14th European Conference on Mixing Warszawa, 10-13 September 2012

AN EFFECT OF GRID QUALITY ON THE RESULTS OF NUMERICAL SIMULATIONS OF THE FLUID FLOW FIELD IN AN

AGITATED VESSEL

Joanna Karcz, Lukasz Kacperski

West Pomeranian University of Technology, Szczecin, Department of Chemical Engineering, al. Piastow 42, 71-065 Szczecin, Poland

Joanna.Karcz@zut.edu.pl

Abstract. The effects of the grid density and geometrical mesh quality on the results of numerical simulations of the fluid flow field in an agitated vessel equipped with up-pumping pitched blade turbine PBT 6 are presented in the paper. The computations were performed for turbulent range of a Newtonian liquid flow in a vessel of inner diameter T = 0.295 m. Simulations were carried out for six numerical grids using ANSYS CFX 13 software. Shear Stress Transport (SST) model was used as turbulence model. Multiple Reference Frame or Sliding Mesh approaches were applied to model the movement of the impeller. For each numerical grid, the following geometrical measures of mesh quality were compared: orthogonality factor, mesh expansion factor and mesh aspect ratio. The study revealed that the results of the numerical modelling of the fluid flow field in the agitated vessel depend on the grid density and geometrical mesh quality.

Keywords: numerical simulation, agitated vessel, grid density, mesh quality, kinetic energy of turbulence, flow field

1. INTRODUCTION The density of cells in a computational grid needs to be fine enough to capture the flow

details, but not so fine that the overall number of cells in the domain is excessively large, because problems described by large numbers of cells require more time to solve. Using a mesh of adequate geometrical mesh quality is an important part of controlling discretization error. Significant measures of mesh quality may be classified as measures of mesh orthogonality, expansion and aspect ratio [1]. The computations carried out on excessively coarse grids or grids of improper geometrical mesh quality may have considerable influence on the propagation of numerical errors and result in an imprecise solution.

In literature concerning numerical modeling of various processes occurring in agitated vessels, the information on the influence of the computational grid used on the results of the numerical solution tends to be presented in a scarce and general way. In papers [2-5], the authors mention the considerable sensitivity of the value of kinetic energy of turbulence to the size of computational elements and mesh quality parameters. Mackiewicz [6,7] performed calculations for two numerical grids of density about 189000 and 215000 cells, using a denser grid for further calculations as it enabled to obtain more reasonable values of the modeled variables. Zakrzewska [8] found out the dependence between predicted values of kinetic energy of turbulence and grid density and y+ values, while using k-ε model.

The aim of this research work was to analyze the effects of the grid density and geometrical mesh quality on the results of numerical simulations of the fluid flow field in an agitated vessel equipped with high-speed impeller.

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2. RANGE OF THE NUMERICAL SIMULATIONS Numerical modelling of the flow field was carried out for a baffled agitated vessel

equipped with a pitched blade turbine with six blades (PBT6, pitch of the blade 450). The flat bottomed vessel of inner diameter T = 0.295 m was filled with a Newtonian liquid up to the height H = T. Up-pumping pitched blade turbine of diameter D = 0.33T was located on the height h = 0.67H from the vessel bottom. Four planar baffles of width B = 0.1T were symmetrically arranged inside the agitated vessel. Such impeller – baffles configuration was proposed by Mackiewicz [6] for the effective production of floating suspensions. Computations were performed for distilled water of temperature 25oC (density ρ = 997 kg/m3 and viscosity η = 0.89·10-3 Pa·s), agitated within the turbulent range of the liquid flow (impeller speed n = 4.17 1/s, Re ≈ 45000).

Simulations were carried out for six numerical grids of density 153, 328, 447, 659, 713 or 945 thousands of computational cells (tetrahedral elements) using ANSYS CFX 13 software. Shear Stress Transport (SST) model [9, 10] was used as turbulence model. Multiple Reference Frame (MRF) or Sliding Mesh (SM) approaches were applied to model the movement of the impeller. For each numerical grid, the following geometrical measures of mesh quality were compared: orthogonality factor, mesh expansion factor and mesh aspect ratio.

3. NUMERICAL MODELLING

3.1 Geometry, grid generation and mesh quality Two geometries of the agitated vessel were prepared for the computations. One of the

geometries was drawn considering the real thickness of the blades and baffles. In second case, the blades and baffles were implemented as surfaces, i.e. they had “zero thickness”. This solution enables to limit the number of computational cells alongside the elements of a computational small-size domain and to improve the quality of the grid.

Fig. 1. Numerical grids in radial cut-plane for the dimensionless axial coordinate z/H=0.67; number of computational cells: a) 153000; b) 328000; c) 447000; d) 659000; e) 713000; f) 945000

a) b) c)

d) e) f)

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The spatial discretization of the computational domain was carried out with the Patch Independent method (Octree algorithm), and as an exception, a grid of density about 447000 cells was generated with Patch Conforming method (Delaunay’s algorithm). The grids differed in initial mesh size in global and local scale. The initial mesh size equaled 0.002, 0.004, 0.006 or 0.008 m, the growth rate was changed from 1.15 to 1.4. Radial planes of the generated computational grids have been presented in Fig. 1 for the dimensionless axial coordinate z/H = 0.67. The initial mesh size was defined in various combinations for the vessel wall and the bottom, free surface of the liquid, baffles, blades, the impeller shaft and/or the whole volume of the vessel. Computational grids of element number 447000 and 659000 were generated for the “simplified geometry”, the remaining grids covered the geometry with blades and baffles of the same thickness as in the real agitated vessel.

Table 1. Mesh geometrical quality parameters

*- recommended values As it can be concluded from the data on geometrical quality parameters shown in Table 1,

the best grid seems to be the one of density 659000 elements. All of its parameters are within the acceptable and recommended range of values [1]. As to the grid of 447000 elements, there are deviations from the acceptable values for the orthogonal quality and mesh expansion factor; especially the later one exceeds the acceptable value. The acceptable values for the mesh expansion factor were exceeded by grids of density 328000 and 954000 cells, and only the grid of density 659000 cells fits within the recommended aspect ratio values (max ratio). In Table 1 the dimensionless values y+ (dimensionless wall distance to first node of computational grid) have also been set together. However, it is difficult to say for certain if the values can be regarded as appropriate. It is generally assumed that the higher the Re values the smaller the values y+ must be. Certain recommendations can be found in literature [1, 11], for example, when it is required to precisely model the flow in the near-wall boundary layer (heat transfer), it is recommended to obtain value y+ < 1 [1]. During the modeling of the majority of the near-wall flows it is sufficient to place the grid’s nodes in the transition layer, i.e. in the range of 5 < y+ < 60 [11].

3.2 Computational model The computations of the turbulent flow in the baffled vessel were carried out using Shear

Stress Transport model of Menter [9]. The model enables to compute the turbulent flow in the liquid core just like the k-ε model. Near the walls of the vessel the k-ω model is used, which is

Mesh quality parameters

Grid a) (153000)

Grid b) (328000)

Grid c) (447000)

Grid d) (659000)

Grid e) (713000)

Grid f) (945000)

accept.range

Orthogonal Quality

(ICEM-CFD)

min. 0.44 av. 0.88

Min. 0.46av. 0.88

min. 0.30av. 0.79

min.0.47 av. 0.91

min. 0.43av. 0.91

min. 0.43 av. 0.86

<1/3;1>

Mesh Expansion Factor

(CFX-SOLVER) max. 17.2 Max. 21.4 max. 150.5 max. 17.8 max. 15.1 max. 29.6 <1;20>

Max ratio (ICEM-CFD)

max. 4.99 av. 1.80

Max. 4.76av. 1.81

max. 3.91av. 1.77

max. 2.69av. 1.30

max. 4.96av. 1.70

max. 5.03 av. 1.86

≤ 100or

<1;3>*y+ at vessel wall

and bottom max. 163

av. 60 Max. 130

Av. 42 max. 119

av. 33 max. 148

av. 46 max. 129

av. 41 max. 123

av. 27 undef.

y+ at baffles max. 122 av. 58

Max. 83 Av. 42

max. 52 av. 17

max. 54 av. 22

max. 86 av. 42

max. 20 av. 9 undef.

y+ at impeller max. 130 av. 62

Max. 101Av. 39

max. 87 av. 36

max. 156 av. 55

max. 64 av. 26

max. 133 av. 35 undef.

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more appropriate for modeling near-wall flows. Both models are related with a proper transformation (blending) function. In the ANSYS CFX 13.0, the SST model is implemented together with the near-wall functions, which partly compensate the inconveniences related with the necessity to maintain a small distance of the computational grid’s first node y+ [1].

For the computational grids of resolution 153, 328, 447, 713 and 945 thousands, the stream generated by the impeller was modeled using the MRF approach. An automatic timescale was used. The computations for the grid of 659000 number of cells were carried out using the SM approach with time step Δt = 0.01 s. The computations were continued by the mixing period equal to 20 s.

The moving zone of height equal to 0.082 m had the same dimension for all the computational grids. The radius of the moving zone was equaled to half distance between impeller blade and baffle edge. The moving zone was connected with the stationary zone by the GGI (General Grid Interface).

Stationary no slip wall boundary conditions were implemented for the walls of the vessel, baffles, blades, and the part of the shaft placed in the moving zone. In the case of the grids of density 447 and 659 thousands, the boundary conditions of wall type were implemented on both sides of the impeller’s blades and the “zero thickness” baffles. For the moving zone and fragment of the impeller’s shaft placed in the stationary zone, the rotational speed was assumed n = 4.17 1/s. The free surface of the liquid was defined as a wall without shear stresses. Discretization of all the transport equations for all the grids was performed with the high resolution discretization scheme and the computations were carried out with single precision. Other process and numerical parameters of the simulation system remained unchanged, in order to enable the mutual comparison of the computation results.

For the computational grids of resolution 153, 328, 447, 713, 945 thousands, between 2000 to 2500 iterations were carried out. A level of residual functions for the momentum equation (axial component) was equal to 6.5·10-4. For the other equations the residual functions maintained on the level below 9·10-5. Because of the high values of the residues the domain imbalance was checked for each transport equation and its values did not exceed 0.0394%. As to the numerical grid of density 659000 the best RMS residual function level was reached for all transport equations in range between 2·10-5 to 9·10-7 and max domain imbalance did not exceed 6·10-4 %.

4. NUMERICAL RESULTS

The contours of the flow field in axial cross-sections of the agitated vessel as well as the contours of the kinetic energy of turbulence and energy dissipation are presented in Figs. 2 – 4. As Fig. 2 shows, visually velocity fields are quit similar. However, differences occur in the dimension of the regions near the impeller blades where mean velocity of the liquid has high values. The regions are slightly smaller for computational grids based on a simplified geometry (Fig. 2c, d). It may be concluded that the increase in the number of grid elements results in a more detailed map of the local features of the flow field.

Fig. 3 presents contours of turbulence kinetic energy in axial cross sections of the agitated vessel for all the tested computational grids. Considerably lower values of turbulent kinetic energy k are observed for the grid of density 447000 (Fig. 3c) compared to the others cases. It may arise from the fact that the acceptable value of the mesh expansion factor has been exceeded (Table 1, value equal to 150.5).

From comparison of the contours shown in Fig. 4, it results that turbulence kinetic energy dissipation ε increases with the grid refining (Fig. 4d, e, f), as well as reducing of the y+ value (Fig. 4f). For all computational grids a small discontinuity is observed for values of the kinetic energy of turbulence (Fig. 3) and its dissipation (Fig. 4) between moving and stationary zones. The smallest differences are present for the grid with number of cells equal to 659000.

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Fig. 2. Contours of mean velocity w in a vertical plane midway between two baffles; number of computational cells: a) 153000; b) 328000; c) 447000; d) 659000; e) 713000; f) 945000

Fig. 3. Contours of turbulent kinetic energy k in a vertical plane midway between two baffles; number of computational cells: a) 153000; b) 328000; c) 447000; d) 659000; e) 713000; f) 945000

5. CONCLUSIONS The results of the numerical simulations show that within the range of the performed

computations: 1. Density of the computational grid significantly affects the distribution of the turbulence kinetic energy and its dissipation in the agitated vessel equipped with PBT 6 impeller. More detailed features of the flow field can be captured using denser numerical grid.

d) e) f)k, m2/s2

a) b) c)

w, m/s d) e) f)

a) b) c)

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Fig. 4. Contours of dissipation of turbulent kinetic energy ε in a vertical plane midway between two baffles; number of computational cells: a) 153000; b) 328000; c) 447000; d) 659000; e) 713000; f) 945000 2. All tested numerical grids have orthogonal quality and max ratio parameters within the acceptable range. Mesh expansion factor is exceeded for the grid with 447000 cells, only. 3. Sensitivity of the grid density and mesh quality on turbulence kinetic energy and its dissipation in the agitated vessel is revealed. 4. Modeling of flow field for a simplified geometry of the agitated vessel can cause under-prediction of local values of liquid velocity modeled near impeller blades.

6. REFERENCES [1] ANSYS CFX-Solver Modelling Guide 12.1. Release 12.1, ©2009 ANSYS, Inc. [2] Aubin J., Fletcher D.F., Xuereb C., 2004. “Modelling turbulent flow in stirred tanks with CFD“, Exp. Therm Fluid Sci., 28, 431–445. [3] Deglon D. A., Meyer C. J., 2006. „CFD modeling of stirred tanks Numerical considerations“, Miner. Eng., 19, 1059–1068. [4] Gentric, C., Mignon, D., Bousquet, J., Tanguy, P. A., 2005. „Comparison of mixing in two industrial gas-liquid reactors using CFD simulation“, Chem. Eng. Sci., 60, 2253 – 2272. [5] Khopkar A. R., Kasat G. R., Pandit A. B., Ranade V. V., 2006. “CFD simulation of mixing in tall gas–liquid stirred vessel: Role of local flow patterns”, Chem. Eng. Sci., 61, 2921 –2929. [6] Mackiewicz B., 2008. “Studies on suspension of floating particles in an agitated vessel with a high-speed impeller”, PhD Thesis, Technical University of Szczecin, Szczecin. [7] Mackiewicz B., Karcz J., 2009. “CFD modelling of suspension of floating particles”, Chemical and Process Engineering, 30, 111–123. [8] Zakrzewska B., 2003. “Numerical heat transfer modelling in stirred tanks”, PhD Thesis, Technical University of Szczecin, Szczecin. [9] Menter F. R., 1994. “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications”, AIAA Journal, 32(8), 1598-1605. [11] Wilcox D. C., 1994. Turbulence modeling for CFD, DCW Industries, Inc.

ε, m2/s3 d) e) f)

a) b) c)

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