17. Reactive Cleaning and Active Filtration in Continuous Steel Casting

To validate the hydrodynamic part of the CFD model, a model experiment is used [5]. The experiment is designed to mimic the typical flow in an ICF. The basic setup of the experiment is shown in Fig. 17.1. It mainly consists of a cylindrical vessel (180 mm diameter) filled with a water-glycerin mixture for refractive index matching, an impeller and eight guiding plates. This configuration is intended to produce the double vortex structure, which is typical for the flow in an ICF [6, 7]. The impeller rotates at \(n=400\,{\text{min}}^{-1}\) to prevent cavitation and air bubbling. The water-glycerin mixture has a density \({\rho }_{{\text{f}}}=1145\, \mathrm{kg }{{\text{m}}}^{-3}\) and a dynamic viscosity \(\mu_{{\text{f}}} = 9.6 \cdot 10^{ - 3} \, {\text{kg s}}^{ - 1} {\text{m}}^{ - 1}\). This results in a Reynolds number of approximately \({\text{Re}}\sim 1500\).

a. A 2 D diagram of a cylindrical vessel with an impeller and 8 guiding plates. Dimensions are marked. b and c are color graded contour plots of time averaged flow field in the stirred vessel for experiment and I L E S simulation. Both present two toroidal vortices. In b, the lower vortex is partially visible.

Particle Image Velocimetry (PIV) measurements were performed at this model. Polyamide particles with an average diameter of 22 µm and a density of \(\rho_{{\text{p}}} = 1060 {\text{kg}}/{\text{m}}^{3}\) were used as tracers and the central cross-section was illuminated with a laser light sheet (Raypower 2000, Dantec). Images were acquired using the Phantom V12.1 High Speed camera (400 fps, 2.5 ms exposure time, 1280 × 800 pixels) and a Nikkon Nikkor 35 mm lens. DaVis 8.4.0 software [8] was used to evaluate the velocity fields. Only the gray shaded area in Fig. 17.1a was used for evaluation in order to prevent a distortion of the evaluation by the rotating impeller. During a measurement, a total of 3200 images were recorded over a period of 8 s.

The hydrodynamic CFD model does not include the three additional terms on the right-hand side of the momentum Eq. (17.2). In order to consider the rotation of the impeller in the numerical simulation, the sliding mesh approach was used. Therefore, the pimpleDyMFoam solver was used for the simulation of the flow in the cylinder. For the turbulence modeling the Implicit Large Eddy Simulation [9] (ILES) approach were applied.

The grid created with cfMesh consists of a total of 6 million cells, with the largest cells being 1.1 mm in the fixed zone and 0.6 mm in the rotor zone. In addition, the mesh is refined around the impeller. There are 3 prism layers on the walls to resolve the boundary layer. The CFL number was set to 1. More information on the numerical setup, such as discretization schemes, can be found in Asad et al. [5]. Unless otherwise mentioned, the same discretization schemes are used in all subsequent simulations discussed here.

Figure 17.1b and c gives the mean velocity field in the vessel found in the PIV measurement and the ILES simulations. The velocity field is time averaged over a period of \(8 {\text{s}}\). In both cases, the formation of two toroidal vortices is clearly visible. This pattern is also typical for the flow in an ICF. However, due to the smaller evaluation area in the experiment, the lower vortex in Fig. 17.1b is only partially visible. The shape of the upper vortex differs only slightly between experiment and simulation. Deviations in the flow field occur especially in the area of the guiding plates and the impeller. Due to this acceptable agreement, ILES approach will be used in all the following sections.

The Lorentz force \({\varvec{f}}_{{{\text{lor}}}}\) acting on the fluid, which drives the flow in the ICF, is determined by Maxwell's equations [10]. These are as follows in a simplified form for magnetohydrodynamic (MHD) problems:

Here, \({\varvec{E}}\) is the magnetic induction, \({\varvec{B}}\) the intensity of the electrical field and \(\mu_{0}\) the magnetic permeability. In addition, Ohm's law in a simplified manner applies to the eddy current density \({\varvec{j}}\):with the electrical conductivity of the steel melt \(\sigma_{{\text{f}}}\) [11]. Combining Maxwell's equations and Ohm's law gives the induction equation for \({\varvec{B}}\):

Finally, the Lorentz force \({\varvec{f}}_{{{\text{lor}}}}\) results from:

The oscillating part of \({\varvec{f}}_{{{\text{lor}}}}\) is neglected due to the high frequency of the time harmonic magnetic field \(f\) (50–1000 \({\text{Hz}}\)). Because of the inertia of the melt, the flow would not be able to follow the oscillating part [11‐13]. In order to calculate the mean part of \({\varvec{f}}_{{{\text{lor}}}}\), the MaxFEM software and a 2D axial symmetrical model of the ICF was used [14, 15]. The resulting field of \({\varvec{f}}_{{{\text{lor}}}}\) is interpolated to the 3D CFD mesh in preparation of the CFD simulations using the “KDTree algorithm for nearest neighbor lookup” [16]. Further details regarding the MHD model can be found in the publication of Asad et al. [5].

The full MHD CFD model of the flow in an ICF is validated using experimental data from Baake et al. [17]. The dimensions and operating conditions of the configuration used by Baake et al. (ICF1) can be found in Fig. 17.2a and Tables 17.1 and 17.2. These also contain the data for the ICF2 investigated in CRC 920, which is discussed in Sect. 17.3. Unlike the ICF2, the ICF1 is run with Wood’s metal. The 3D grid of ICF1, created with cfMesh, includes about 4 million cells with a maximum cell size of 2.3 mm. Three prism layers were generated on the walls and top surface. This leads to an average \(y^{ + } = 1\), and therefore the use of wall functions can be omitted [18, 19]. The no-slip boundary condition is applied to the walls. The free surface of the fluid is modeled with a slip-wall, which means that its deformation is neglected. After a settling time of 30 s, the results are averaged over a period of 300 s of flow time. The time step \({\Delta }t\) equals 0.001 s and the ILES approach is used for turbulence modeling.

a. A sketch of right half of furnace around the crucible. Along the symmetry axis is the filter layer, with 6 coils on the right. Different radii values from the axis are marked. b. A scatterplot of u dot y in meters per second versus x i in millimeters plots a dataset fit by a concave upward ascending curve till the wall.

The Lorentz force \({\varvec{f}}_{{{\text{lor}}}}\) calculated with MaxFEM software is highest at the side walls of the ICF1, which is caused by the so-called skin effect [5, 20]. The averaged flow field is dominated by two toroidal vortices, with the vortices rotating such that the melt flows along the side wall coming from the bottom and the top. Since the Lorentz force is distributed almost symmetrically, both vortices are also nearly symmetrical to each other. However, because of typical long-term fluctuations in the flow, a slight asymmetry can be observed despite averaging over 300 s [17, 21‐23]. The highest velocities occur at the side walls due to the skin effect.

To validate the full MHD CFD model, the time-averaged vertical velocity \(\overline{u}_{y}\) in experiment and simulation is compared [17]. Figure 17.2b shows its distribution along a horizontal line running from the axis of symmetry through the center of the lower vortex to the side wall. The full MHD CFD model is thus able to reproduce the flow in the ICF1 with good accuracy. Therefore, the model is also applied in the following to the ICF2 investigated in CRC 920. A more detailed description of the model and the validation can be found in Asad et al. [5] and Asad [20].

To model the influence of the ceramic foam filter on the flow in the CFD simulations, it is assumed to be a homogeneous, isotropic, porous medium. The filter causes an additional pressure loss, which is considered in the Navier–Stokes equations (17.2) by the source term

This relationship corresponds to the Darcy-Forchheimer law, where \(\mu_{{\text{f}}}\) corresponds to the dynamic viscosity of the fluid, \(\kappa_{1}\) to the permeability and \(\kappa_{2}\) to the Forchheimer coefficient. Both, \(\kappa_{1}\) and \(\kappa_{2}\) were determined using a Direct Numerical Simulation (DNS) by subproject B02 [27]. The values for the filter with 10 pores per inch (ppi), which is used in the following, are \(\kappa_{1} = 1.96 \cdot 10^{ - 7} {\text{m}}^{2}\) and\(\kappa_{2} = 2.78 \cdot 10^{ - 3} {\text{m}}\).For each inclusion that passes through the filter, a filtration probabilityis calculated using the law of depth filtration. Here, \({\Delta }s\) is the penetration depth and \(\lambda\) is the filtration coefficient, which is a function of the local Reynolds number. The filtration probability is accumulated over several time steps and an effective filtration probabilityis calculated [27]. The indices n and o denote the current and the previous time step and \(\psi^{{\text{i}}}\) the instantaneous filtration probability according to (17.11). When the inclusion leaves the filter, \(\psi_{{{\text{eff}}}}\) is compared to a random number \(\zeta \in \left[ {0,1} \right]\). If \(\zeta < \psi_{{{\text{eff}}}}\) the inclusion is filtered and therefore removed from the simulation.

The discrete phase model (DPM) is used to model the transport of a disperse phase in the melt flow within a Lagrangian framework [28, 29]. The motion of the inclusions and bubbles with mass \(m_{{\text{p}}}\) and velocity \({\varvec{v}}\) is based on Newtons second law:

The total force \({\varvec{F}}_{{{\text{tot}}}}\) is the sum of the forces acting on the particle. In the simulations performed, the forces considered include buoyancy force \({\varvec{F}}_{{\text{B}}}\), gravitational force \({\varvec{F}}_{{\text{G}}}\), drag force \({\varvec{F}}_{{\text{D}}}\), virtual mass force \({\varvec{F}}_{{{\text{VM}}}}\), Saffman lift force \({\varvec{F}}_{{\text{L}}}\) and electromagnetic pressure force \({\varvec{F}}_{{{\text{EM}}}}\). Since the filtered velocity \(\overline{\user2{u}}\) is approximately the same as the melt velocity \({\varvec{u}}\) in the case of ILES, the use of a dispersion model is omitted.

Since the particles can be relatively large (see Sect. 17.3), they have an influence on the melt flow. Therefore, the source termis included in the momentum Eq. (17.2) to account for the two-way coupling between particles and fluid. Details on the implementation of the DPM model in OpenFOAM can be found in Asad [20]. Further investigation on the influence of the turbulence model and drag closure on particle motion are provided by Asad et al. [30, 31].

Before the start of the simulation, the carbon concentration \({\text{c}}\) in the entire melt is \(c = 0\). The carbon required for bubble formation is dissolved from the filter, making it the only source of carbon in the simulation. The transport of the dissolved carbon within the melt is subject to the transport equation

The Schmidt number \({\text{Sc}} = {\upnu }_{{\text{f}}} /D\) is obtained from the diffusion coefficient for carbon in the molten steel \(D = 10^{ - 8} {\text{ m}}^{2} s^{ - 1}\) [35]. The turbulent Schmidt number is assumed to be \({\text{Sc}}_{{\text{t}}} = 1\) and the eddy viscosity \(\nu_{{\text{t}}}\) is estimated according to Smagorinsky [36, 37]. The sink term \(S_{{\text{k}}}\) considers the reaction and the associated reduction of the carbon concentration.

The reaction between oxygen and carbon is limited to the inclusion surface and therefore cells in which inclusions are present. It is assumed that there is always enough oxygen available in the melt for the reaction. Thus, the oxygen concentration is not considered in the simulations. The reaction reduces the initial amount of carbon \(n_{0} = cV_{{{\text{cell}}}}\) in the cell with volume \(V_{{{\text{cell}}}}\) to

Here, \(k_{{\text{r}}} = 14.9 {\text{mol m}}^{ - 2} s^{ - 1}\) is the reaction rate constant, \(A_{{\text{p}}}\) the surface area of the particle and \(n_{{\text{p}}}\) the number of particles in the computational cell. The increase of volume of each bubble at an inclusion is, according to ideal gas lawwith ideal gas constant \(R\) and Temperature \(T\). The corresponding sink term is evaluated as

The model assumes, that the CO bubble creates a gas layer around the inclusion, which can grow due to the reaction. Therefore, inclusion and surrounding gas form an equivalent particle with the effective volume \(V_{{{\text{eff}}}} = V_{{\text{b}}} + V_{{\text{p}}}\) and mass \(m_{{{\text{eff}}}} = \rho_{{\text{b}}} V_{{\text{b}}} + \rho_{{\text{p}}} V_{{\text{p}}}\). The CO density in the bubble \(\rho_{{\text{b}}}\) is calculated according to the ideal gas law. The size of the inclusion remains constant throughout the simulation, allowing the equivalent particle to grow only as the CO bubble grows. In addition, effective density \(\rho_{{{\text{eff}}}}\) and effective diameter \(d_{{{\text{eff}}}}\), which is the diameter of a sphere with the same volume, are calculated. The effective quantities are used to calculate the forces acting on the equivalent particle and the particle motion (17.13). Because of the low void fraction of particles and therefore the low contact probability, the agglomeration of inclusion-bubble aggregates is neglected.

The computational grid for the simulation of the ICF2 was created with cfMesh and has the same properties and boundary conditions as the grid of the ICF1 (see Sect. 17.2.2). ILES was used for turbulence modelling and the same schemes as in Sect. 17.2.1 are applied to the simulation. The time step is \({\Delta }t = 0.001 {\text{s}}\).

First, the flow is simulated without considering the NMIs for 30 s to allow the flow to fully develop. Then \(2 \cdot 10^{6}\) inclusions with a diameter \(d_{{\text{p}}} \, = \,5\,{\upmu m}\) and a density \(\rho_{{\text{p}}} = 3200 {\text{kg}}/{\text{m}}^{3}\) are randomly injected close to the top surface. To allow the particles to disperse in the ICF2, the simulation is run again for 30 s. Subsequently, the filter is inserted for 10 s so that carbon can distribute in the melt and bubble formation can begin. The carbon concentration within the filter is \(c = 1 {\text{mol}}/{\text{m}}^{3}\) and remains constant throughout the immersion phase. After this time, the filter is removed and the magnetic field is deactivated, analogous to the experiments of Dudczig et al. [38] and Storti et al. [32]. Finally, the simulation is run for another 100 s to allow the reaction to continue and the particle-bubble agglomerates to rise to the surface.

The flow field at the vertical midplane and horizontal plane at \(y = 0\) shown in Fig. 17.3 is averaged over the immersion time of filter (10 s). It can be seen that a large toroidal vortex dominates the flow in the ICF2 due to the different setup of the magnetic field compared to ICF1. The filter also significantly influences the flow. Inside the filter, the flow velocities are very low due to the high flow resistance. At its surface, on the other hand, the velocity is very high. The rising particle-bubble agglomerates accelerate the melt in this area. The high velocity at the filter surface also ensures improved transport of the carbon away from the filter. Despite time averaging, asymmetric structures and secondary vortices can be seen in the horizontal plane. In addition, when instantaneous flow fields are examined, many turbulent structures are visible, which enhances the mass transport.

2 simulation plots of velocity fields and streamlines in the vertical midplane and horizontal plane. The former presents a vertical rectangle in the center in the rectangular plane. The latter presents a circle with a small square representing a filter at the center. The flow in the filter is low, and at the surface it is higher.

Since the filter is the only source of carbon, the concentration \(c\) is highest in it and on its surface. With advancing time, the carbon is better distributed in the ICF2, as can be seen in Fig. 17.4. After removing the filter, the concentration subsequently decreases, since the dissolved carbon is used for bubble formation. Figure 17.5 shows the particle distribution at times \(t = 1 {\text{s}}\) and \(t = 10 {\text{s}}\). Only particles with \(d_{{{\text{eff}}}} > d_{{\text{p}}}\) are included there. Because of the higher carbon concentration directly around the filter, the largest equivalent particles are located in this area. These rise quickly to the surface because of their low effective density. The formation of bubbles is also possible further away from the filter. However, it takes a certain amount of time from the immersion of the filter until particle-bubble agglomerates are also present near the side walls. There, the equivalent particles are smaller because of the lower carbon concentration \(c\). Due to the wetting properties, once NMIs have risen to the free surface, they will remain there and are not redispersed back into the melt. Therefore, as they reach the free surface, the particles are deactivated, preventing them from moving and growing any further.

2 simulation plots of carbon concentration field in the vertical midplane and horizontal plane. The former presents a vertical rectangle in the center in the rectangular plane. The latter presents a circle with a small square representing a filter at the center. Streaks emanate from the filter in the melt.

a and b are 3 D diagrams of the cylindrical vessel plot the particle growth around the filter at 1 and 10 seconds after immersion of filter. a presents diffusion around the cuboidal filter and on the top surface. b presents the spread of growth extending towards the walls. D effective is smaller near the walls and larger near the center.

Due to the ongoing bubble formation, the average particle size increases continuously over the simulation time. In addition, the number of particles with \(d_{{{\text{eff}}}} \, = \,5\, {\upmu m}\) decreases as bubbles form on an increasing number of inclusions.

The formation of in-situ layers on the filter wall can cause the carbon concentration \(c\) on the filter wall to decrease [39]. In addition to the previously discussed case with a constant concentration in the filter, four cases with decreasing concentration in the filter are investigated. The concentration decrease during the immersion phase is either linear or exponential. The concentration in the filter at the end of the immersion time is either \(c_{{{\text{min}}}} = 0\) or \(c_{{{\text{min}}}} \, = \,0.5\, {\text{mol}}/{\text{m}}^{3}\). Considering the decrease in concentration does not significantly affect the flow in ICF2, it will not be discussed in more detail.

With the combination of \(c_{{{\text{min}}}} = 0\) and exponential decrease, there are very small amounts of dissolved carbon in the melt after the filter is removed. This is also reflected in the barely noticeable increase in size of the equivalent particles. With the linear decrease to \(c_{{{\text{min}}}} = 0\), however, there is a higher carbon concentration around the filter. This leads to an improved bubble growth compared to the exponential decrease.

For \(c_{{{\text{min}}}} \, = \,0.5\, {\text{mol}}/{\text{m}}^{3}\), the carbon distribution in the melt is very similar for the linear and exponential decrease. The amount of dissolved carbon is significantly larger compared to \(c_{{{\text{min}}}} = 0\), which leads to a larger effective diameter of the equivalent particles.

The cleanliness of the melt \(\eta\) is defined by the ratio of particles on the surface at the end of the simulation \(n_{{{\text{top}}}}\) to the initial number of particles \(n_{{{\text{init}}}}\):

Figure 17.6 shows the evolution of \(\eta\) over time starting from the immersion of the filter for all cases. It can be seen that the formation of the CO bubbles has a significant effect on the cleanliness of the melt. With increasing time, more and more inclusion-bubble-agglomerates reach the melt surface. At constant carbon concentration in the filter, about 30% of the NMIs have reached the surface and thus are filtered out of the melt 10 s after insertion of the filter. Even after removal of the filter, NMIs continue to be transported to the melt surface. At the end of the simulation, about 63% of the particles have been filtered out of the melt. This result is in the order of the experiments of Storti et al., where 60–95% of the NMIs were removed [32]. Reactive filtration could thus explain the difference between the active filtration simulations of Asad et al. [27] and the experimental results mentioned before.

A line graph of eta percent versus time in seconds. It plots 4 concave downward ascending trends for constant c = 1 moles per cubic meter, linear c min = 0.5 moles per cubic meter, exponential c min = 0.5 moles per cubic meter, and linear c min = 0 moles per cubic meter from top to bottom. A horizontal line is plotted for exponential c min = 0.

Reducing the C concentration in the filter during the immersion phase to \(c_{{{\text{min}}}}\) provides a reduced cleaning effect. At \(c_{{{\text{min}}}} = 0\), there are large differences between linear and exponential decrease. For the linear decrease, a cleanliness level of 38% is reached at the end of the simulation. With the exponential decrease, almost no cleaning effect is observable. The poorer cleaning effect is closely related to the reduced carbon concentration in the crucible. For \(c_{{{\text{min}}}} = 0.5 {\text{mol}}/{\text{m}}^{3}\) there is little difference in the cleanliness of the melt between the two variants, analogous to the carbon concentration field. For both linear and exponential decay, the degree of cleanliness at the end of the simulation is \(\eta \approx 50 \%\).

The second possibility for bubble formation besides the formation at inclusions is the formation directly at the filter. Since the filter is considered in the simulation by means of the Darcy Forchheimer law and thus there are no explicit filter walls, the bubble growth at the filter cannot be represented correctly. Therefore, the bubbles are initialized at 32 different positions in the lower half of the filter with a diameter of \(1 {\upmu m}\). Injection in the lower half of the filter should ensure a long residence time. At each of the points 100 bubbles per second are injected. After injection, the bubble grows to diameter \(d_{{{\text{b}},{\text{max}}}}\) over a period of \(\tau = 10 {\text{s}}\). Two types of bubble growth are examined: the bubble volume \(V_{{\text{b}}}\) increases either linearlyor exponentially

\(V_{{{\text{b}},0}}\) is the initial volume of the bubble and \(V_{{{\text{b}},{\text{max}}}}\) its maximum volume. The maximum diameters investigated are \(d_{{{\text{b}},{\text{max}}}} \, = \,100, \,300, \,500\, {\upmu m}\), which ensures that the bubbles remain spherical [40].

In order to account for the cleaning effect of the bubbles in the simulation, the bubble-inclusion attachment is modeled. The collision probability \({\Psi }\) between bubbles and inclusions in the same computational cell is calculated as follows [41]:

Here \({\varvec{U}}_{{{\text{rel}}}}\) is the relative velocity between bubble and inclusion and \(n_{{{\text{min}}}}\) is the minimum number of bubbles or inclusions in the cell. The collision probability is compared to a random number \(\xi\) to determine a collision occurs. In case of a collision, the corresponding inclusion is removed from the simulation. Two-way coupling between bubbles and melt is considered, but not the collision and agglomeration of NMIs with each other. The filter only influences the flow field and does not actively filter inclusions. The simulation duration is \(t = 70 {\text{s}}\). Further information can be found in Asad et al. [34].

The formation of the bubbles directly at the filter again leads to high flow velocities around the filter. Therefore, the flow field is similar to that shown in Fig. 17.3 and will not be discussed in more detail [34].

The distribution of bubbles at time \(t\, = \,10\, {\text{s}}\) and for exponential growth with maximum diameter \(d_{{{\text{b}},{\text{max}}}} \, = \,500\, {\upmu m}\) is shown in Fig. 17.7. The bubbles have a small size at first, causing them to follow the melt flow after their injection. The size of the bubbles increases as the residence time increases. As a result, they experience a greater buoyancy force and begin to rise upward. It can be observed that the bubble ascent starts sooner the larger \(d_{{{\text{b}},{\text{max}}}}\) is. With linear growth, the bubble size increases faster in the early stages, causing them to rise earlier. At \(d_{{{\text{b}},{\text{max}}}} = 500 {\upmu m}\) and linear increase, the bubbles grow so fast that they cannot distribute in the ICF2 and rise to the surface right at the filter.

A 3 D diagram of the cylindrical vessel plots the bubble distribution around the filter surface on exponential bubble growth. D b is plotted using a color gradient scale. The bubbles have a larger diameter near the filter that decreases as we approach the wall.

The cleanliness of the melt at the end of the simulation (70s) for the different cases is summarized in Table 17.3. A visible cleaning effect occurs only for exponential bubble growth. The best cleanliness of the melt is achieved at exponential growth and \(d_{{{\text{b}},{\text{max}}}} \, = \,500 \,{\upmu m}\). In this case, the bubbles can distribute well due to the initially slow growth. As a result, during the subsequent ascent of the larger bubbles, almost 6% of the inclusions are removed. In the case of linear growth, especially for \(d_{b,\max } \, = \,500 \,{\upmu m}\), the bubbles grow faster in the early stage and thus move upwards faster. Therefore, they are not distributed well in the ICF2 and cannot collect many inclusions [34].

Overall, the influence of bubble formation on the filter on the cleanliness level is negligible compared to bubble formation on NMIs. The reason for this could be that small bubbles have a relatively low probability of contact with an inclusion. Larger bubbles, on the other hand, rise quickly to the surface and thus have little time for contact with inclusions.

Simulations of active filtration have shown that the melt cleanliness \(\eta = 4 \%\) with a single filter (10 ppi) is significantly lower than that of reactive filtration with bubble formation at the NMIs [27, 33]. Therefore, in this section, the cleaning efficiency with combined active and reactive filtration is investigated. For this purpose, one or three filters are placed in the ICF2. The filters act either actively only or actively and reactively. An overview of the different configurations is provided in Table 17.4 and Fig. 17.8. CO bubble formation on the filter is not considered in this case, because of the better cleaning performance of the bubble formation on inclusions.

A front view sketch of I C F 2 with 3 filters in line separated by a distance of 35 millimeters. The length from top to bottom is 60 millimeters, while the width is 20 millimeters.

The combined filtration is simulated for 300 s, with the respective filters acting reactively and releasing carbon to the melt only in the first 10 s. While the filters act reactively, there is a constant concentration of \(c = 1 {\text{mol}}/{\text{m}}^{3}\) in them. Unlike in Sect. 17.3.1, the magnetic field remains active for the entire simulation. The NMIs have the same properties as in Sect. 17.3.1. Also, the same grid and numerical schemes are used. The time step size is \({\Delta }t = 0.001 {\text{s}}\) and ILES is used for the turbulence modeling.

The flow field of case (2) at the vertical midplane and horizontal plane at \(y = 0\) in Fig. 17.9 is averaged over the entire simulation duration of 300 s. As expected, the number and position of the filters has a significant influence on the flow field. The toroidal vortex is again dominant. However, it is confined to the space between the side wall and the two outer filters. Compared to Fig. 17.3, the velocity at the filter surface is lower. Since bubble formation only occurs in the first 10 s and the averaging is done over the entire simulation, the mean flow field is barely influenced by the bubble rise. The flow field of case (1) is very similar to that of Fig. 17.3 except for the lower velocity at the filter surface.

2 simulation plots of velocity fields and streamlines in the vertical midplane and horizontal plane for 3 filter case. The former presents 3 vertical rectangles in the rectangular plane. The latter presents a circle with 3 squares representing filters. Vortices are toroidal and confined to the space between the side wall and 2 outer filters. Compared to single filter case, the velocity at filter surface is lower.

The flow fields of cases (2) and (3) are also almost identical with only small differences because of the bubble formation. Due to remaining residual carbon concentration, bubble formation can continue even after the filter is removed from the melt. With a longer simulation time, the differences should disappear, since the flow is then determined by the Lorentz force.

The carbon concentration field of case (1) was already shown in Fig. 17.4. Figure 17.10 shows the carbon distribution at the end of the reactive phase (\(t = 10 {\text{s}}\)) for case (2). Since there are two reactive filters, the carbon can be better distributed in the crucible. This should be beneficial for the filtration effect due to bubble formation at inclusions. It can also be seen that the carbon practically cannot get into the middle filter. The carbon field of case (3) differs only slightly from case (1) because the additional two filters impede the transport of carbon to the side wall [20].

2 simulation plots of velocity fields and streamlines in the vertical midplane and horizontal plane for 3 filter case. The former presents 3 vertical rectangles in the rectangular plane. The latter presents a circle with 3 squares representing filters. Streaks emanate from the two outer filters in a more prominent manner than the middle filter.

The effect of reactive cleaning by bubble formation is again evaluated by the cleanliness of the melt \(\eta\). Figure 17.11 shows the temporal development of \(\eta\) for all three cases with combined filtration. The effect of active filtration is not considered here. The best results in terms of reactive filtration are provided by case (2), followed by case (1) and case (3). As shown in Sect. 17.3.1, further cleaning of the melt takes place even after the end of the reactive phase. Through carbon distributed in the ICF2, bubbles can still form on inclusions. Because of the two reactive filters in case (2) and therefore better distributed carbon, the cleanliness of the melt at the end of the simulation is \(\eta = 83 \%\). Case (3) reaches the lowest cleaning effect with \(\eta = 63 \%\) because of the blocked carbon transport and therefore reduced bubble formation.

A line graph of eta percent versus time in seconds. It plots 3 concave downward ascending trends for case 2, case 1 and case 3 from top to bottom.

Figure 17.12 shows the fraction of inclusions removed at the end of the simulations for both reactive and active filtration. When combining both filtering mechanisms, reactive cleaning again has a stronger effect on the cleanliness of the melt than active filtration. This is true for all three configurations and despite the longer duration of the active filtration. This supports the assumption that the large differences between the experiments of Storti et al. [32] and the active filtration simulations of Asad et al. [27] are caused by the formation of CO bubbles on inclusions. Regarding the combined filter performance, case (2) also provides the best results. In addition, it can be seen that a larger number of active filters also allows more particles to be actively filtered. However, purely active filters can impede reactive filtering, as observed in case (3). Therefore, only slightly more NMIs were filtered in case (3) than in case (1), although only one filter is present there. Further configurations for combined cleaning can be found in Asad and Schwarze [42].

A horizontal stacked bar graph compares the cleaning efficiencies for cases 1 to 3 based on active and reactive cleaning. The efficiency is highest in case 2, followed by cases 3 and 1. Reactive cleaning has a stronger effect on efficiency than active filtration.

To study the interaction between bubbles and particles during reactive cleaning in more detail, a new experimental setup was developed. The setup is shown in Fig. 17.13 and consists of a thin column (1) of dimensions 300 × 120 × 30 mm. The column is filled with pure water in order to avoid additional influences on the bubble formation and motion. The height of the water level is \(110 \,{\text{mm}}\). An automatic bubble injector is located centrally at the bottom of the column and is supplied with compressed air. The bubble injector has the advantage of being able to generate reproducible bubbles at an adjustable frequency. Here, the bubble generation frequency was set to \(f\, = \,2\, {\text{Hz}}\). The bubble injector device is presented in detail in Ostmann and Schwarze [43].

A diagram of the experimental setup for particle bubble interaction investigation. It has a cuboidal column with a bubble injector at the bottom center. A high speed camera mounted on a guide block is used to capture ascending bubbles. A motor moves the camera over a linear toothed belt.

Polyester particles with a density \(\rho_{{\text{p}}} \, \approx \,1300 \,{\text{kg}}/{\text{m}}^{3}\) were dispersed in the water. The particles are hydrophobic, polydisperse and have a diameter \(d_{{\text{p}}} \, < \,315 \,{\upmu m}\). Moreover, the particles are not spherical.

A MIKROTRON Motion BLITZ EoSens mini2 (Mikrotron GmbH, Unterschleissheim, Germany) high-speed camera (2) was used to capture the ascending bubbles. The camera is mounted on the guide block (3), which can move vertically along the linear toothed belt axis (4) (Igus drylin R © ZLW-1040) in order to track the rising bubble at sufficiently large resolution in a small field of view (12.8 × 13.5 \({\text{mm}}^{2}\)). A motor (5) with motor control system (Igus drylin R © dryve D1) moves the camera upwards at approximately \(300\, {\text{mm}}/{\text{s}}\) to match the velocity of the rising bubble. The image size was 992 × 1040 pixels which allowed a frame rate of 1200 fps, i.e., an interframe delay of \(833\, {\upmu s}\). In combination with a \(100 \,{\text{mm}}\) lens the camera system provides a resolution \(13 \,{\upmu m}\) per pixel. The water column was illuminated with a diffuse LED-based light source (6). A total of 62 image series of rising bubbles were acquired.

The Open Source Computer Vision Library (OpenCV) is used to evaluate the images. First, the bubble is detected in each image of the series and its properties such as diameter \(d_{{\text{b}}} \, \approx \,2.2 \,{\text{mm}}\), position \({\varvec{x}}_{{\text{b}}}\), and velocity \({\varvec{v}}_{{\text{b}}}\) are determined. Selected bubble trajectories are shown in Fig. 17.14a. The trajectories of the bubbles were detected until the bubble touches an edge of the raw image. Up to a height of 20 mm from their injection point, the bubbles follow approximately the same trajectory. After that, the bubbles start to leave the nearly vertical path and the trajectories of the individual bubbles begin to differ significantly, i.e. the particle motion becomes unstable. During the horizontal dispersion, the bubble also tilts in the direction of its motion.

a. A graph of y versus x plots bubble trajectories originating from (0, 0), rising vertically and then diverging. b. A photo of a bubble with 2 attached particles at the downstream side. The bubble has a ring-shaped periphery.

In the next step, the image is cropped to a size of 280 × 440 pixels such that the bubble stays in the center of the image. An example of a cropped image with a bubble in the center can be seen in Fig. 17.14b. The cropped images are used to study the interaction of the rising bubble with the dispersed particles. To evaluate the particle’s trajectories, they are tracked using OpenCV's Channel and Spatial Reliability Tracking (CSRT) [44]. For this purpose, the particles to be tracked are manually selected. Thereby, a distinction is made whether the particles are will attach to the bubble or not.

The resulting particle trajectories \({\varvec{x}}_{{\text{p}}} \left( t \right)\) are shown in Fig. 17.15. The center of the bubble is used as the origin of the coordinate system. The particle velocity \({\varvec{v}}_{{\text{p}}}\) is also specified relative to the center of the bubble. All particles which attach to the bubble approach the bubble at \(x_{{{\text{p}},0}} \, < \,r_{{\text{b}}}\), where \(x_{{{\text{p}},0}}\) is the first tracked position of a particle and \(r_{{\text{b}}}\) is the bubble radius. It is noticeable that the particles, especially those near \(x_{{\text{p}}} \, = \,0\) and thus the stagnation point, are decelerated before contact with the bubble. As they slide around the bubble, the particles are accelerated by the flow around the bubble until they finally reside on the downstream side of the bubble.

a and b are graphs of y p versus x p that plot the velocity field lines. a presents lines of higher velocities approaching the bubble, resulting in further reduction of velocity at the bubble surface. b presents field lines on the left and right periphery of the bubble.

There are also many particles which approach the bubble at \(x_{{{\text{p}},0}} \, < \,r_{{\text{b}}}\), but do not attach to the bubble (Fig. 17.15b). It should be noted that only those particles are shown which were visible on their entire path around the bubble and were not covered by the bubble. The particles are deflected and accelerated by the flow around the rising bubble. Only the particles closest to \(x_{{\text{p}}} \, = \,0\) are slowed down as they approach the bubble and are influenced by the stagnation point.

In order to assess the dependence of the attachment on the particle position, the attachment probability \(P_{{{\text{att}}}}\) is evaluated as a function of the initial distance of the particles from the symmetry axis of the bubble \(\left| {x_{{{\text{p}},0}} } \right|\) (Fig. 17.16). For this purpose, nine bins with the width \({\Delta }x_{{{\text{p}},0}} \, = \,0.125 \,{\text{mm}}\) are analyzed. For each bin, the attachment probability \(P_{{{\text{att}}}}\) is calculated as follows:

A histogram plots the variation in attachment probability with relative position to the axis of symmetry. The probability decreases with an increase in the relative position value.

Here, \(n_{{\text{p}}}\) is the total number of particles in the bin and \(n_{{{\text{p}},{\text{att}}}}\) is the number of attaching particles in the bin. A total of 1094 particles were evaluated for this purpose.

The attachment probability decreases with increasing distance from the bubble's symmetry axis. If a particle approaches the bubble along the symmetry axis, the attachment probability is \(P_{{{\text{att}}}} \, \approx \,21 \%\). This value decreases for \(0.875 \,{\text{mm}}\, < \,\left| {x_{{\text{p}}} } \right|\, < \,1{ }\,{\text{mm}}\) to \(P_{{{\text{att}}}} \, \approx \,2 \%\). For \(\left| {x_{{\text{p}}} } \right|\, > \,1 \,{\text{mm}}\), which is less than the bubble radius \(r_{{\text{b}}} \, \approx \,1.1 \,{\text{mm}}\), no more attachment of particles was observed.

However, the current experimental setup has disadvantages because the particle paths can only be determined within the x–y plane. Due to the lack of information in the z-direction, particles located in front of or behind the bubble are also taken into account when determining \(P_{{{\text{att}}}}\). In addition, the evaluation of the distance to the symmetry axis of the bubble should ideally be done within the x–z plane and not only in the x-direction. In this case, the attachment probability around the symmetry axis would be expected to be greater than 21%. Therefore, a new experimental rig has already been set up, on which 3D observations are possible by implementing three cameras from three different viewing angles (Fig. 17.17).

A photo of the test rig for particle bubble interaction. It has 3 cameras around the fluid column.

In addition to the experimental studies, the bubble rise is also be investigated numerically. For this purpose, the interFoam solver of OpenFOAM, which is based on the volume of fluid (VOF) method, is used [45]. With the VOF method, the interface is implicitly represented by the phase fraction \(\alpha\). The VOF-method was also used by Asad et al. in order to investigate the immersion process of CFFs [46].

The dimensions of the mesh are 60 × 63 × 23 mm and the base mesh consists of 1.6 million cells with a maximum cell size of 1 mm. Adaptive mesh refinement is used to capture the interface as accurately as possible. The cells at the phase interface are refined to a size of \(62.5\, {\upmu m}\).

The bubbles are generated with a frequency \(f\, = \,20\, {\text{Hz}}\) or \(f\, = \,40\, {\text{Hz}}\) and a diameter \(d_{{\text{b}}} \, = \,2\, {\text{mm}}\) at the bottom of the domain using setFields. The surface tension between air (\(\rho_{{\text{g}}} \, = \,1.22\, \frac{{{\text{kg}}}}{{{\text{m}}^{3} }},\nu_{{\text{g}}} \, = \,14.8 \cdot 10^{ - 6} \, {\text{m}}^{2} /{\text{s}}\)) and water (\(\rho_{{\text{g}}} \, = \,1000 \,\frac{{{\text{kg}}}}{{{\text{m}}^{3} }},\nu_{{\text{g}}} \, = \,1.0 \cdot 10^{ - 6} { }\,{\text{m}}^{2} /{\text{s}}\)) is \(\sigma \, = \,0.07 \,{\text{N}}/{\text{m}}\). The bottom wall and the side walls are modelled as no slip walls. The simulated time is \(t = 1 {\text{s}}\).

Figure 17.18 shows the velocity in a cutout of the vertical midplane and the bubble contours (\(\alpha \, = \,0.5\)). For \(f\, = \,20\, {\text{Hz}}\), the bubbles initially rise in a straight line along the same trajectory, as the wake of the bubbles shows. The bubble velocity at the end of the straight-line ascent is approximately 320 mm/s. After about 30 mm of ascent, the bubbles become unstable and leave the straight-line path. At \(f\, = \,40\, {\text{Hz}}\), the trajectories of the individual bubbles already start to differ after about 25 mm. The rising velocity of the bubbles is also slightly increased in the case of the higher bubble frequency.

2 V O F simulations plot the velocity field and bubble contours of a rising bubble chain in vertical midplane at 2 different generation frequencies. The bubbles rise initially along a vertical line till 30 millimeters and divulge. The velocity of bubble increases, and the rise is higher for a higher frequency.

In future simulations, particles will also be taken into account using the Lagrangian approach. In addition, the goal is to implement a model for the particle-bubble interaction and the attachment probability of the hydrophobic particles to the bubble. The implemented model is expected to reflect the experimental results well, so that the new model can be applied to real melt-NMI systems.