(2d/D - 1)-0.5 if d/D g 0.536 (11c) and for cylindrical particles (eqs 12a-12c): ε ) 0.36 + 0.1(dp /D) + 0.7(dp /D)2 if d/D e 0.6

(12a)

ε ) 0.677 - 9(dp /D - 0.625)2 if 0.6 e d/D e 0.7 (12b) ε ) 1 - 0.763(dp /D)2 if d/D g 0.7

(12c)

In eqs 12a-12c, dp, the effective particle diameter, is defined as the diameter of a sphere with the same volume as the particle, that is

dp ) √(3/2)d2h 3

for cylindrical particles

(13)

The correlations in eqs 8-13 are used to calculate the pressure drops for the cases in Table 1 except the first three cases, for which the square duct tubes were used and the correlations only apply to cylindrical tubes. The calculation results are compared with the predictions using the DEM/CFD simulations as well as the measurements. 5. Results and Discussion 5.1. Structured Packing and CFD Model Validation. The experimentally measured pressure drops in Table 1 are plotted together with the corresponding CFD predictions for a direct comparison. Figure 9 shows the results for the three laboratoryscale cases with structured packing of spherical particles (LS1, LS-2, and LS-3). The entire packed beds were directly modeled in the CFD without DEM for the structured packing seen in Figure 9. The only difference between the CFD packing and the experimental packing is that the particle diameter in CFD was shrunk by 0.5%. The predicted pressure drops match well the experimental measurements, although the CFD predictions are consistently lower than the measurements. The average error (3.1%) is within the desired range (10%). This good agreement between the predictions and measurements validates the CFD model. Also shown in Figure 9 are the CFD predictions with the porosity correction factor k to compensate the porosity deviation due to the particle shrinkage (marked as “CFD-k”). The packing porosity for the experimental packing with 10 mm spheres is 0.476. For the CFD packing with the shrunken particle diameter d ) 9.95 mm, the porosity becomes 0.484, which is 1.68% larger than the actual packing. It is this slightly larger porosity that caused the consistently lower predicted pressure drops than the measurements. By including the porosity correction factor, the model prediction errors are reduced to less than 2%. 5.2. Random Packing of Spherical Particles (Laboratory Scale). Figure 10 shows the results for the laboratoryscale case with random packing of 153 spherical particles (LU1). The entire packed bed, simulated by DEM, was included in the CFD simulation. The particle diameter in the CFD simulation was shrunk by 0.5% to 9.95 mm. The predicted pressure drops, as shown in Figure 10, are noticeably lower than the measurements with average error of 13.7%, which is beyond the desired error limit (10%). This discrepancy is mainly caused by two factors: the DEM simulation and the particle shrinkage. The DEM-predicted packed bed height is about 1.8% taller than the measured packing height, which makes the DEM-simulated packing looser than the actual packing. Combined with the particle shrinkage, the packing porosity deviation between the experimental packing and the CFD model is 3.52%. Applying the porosity correction factor, the model predictions match the experimental data with an average error less than 4%. Also shown in Figure 10 are the pressure drops calculated using the empirical correlations (eqs 8-11c), which significantly overpredicted the pressure drops with an average error of 61.5%. 5.3. Random Packing of Cylindrical Particles (Laboratory Scale). For the laboratory-scale case with 82 randomly packed cylindrical particles (LU-2), the entire packed bed simulated by DEM was included in the CFD simulations. Figure 11 shows the flow field in the packed bed corresponding to the median flow condition (16.65 kg/h). The flow field is illustrated by the flow path lines (Figure 11a) and the velocity vectors on the central plane (Figure 11b). The flow path lines are the trajectories of tracers released at the top inlet and colored by initial


Figure 9. Comparison of predicted and measured pressure drops in structured-packed beds.

Figure 10. Comparison of predicted and measured pressure drops in laboratory-scale packed bed with random packing of spherical particles (LU-1).

locations, thus providing a three-dimensional overview of the gas flow through the packed bed. The velocity vectors can reveal the flow structures, such as eddies behind/between the particles, and local channeling. The local flow velocity in the packed bed can be a few times higher than the superficial flow velocity in the packed bed. Figure 11c shows the pressure contours on the particles and tube wall, depicting local pressures and gradual variations through the packed bed. The CFD-predicted pressure drops across the packed bed are plotted in Figure 12 to compare with the measurements as well as the calculations with the empirical correlations (eqs 8-13). The CFD predictions (marked as “CFD” in Figure 12) are consistently lower than the measurements. The average deviation is 8.3%. Also shown in Figure 12 are the pressure drops corrected by the porosity correction factor (marked as “CFDk”). The DEM-simulated packing, calculated using eq 2, has a porosity of 0.5573 that is 1.3% less than the packed bed in the experiment, which has a porosity of 0.5646, calculated using eq 2 also. This means that the DEM-simulated packing is more compact than the actual packing. The porosity of the packed

bed used in the CFD simulations (εCFD) is 0.5704, which is 1.05% larger than the porosity of the packed bed in the experiment (0.5646). The difference between CFD and DEM is caused by the particle shrinkage (1%) and rounded edges (r ) 1 mm) of the particles during the CFD grid generation, which turns the slightly more “compact” than actual packing into a slightly less “compact” than actual packing. Plugging in σ ) 1.05% (the packing porosity deviation between the CFD and the experiments) into eq 7 gives the porosity correction factor k ) 1.04. After the k-correction, the average error (from the measured pressure drops) now drops from 8.3% to 6.1%. The CFD predictions before and after the k-correction are both within the acceptable error range (10%). On the other hand, the empirical correlations significantly underpredicted the pressure drops with an average error of 40.1%. 5.4. Random Packing of Spherical Particles (Plant Scale). Figure 13 shows the results for the three plant-scale cases with randomly packed 10 mm spherical particles (PU-1, PU-2, and PU-3). The packed beds in the experiment contain 799, 916,


Figure 11. Sample CFD simulation results for packed bed with random packing of cylindrical particles: (a) flow pattern shown as flow path lines, (b) velocity vector on central plane, and (c) pressure contours on particles and tube wall.

Figure 12. Comparing model predictions with measured pressure drops for packed bed with random packing of cylindrical particles.

and 1545 particles for tube-to-particle diameter ratios of 2.14, 2.61, and 3.23, respectively. The entire packed beds were simulated by DEM. However, limited by the available computational resource, only 125 particles (taken from the bottom section of each entire packed bed, shown in Figure 13a) were

included in the CFD simulations. The particle diameter was shrunk by 0.5% in the CFD model. As seen in Figure 13b, the agreement between the CFD-predicted pressure drops (without the porosity correction factor) and the measurements varies significantly, from relatively good (for PU-1, average error 4.9%) to slightly over the limit (for PU-2, average error 10.4%), and then to very poor (for PU-3, average error 26%). Comparing the packing porosity in the CFD model with the experimental packing, it is clear that the porosity deviations are the main contributor to the mismatch. For the three plant-scale experiment cases (PU-1, PU-2, and PU-3), the packing porosities in the CFD model are 1.36%, 3.09%, and 9.25% larger than the experiments, respectively, resulting from the combination effect of the DEM simulation and the use of only 125 particles in the CFD model and the particle shrinkage. For the case with the largest tube-to-particle diameter ratio (PU-3, D/d ) 3.23), the selected segment is more loosely packed than the experimental bed. In addition to the position of the selected segment, an insufficient number (125) of particles included in the CFD simulation is the main cause for this large deviation of packing porosity, as suggested in Figure 6 that more that 1000 particles


Figure 13. Comparison of predicted and measured pressure drops in plant-scale randomly packed beds.

are required in order to have the segment porosity close to the entire packed bed. After compensation by the porosity correction factor, the match between the CFD predictions and the measurements is significantly improved as shown in Figure 13, and the average error is reduced to less than 4%. The empirical correlations (eqs 8-13) overpredicted the pressure drops in all cases. The deviations from the measured pressure drops vary dramatically, from pretty accurate (average error 3.8% for PU-1) to acceptable (average error 9.3% for PU2) to considerable (average error 93.4% for PU-2). Comparing the measurements and the calculations shown in Figures 10, 12, and 13 confirms that the empirical correlations are unreliable in predicting pressure drop across a packed bed with a low tube-to-particle diameter ratio (D/d < 4). Their deviations are inconsistent and can be significantly beyond the acceptable error limit. In contrast, the predictions using the

DEM/CFD approach consistently matched well with the measured pressure drops. 6. Conclusions The ability to a priori predict void fraction and pressure drop in a packed bed would significantly improve reactor design as well as allow for optimization around catalyst performance, catalyst design, and the resulting process pressure drop. Traditionally, the packed bed reactor designs are based on a homogeneous assumption with averaged empirical correlations along with a sophisticated process of trial and error. Those correlations often become invalid or unreliable for low tubeto-particle diameter ratios (D/d < 4), where the tube wall effect and local phenomena dominate. In this paper, the discrete element method (DEM) and computational fluid dynamics


(CFD) are coupled to model the flow field in packed beds. DEM is used to generate realistic random packing for a packed bed of spherical or cylindrical particles, which is then converted into a packing geometry to generate mesh for CFD simulation. Scripts have been developed to automate the tedious process. Only a segment of entire packed bed would be included in CFD simulation if modeling the entire packed bed becomes unmanageable or beyond the limit of available computational resources. Pressure drop across the bed is part of the CFD simulation results. Two types of experiments were conducted: laboratory-scale experiments with up to 153 particles packed to allow modeling of entire packed beds, including random packing and structured packing, and plant-scale experiments conducted with up to 1545 randomly packed particles. A concept of the “porosity correction factor” was introduced to compensate the effect of porosity deviation between actual packing and the CFD model since pressure drop is very sensitive to packing porosity. The porosity deviation can be caused by DEM simulation, use of a segment of entire packed bed, and particle shrinkage in CFD. They are inherently part of the DEM/CFD methodology and cannot be eliminated. The random nature of packing naturally reflects as the difference between DEM-simulated packing and actual packing. The number of particles included in CFD simulations is limited by available computational resources. Shrinking particles is necessary to prevent highly skewed mesh at those single-point contacts between particles or between a particle and the reactor wall. The CFD model was validated by good agreement between the CFD predictions and the measurements for the structured packing cases in which the error due to the DEM simulation and use of a segment of packed bed were eliminated. For the packed beds with randomly packed particles, the agreement between the CFD-predicted pressure drops and the measurements varied, depending on the porosity deviation. After compensated by the porosity correction factor, the match between the CFD predictions and the experimental data is improved and the average errors are reduced to within the desired limit for industrial design (10%). The pressure drops calculated by the empirical correlations confirmed the advantage of the DEM/CFD approach as well as the inconsistency and unreliability of the empirical correlations for the packed beds with low tube-to-particle diameter ratios (D/d < 4). The DEM/CFD approach is being extended to study fixed beds packed with more complicated designs of catalyst particles such as ring or penta-ring as well as with more complicated physical models such as heat transfer, mass transfer, and chemical reactions. The DEM/CFD approach can be used to help design packed bed reactors such as optimizing the tubeto-particle diameter ratio, optimizing the tube diameter, and optimizing the size and shape of catalyst particles. Furthermore, the DEM/CFD simulations provide detailed information about local voidage, local head loss, and local heat transfer rate through the packed bed, which can be critical for stability analysis and reactor control. Acknowledgment Dan Friedhoff, Billy Smith and Mike Cloeter in The Dow Chemical Company are thanked for their contributions in conducting the experiments. Nomenclature a ) Ergun constant in Ergun equation for pressure drop (dimensionless)

b ) Ergun constant in Ergun equation for pressure drop (dimensionless) D ) tube inner diameter (mm) d ) spherical particle diameter; cylindrical particle diameter (mm) dCFD ) particle diameter used in CFD (mm) D/d ) tube-to-particle diameter ratio (dimensionless) dp ) effective particle diameter (mm) g ) gravitational constant (9.81 m/s2) G ) mass flux (kg/m2 · s) h ) height of cylindrical particle (mm) H ) height of packed bed (mm) k ) porosity correction factor (dimensionless) Mw ) molecular weight of gas through packed bed (g/mol) N ) total number of particles in a packed bed NS ) total number of particles in a segment of the packed bed Po ) inlet pressure to a packed bed (Pa) R ) gas constant (8.314 m3 Pa/mol · K) Re ) Reynolds number (dimensionless) ∆P ) pressure drop across packed bed (Pa) ∆PS ) pressure drop across a segment of the packed bed (Pa) ∆PC ) pressure drop prediction corrected by porosity correction factor (Pa) T ) temperature (K) ε ) void fraction of packed bed, packing porosity; packed bed porosity (dimensionless) εCFD ) void fraction of packed bed in CFD (dimensionless) σ ) deviation of packing porosity between actual and CFD model (%) η ) particle shrinkage (%)

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ReceiVed for reView October 14, 2008 ReVised manuscript receiVed January 9, 2009 Accepted January 15, 2009 IE801548H

A Coupled DEM and CFD Simulation of Flow Field ... - ACS Publications

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