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Energy Dissipation Rate in an Agitated Crucible Containing Molten Metal

  • Tao Li EMAIL logo , Shin-ichi Shimasaki , Shunsuke Narita and Shoji Taniguchi
Published/Copyright: January 6, 2017
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High Temperature Materials and Processes
From the journal High Temperature Materials and Processes Volume 36 Issue 10

Abstract

The energy dissipation rate (EDR) is an important parameter for characterizing the behavior of inclusion coagulation in agitated molten metal. To clarify the inclusion coagulation mechanism, we review previous water model studies by particularly focusing on the relation between the impeller torque and the EDR of the fluid, which indicates the ratio of energy dissipated in the viscous medium to the energy inputted by the rotating impeller. In the present study, simulations coupled with experiments were performed to determine the relation between the torque and the effective EDR for water and liquid Al in crucibles with and without baffles.

Keywords: energy dissipation rate; turbulence; torque; particle coagulation; simulation

Introduction

Agitated vessels are used in various industrial processes. In the steel industry, the KR process [1] is currently used to enhance the desulfurization rate of molten steel. In this process, a rotating impeller is used to disperse granular refining agents in molten steel. Several other industrial processes employ solid particles in a liquid phase in a mechanically agitated vessel. Furthermore, agitated vessels are often used in fundamental studies to generate specific agitation intensities and for investigating correlations with metallurgical phenomena, which are influenced by agitation. Several water model studies have investigated particle coagulation by using agitated vessels to study the mechanism of inclusion coagulation in turbulence [2, 3, 4, 5, 6]. Smoluchowski [2] established an ideal particle coagulation model in shear flow that assumes straight streamlines in which particles collide with each other through the velocity gradient. Camp and Stein [3] extended the Smoluchowski model by describing the average velocity gradient in turbulence by the square root of the ratio of the energy dissipation rate (EDR) to the kinematic viscosity in turbulent flow. Saffman and Turner [4] developed a more rigorous model based on a statistical turbulence model, but it did not consider the interaction force between the particles. Higashitani et al. [5, 6] improved the Saffman–Turner model by introducing a coagulation coefficient that depends on London–van der Waals forces and the viscous resistance force, which is related to the EDR. They performed particle turbulent coagulation experiments in an agitated vessel containing an aqueous solution and latex particles. Their model assumed that all the energy of the agitating impeller is dissipated in the bulk of the liquid; it gave considerably better agreement with experimental results than the Saffman–Turner model. The experimental EDR,ε0, was evaluated by assuming that all the input energy from the agitating impeller is dissipated in the bulk of liquid in the vessel. This assumption gives ε0=2πnTW, where T is the measured impeller torque, n is the impeller agitation speed, and W is the mass of the fluid in the vessel.

However, some of the energy inputted by rotating impeller may be dissipated on the impeller surface and the vessel wall directly. It is difficult to estimate the proportion of the input energy dissipated in the smallest eddies in the bulk of the liquid in which particle coagulating occurs. Investigations of turbulence in agitated vessels [7, 8, 9, 10] indicate that the effective EDR, ε=kε0 which is obtained by taking the volume-weighted average of the local EDR and is given by ε=kε0(where k varies between 0.15 and 1), indicates the proportion of the input energy from the rotating impeller that is dissipated in the bulk of the liquid. The effective EDR is a significant input parameter in the particle coagulation model developed by Saffman and Turner [4], which is critical for investigating the fundamentals of inclusion behavior in the steel making process. However, the effective EDR could not be directly measured in the experiments. The experimental measured torque is used to estimate the effective EDR for a mechanically agitated vessel. Taniguchi et al. [11] performed water-model experiments of particle coagulation using various kinds of particles and obtained improved agreement between the observed and calculated coagulation rates by using the Higashitani model with ε=0.15ε0. Nakaoka et al. [12] obtained the same result, k=0.15, although they incorrectly modified the Smoluchowski population balance equation (PBE). Arai et al. [13] revised Nakaoka’s PBE and determined the value of k by performing particle coagulation experiments and numerical simulations of aqueous solution flows in four-baffled vessels whose sizes were varied proportionally. The simulated torque agreed well with the measured torque and the value of k was determined to be 0.5, which gave good agreement between the numerical results for particle coagulation using the revised PBE and the experimental results.

In the present study, the torque of a two-paddle impeller rotating in a two baffled crucible containing molten Al was measured and compared with simulation results. Numerical simulations using the multiple reference frame (MRF) model based on ANSYS Fluent 12.0 were performed to obtain the impeller torque and ε. The relation between these parameters is discussed.

Experimental and simulation conditions

Figure 1 shows a schematic of the experimental apparatus. Molten Al was agitated by a two-paddle impeller in a crucible (31 % Al2O3–69 % SiO2) at 1073 K. The torques in baffled and non-baffled crucibles were measured using a rotational torque meter. To realize acceptable working conditions for the torque meter, an isolator was inserted between the impeller and the torque meter to prevent heat transfer from the molten Al and a fan was set beside the torque meter to cool it by blowing air. The impeller agitation speed was varied in the range of 200–400 rpm and the impeller torque in air at 1073 K was measured in advance as a blank test of the torque meter. Table 1 shows the experimental and simulation conditions.

Figure 1: Schematic of experimental apparatus.
Figure 1:

Schematic of experimental apparatus.

Table 1:

Experimental and simulation conditions.

CasesMass of Al (g)Number of bafflesAgitation speed (n/rpm)
AirAirNo baffle400
NB350350No baffle200–400
BF3502 baffles
NB500500No baffle
BF5002 Baffle

Simulation by MRF model

The simulation was performed in ANSYS Fluent 12.0 by using the 3D, double precision, pressure-based steady solver. Turbulence is modeled by the realizable k–ε model with standard wall functions. The MRF model proposed by Luo et al. [14] was applied to model. A rotating frame was used for the region containing the rotating components (i. e., the impeller), while a stationary frame was used for stationary regions. Steady data transfer is made at the MRF interface as the solution progresses. The solution of the flow field in the rotating frame in the region surrounding the impeller imparts the impeller rotation to the region outside this frame; the impeller itself does not move during this calculation. The boundaries of the impeller are treated as rotating walls; the top surface of the liquid is considered as a standard wall with zero shear stress; all other walls are set as non-slip standard walls.

The pressure velocity coupling is performed by means of the SIMPLE-scheme. Gradients are calculated based on a Green-Gauss cell-based scheme. The pressure is discretized using the PRESTO! Scheme; while the QUICK scheme is used for the momentum and turbulence. Figure 2 shows the meshes used for simulations of the agitated crucible with and without baffles, which are labeled by NB500 and BF500 respectively.

Figure 2: Meshes used in simulation models.
Figure 2:

Meshes used in simulation models.

Results and discussion

Turbulent flow in agitated vessels

The Saffman–Turner model [4] focuses on particle coagulation in the smallest eddies whose sizes are given by (as proposed by Kolmogorov):

(1)η=ν3ε14

Figure 3 shows a schematic diagram of the turbulent energy spectrum and the transport process in an agitated vessel [15]. The energy spectrum is obtained by calculating the kinetic energies of different sized eddies. Large eddies are produced by the rotating impeller. Their kinetic energy is then transferred to smaller eddies in a cascade process that is independent of the viscous force that is dominant only in the smallest eddies in which all the flow energy is dissipated as heat. A part of the energy of the fluid directly dissipates as heat near solid surfaces (e. g., the wall and the impeller). The collision frequency of particle groups i and j is given by eq. (2) when the particles are smaller than the Kolmogorov microscale (see the viscous subrange in Figure 3), as recommended by major studies [5, 6, 11, 12, 13, 17] since the particle size is much smaller than the turbulence microscale in practice. Additionally, the particle collision frequency is given by eq. (3) for turbulent eddies in the inertial subrange.

(2)Jij=A1(εl/ν)1/2(Rij)3ninj
(3)Jij=A2(εl/ν)1/3(Rij)7/3ninj

where Jij is the particle collision frequency, εl is the local EDR, ν is the kinematic viscosity of the fluid, Rij is the radius of the coagulating cluster, ni and nj are the particle number concentrations, and A1 and A2 are coefficients.

Figure 3: Schematic diagram of turbulent energy spectrum.
Figure 3:

Schematic diagram of turbulent energy spectrum.

Figure 4 shows the calculated local fluid velocity distribution in the crucible. The region immediately around the impeller (particularly in front of the blade) is a small zone with high kinetic energy, a strong vortex, rapid pressure changes, and fluctuating shear stresses. The tangential velocity of the liquid metal in non-baffled crucible (see Figure 4(a)) is much larger than that in the baffled crucible (see Figure 4(b)), whereas the radial velocity is much larger in the baffled crucible. Figure 5 shows the distribution of εl of the liquid metal in the crucible. Figure 5(a) shows that εl varies in the horizontal (XZ) plane in the non-baffled crucible, whereas εl is very low close to the crucible wall without much variation in the vertical (XY) plane. Figure 5(b) shows more uniform distributions of εl in the vertical and horizontal planes, which is suitable for particle coagulation experiments.

Figure 4: Fluid velocity distribution in crucible. (a) NB500, 400 rpm (velocity m/s). (b) BF500, 400 rpm (velocity m/s).
Figure 4:

Fluid velocity distribution in crucible. (a) NB500, 400 rpm (velocity m/s). (b) BF500, 400 rpm (velocity m/s).

Figure 5: Local EDR distribution of liquid Al in crucible. (a) NB500, 400 rpm (ε, m2/s3). (b) BF500, 400 rpm (ε, m2/s3).
Figure 5:

Local EDR distribution of liquid Al in crucible. (a) NB500, 400 rpm (ε, m2/s3). (b) BF500, 400 rpm (ε, m2/s3).

Relation between torque and ε in liquid metal

Since the experimentally measured torque of the impeller rotating at 400 rpm in air at 1073 K is almost zero, it is ignored in the following discussion. Figure 6 compares the measured and simulated torques in liquid metal. It shows that the calculated and measured torques agree well and are not greatly affected by the mass of the liquid metal. The impeller torque increases exponentially with the impeller rotation speed. It is much larger with baffles than without baffles due to the larger resistance force to the tangential fluid flow. The torque meter normally delivers the most accurate measurements when the measured torque is around the middle of the maximum scale. The deviations between the measured and calculated torque at low agitation speed are coming from the experimental errors cause by the torque meter.

Figure 6: Comparison of calculated and measured torques in liquid metal.
Figure 6:

Comparison of calculated and measured torques in liquid metal.

In the following discussion, the effective EDR is obtained from the volume-weighted average of εl. Figure 7 shows the dependence of ε on the simulated torque; ε increases exponentially with the agitation speed and it is affected by the presence of baffles in the crucible. The lower mass of liquid metal (350 g) gives a larger ε than the higher mass (500 g) of liquid metal. This indicates that turbulence dissipates easier in a smaller amount of liquid. The relation between the torque and ε is given by

(4)ε=k2πnTW

To investigate the effects of the fluid properties on k determined by a numerical simulation, 500 g of liquid Al was replaced with 210 g of water, which has the same volume in the crucible. Figure 8 compares the torques in water and liquid Al. They both increase exponentially with increasing agitation speed. At the same agitation speed, the torque in Al is greater than that in water, which indicates that the torque is affected by the properties of the fluid in the agitated vessel. Figure 9 compares ε in liquid Al and water; ε also increases exponentially with increasing agitation speed, whereas it is independent of the fluid properties.

Figure 7: Relation between ε and torque in liquid metal.
Figure 7:

Relation between ε and torque in liquid metal.

Figure 8: Comparison of torques in water and liquid Al.
Figure 8:

Comparison of torques in water and liquid Al.

Figure 9: Comparison of ε in water and liquid Al.
Figure 9:

Comparison of ε in water and liquid Al.

Figure 10 shows k obtained by eq. (4) using ε and the torques in water and liquid Al. Table 2 lists the average k values for crucibles with and without baffles. The k values for the crucible without baffles are slightly smaller than those for the crucible with baffles. The energy is mainly dissipated close to the rotating impeller and the baffles, where the turbulence is well developed. Thus, more energy is dissipated in the bulk of fluid for the baffled crucible. For a crucible with smaller amount of liquid, the turbulence is easier to be developed. Therefore, the k value is slightly larger for smaller fluid amounts in the crucible, since the turbulence dissipates easier into the bulk of the fluid. The fluid properties do not greatly affect k, particularly for the baffled crucible, whose mechanisms have not been reported. The k values increase slightly with increasing rotation speed of impeller, which is caused by the developing of turbulence in the crucible. The Re number in the agitating vessel is given by

(5)Re=ρndimpeller2μ
Figure 11 shows the Re numbers in the crucible with varying agitating speeds. The turbulence is well developed when the Re number is larger than 104. Thus, the k values increase a bit with the developing turbulence. The Re number in the water is much smaller than that in the liquid aluminum. Additionally, the physical meaning of k value is the proportion of the energy dissipated in the turbulence. It is easy to image that the turbulent is much less in the crucible without baffle, particularly with low agitation speed. On this account, k values for water with lower agitation speed are relatively lower in Figure 10.
Figure 10: Values of k as a function of agitation speed in water and Al for crucibles with and without baffles.
Figure 10:

Values of k as a function of agitation speed in water and Al for crucibles with and without baffles.

Table 2:

Average k values obtained in this study.

BaffleAl (350 g)Al (500 g)Water (210 g)*
NF0.74840.72230.6708
BF0.81640.73560.7313

  1. Note: *Same fluid volume as 500 g of Al.

Figure 11: Re numbers in the crucible.
Figure 11:

Re numbers in the crucible.

Comparison with previous studies

Table 3 summarizes k values and related data obtained in previous and present studies (500 g Al or 210 g water), where D is the top diameter of the vessel, H is the height of the vessel, d is the rotational diameter of the impeller, h is the impeller height, and w is the baffle width. Taniguchi et al. [11] adopted k=0.15 without simulating fluid flow by which the agreement between experimental results and theory was improved. However, this may not be the optimal agreement because the calculated result is not very sensitive to the value of ε. Nakaoka et al. [12] obtained the same results for k=0.15 as when the incorrectly modified Smoluchowski PBE was used [16]. Toyama et al. [10] measured the dissipated energy using a hot film probe and found that the values of k were about 0.5 and 0.2 in cylindrical agitated vessels with and without baffles. Arai et al. [13, 17] performed simulations in conjunction with torque measurements and proposed k=0.47 for a non-baffled cylindrical agitated vessel with a two-paddle impeller. The present study found k=0.6708 and 0.7223 for water and liquid Al (500 g) in a non-baffled crucible with a two-paddle impeller, respectively. All the above values for k are larger than that (k=0.2) of Toyama et al. This may be due to Toyama’s model using an impeller with multiple blades so that much more energy is dissipated at the vessel wall due to the tangential velocity being larger than in baffled vessels. Arai et al. [17] proposed the k values in a four-baffled cylindrical vessel with a two-paddle impeller as 0.76 (present study, k=0.7763) and 0.56 for two-pitched-blade impeller, which is smaller than that of a two-paddle impeller. This is because the vertical recirculation flow generated by the pitched-blade impeller dissipates more energy on the crucible wall. Arai et al. [17] also reported that k did not change when the sizes of the vessels, baffles, and impellers were varied, provided their shapes remained similar. The values of k found in the present study are slightly larger than those found by Arai et al. for the cases both with and without baffles in the vessels; this may due to the different vessel structures. Therefore, k for an agitated vessel is independent of the sizes of the vessels, impeller, and baffles when they are varied in proportion. It is also insensitive to the fluid properties. However, the structure and shape of the vessel and the impeller affect k, particularly the impeller shape and the presence of baffles.

Table 3:

Summary of k values and related data in this and previous studies.

Research workskLiquid (solution)Baffles (w)ImpellerVessel D (H/D)Coagulation Agreement
BladeTypeSize(d, h)
Higashitani et al. [6]0.15–1KCl4 (0.1D)2Paddle0.5D, 0.14D140 mm (1.0)Yes (improved)
Toyama et al. [10]0.2WaterNo6/8Paddle0.5D, 0.2D585 mm (1.0)—
0.54 (D)—
Taniguchi et al. [11]0.15NaCl4 (0.13D)2Pitched0.44D, 0.16D95 mm (0.95)Yes
Nakaoka et al. [12]0.15NaCl4 (0.13D)2Pitched0.44D, 0.16D95 mm (0.95)Yes
Arai et al. [13, 17]0.47MgSO4No (0.13D)2Paddle0.35D, 0.2D96 mm or 192 mm (2.0)
—No (0.13D)PitchedYes
0.764 (0.13D)Paddle
0.564 (0.13D)PitchedYes (k=0.5)
This study0.6708Water (210 g)No2Paddle0.43D, 0.21D—
0.73132 (0.14D)70 mm—
0.7223liquid Al (500 g)No(1.0)*—
0.73562 (0.14D)—

  1. Note: *D is the diameter of the top of the crucible in this study (70 mm).

Arai et al. [17] employed k=0.5 since it gave a good fit with the coagulation data, but it is smaller than the simulation values. Additional experiments are required to estimate k for fitting to the particle coagulation in molten metal.

Conclusions

The EDR relating to Kolmogorov microscale eddies is an important factor in the particle coagulation model. The impeller torque and ε were obtained by simulations using the MRF model in which the torque was validated by experimental measurements. The relation between the torque of the rotating impeller and ε in the bulk of fluid was investigated and compared with the results of previous studies concerning the particle coagulation. The following conclusions are drawn:

  1. The fluid properties do not greatly effect on k; it is independent of the sizes of the vessels, impeller, and baffles when they vary proportionally.

  2. The structures and shapes of the vessels and impellers affect k, particularly the impeller shape and the presence of baffles, which greatly affect fluid flow pattern and the proportion of energy dissipated in the bulk of the fluid.

  3. The k for particle coagulation model was lower than that obtained from the simulation according to the results of previous water model study. Additional experiments are required to fit k for the particle coagulation rate in molten metal.

Funding statement: This work was supported in part by a Grant-in-Aid for Scientific Research (A) (No. 22246097) provided by the Japan Society for the Promotion of Science (JSPS).

Acknowledgments

The authors are grateful to Dr. Arai of Kobe Steel Co. Ltd for his valuable suggestions and to Mr. Oka of Tohoku University for assistance with this study.

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Received: 2016-3-2
Accepted: 2016-7-16
Published Online: 2017-1-6
Published in Print: 2017-10-26

© 2017 Walter de Gruyter GmbH, Berlin/Boston

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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  1. Frontmatter
  2. Research Articles
  3. Effects of Rolling and Cooling Conditions on Microstructure of Umbrella-Bone Steel
  4. Synthesis and Characterization of Poly (styrene-co-butyl acrylate)/Silica Aerogel Nanocomposites by in situ AGET ATRP: Investigating Thermal Properties
  5. Processing and Properties of Fire Resistant EPDM Rubber-Based Ceramifiable Composites
  6. Preparation of Direct Reduction Sponge Iron (DRI) Using Pyrite Cinder Containing Nonferrous Metals
  7. Self-healing and Oxidation Resistance of B-Si Doped Carbon Materials Derived from Modification Coal Tar Pitch
  8. Evolution of Voids in Mg/Al Diffusion Bonding Process
  9. Energy Dissipation Rate in an Agitated Crucible Containing Molten Metal
  10. Dynamic Recrystallization Behavior of Vanadium Microalloyed Cryogenic Fine Grain Structural Steel Pipe at High Strain Rate
  11. Effect of High Temperature Hot Corrosion on the Compression Creep Behavior of 12Cr1MoV Alloys
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  13. Solubility of Nitrogen Gas into Molten Copper at Temperature Range of 1,993 K to 2,443 K
https://doi.org/10.1515/htmp-2016-0043
Keywords for this article
energy dissipation rate; turbulence; torque; particle coagulation; simulation
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Effects of Rolling and Cooling Conditions on Microstructure of Umbrella-Bone Steel
  4. Synthesis and Characterization of Poly (styrene-co-butyl acrylate)/Silica Aerogel Nanocomposites by in situ AGET ATRP: Investigating Thermal Properties
  5. Processing and Properties of Fire Resistant EPDM Rubber-Based Ceramifiable Composites
  6. Preparation of Direct Reduction Sponge Iron (DRI) Using Pyrite Cinder Containing Nonferrous Metals
  7. Self-healing and Oxidation Resistance of B-Si Doped Carbon Materials Derived from Modification Coal Tar Pitch
  8. Evolution of Voids in Mg/Al Diffusion Bonding Process
  9. Energy Dissipation Rate in an Agitated Crucible Containing Molten Metal
  10. Dynamic Recrystallization Behavior of Vanadium Microalloyed Cryogenic Fine Grain Structural Steel Pipe at High Strain Rate
  11. Effect of High Temperature Hot Corrosion on the Compression Creep Behavior of 12Cr1MoV Alloys
  12. Phenomenological Models to Predict the Flow Stress up to the Peak of as-Extruded 7050 Aluminum Alloy
  13. Solubility of Nitrogen Gas into Molten Copper at Temperature Range of 1,993 K to 2,443 K
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