Cold Model Experiment and Numerical Simulation of Flow Characteristics of Multi-phase Slag

Abstract

In order to attain a highly efficient dephosphorization, multi-phase slags consisting of liquid and solid phases have been used for the hot-metal treatment processes.However, the viscosity of a multi-phase slag dramatically increases with the fraction of coexisting solid phase and transits to a Bingham fluid, which may cause difficulties in discharging the slag after the refining. In this study, in order to understand flow behaviors, the simulated multi-phase slag discharged from a dam was observed by a high-speed video system. The criterion for the free jet to wall flow was obtained. The travelling time of the simulated slag increased dramatically with the solid fraction in the Bingham fluid region. A three-dimensional smoothed-particle hydrodynamics simulation program for Bingham fluid was developed and its calculations were compared with the experimental observations. The simulation replicated the observed flow in the cold model experiment.

1. Introduction

In order to attain a highly efficient dephosphorization, multi-phase slags consisting of liquid and solid phases have been used for the hot-metal treatment processes.¹⁾ However, it is reported that the viscosity of a multi-phase slag dramatically increases with the fraction of coexisting solid phase and transits to a Bingham fluid,^2,3) which may cause difficulties in discharging the slag after the refining. Therefore, the study of the rheological characterization and flow behavior of the multi-phase slag is important for the development of more efficient refining processes. Since high-temperature experiments are usually difficult and costly, the development of numerical simulation is also necessary. Since the simulation will be applied to the slag-metal separation and the slag treatment processes such as a deslagging and an atomization, the large deformation of slags with free surfaces must be treated, the particle methods are promising ones which can treat the deformations in fluids without the problem of grid tangling. Smoothed particle hydrodynamics (SPH) methods using explicit algorithm has the advantage of reduced calculation costs for the system of a large number of particles.

In this study, a cold model experiment of the simulated multi-phase slag was performed and its flow behavior was observed. A three-dimensional SPH simulation program was developed to describe the flow of very viscous fluids and Bingham fluids. The flow behavior calculated with the SPH program was compared with the experimental observations.

2. Experimental Conditions

2.1. Viscosity Measurement of Simulated Slags

Slurry, composed of polystyrene spheres and silicone oil, was employed for the simulated multi-phase slag as reported by Sukenaga.³⁾ Two types of silicone oil were used as liquids: A (density = 0.965, viscosity = 0.0482 Pa∙s) and B (density = 0.97, viscosity = 0.324 Pa∙s). Spherical polyethylene beads (density = 0.92, average diameter = 180 μm) were uniformly suspended in the liquids with various volume fractions (from 0 to 0.55). The Brookfield rotating cylinder viscometer (Eiko instruments, type-LVT) was used to measure the viscosity. The sample was kept in a chamber (12.7 mmϕ). The spindle (8.74 mmϕ) was immersed into the chamber and rotated from 0.3 to 60 rpm to measure the required torque. The viscosity of the simulated slag was estimated by measuring the relationship between shear stress and shear rate. Experimental conditions are listed in Table 1.

Table 1. Experimental conditions.

Mean diameter of beads (μm)	162.5
Solid fraction (−)	0, 0.1, 0.2, 0.3, 0.40, 0.5, 0.55
Viscosity of silicone oil (Pa∙s)	0.0482, 0.324
Shear rate (s−1)	0.12, 0.24, 0.6, 1.2, 2.4, 4.8, 12.0, 24.0

2.2. Observation of the Flow Discharged from a Dam

Figure 1 shows the schematic drawing of the experimental apparatus. The transparent vessel was made from acrylic resin plates with 10-mm thicknesses. The simulated slag (slurry) with a 20-mm depth was stored in the dam (width = 50 mm, length = 100 mm). Flow under gravity was started by quickly opening the sliding gate made of aluminum. The flow behavior was continuously observed by a high-speed video system recording at 480 frames per second.

Fig. 1.

Schematic drawing of the experimental apparatus.

3. Simulation Model

3.1. SPH Model

The three-dimensional SPH model^4,5) was used for the numerical simulation in this study.⁶⁾ Lucy’s kernel function was employed for the simulation. It is defined as follows:   

$$W\left(r<h\right)=\omega \left(\frac{5}{\pi h^2}\right)\left(1+3\frac{r}{h}\right){\left(1-\frac{r}{h}\right)}^3$$(1)

where r is the distance between the particles, h is the effective radius of the kernel function, and the normalization constant ω is unity in two-dimensional space and 21/16h in three-dimensional one.

The governing equation is the Navier–Stokes equation:   

$$\frac{Dv}{Dt}=-\frac{1}{\rho }\nabla P+ \frac{\eta }{\rho }\mathrm{\Delta }v+F$$(2)

where v is the velocity, ρ is the density, P is the pressure, η is the viscosity, and F is an external force. The space derivatives of a physical property can be expressed by using those of the kernel function. Equation (2) was discretized for the numerical calculation.

3.2. Viscosity Model of Bingham Fluid

The mathematical model of a Bingham fluid is expressed by the following equations.⁷⁾   

$$\{\begin{array}{l}\tau ={\tau }_B+{\eta }_P\dot{\gamma }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(\left|\tau \right|\ge {\tau }_B\right)\\ \dot{\gamma }=0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left(\left|\tau \right|<{\tau }_B\right)\end{array}$$(3)

where τ is the shear stress, τ_B is the yield stress, η_P is the viscosity of yielded fluid, and $\dot{\gamma }$ is the shear rate. Equations (3) imply that the fluid has a solid structure against the flow when $\left|\tau \right|<{\tau }_B$, and exhibits Newtonian flow when $\left|\tau \right|\ge {\tau }_B$. Since Eq. (3) has a discontinuity at τ_B, its direct usage in the simulation program is difficult. Therefore, the smoothed expression for the viscosity of a Bingham fluid proposed by Papanastasiou⁸⁾ was employed:   

$$\tau (\dot{\gamma })=\left[{\eta }_P+\frac{{\tau }_B}{\left|\dot{\gamma }\right|}\left(1-e^{-m\left|\dot{\gamma }\right|}\right)\right]\dot{\gamma }$$(4)

where m is the stress growth exponent, which has a time dimension. The value of m was set to 1000 in the calculation as proposed by Nagai.⁷⁾

3.3. Calculation Scheme

When the viscosity term is solved by an explicit scheme, the following numerical stability condition should be satisfied in the SPH method.⁹⁾   

$$\mathrm{\Delta }\text{t}\le d_i\frac{\rho r^2}{\eta }$$(5)

where Δt is the time step and d_i is the diffusion number, which takes the value 0.2–1.0 empirically.⁹⁾ According to Eqs. (4) and (5), the viscosity at the beginning of the flow is very high and the time step must be reduced or the particle distance must be enlarged, which results in increased calculation time and decreased accuracy. To avoid such difficulties in this study, the viscous term was separated from the others and calculated by the implicit method as proposed by Kim.¹⁰⁾ The change from the explicit scheme to the implicit scheme enables an increase in the time step and results in a stable calculation.

The calculation scheme was as follows: The temporary velocity, u^* was calculated by the external force and pressure gradient according to the following equation:   

$$u^{*}=u^k+\mathrm{\Delta }t\left(F-\frac{1}{\rho }\nabla P\right)$$(6)

where u^k is the actual velocity at k-th step. The Crank–Nicolson method was used for the calculation of the viscous term and the actual velocity at the next step was obtained by the following equation, which was solved using the successive over-relaxation method:   

$$u^{k+1}=u^{*}+\frac{1}{2}\frac{\eta }{\rho }\left({\nabla }^2{u}^{k+1}+{\nabla }^2{u}^{*}\right)$$(7)

4. Results and Discussion

4.1. Viscosity Measurement of Simulated Slags

Figure 2 shows the effect of the solid fraction of the slurry on the apparent viscosity. For both silicone oils, the viscosity increases with the increase in the solid fraction. It has been reported that the viscosity of the slurry obeys Einstein-Roscoe’s equation:¹¹⁾   

$$\eta ={\eta }_0{\left(1-\frac{\varphi }{{\varphi }_c}\right)}^{-n}$$(8)

where η₀ is viscosity of the liquid phase, ϕ is the solid fraction, ϕ_c is the critical solid fraction, and n is a constant.²⁾ When the solids are spheres with the same diameter, it is derived theoretically that ϕ_c = 0.74 and n = 2.5. The calculation by Eq. (8) is shown as dotted lines in Fig. 2, which shows the positive deviation from the experimental data. If the value of n is changed from 2.5 to 2.0, the calculation shows an excellent agreement with the experiment. Since n=2.5 was derived for a complete sphere, n = 2 suggests that a part of particles formed the non-spherical agglomerations in the suspension. Therefore, the experimental confirmation of Eq. (8) will be needed for the actual multi-phase slags.

Fig. 2.

The effect of the solid fraction on the viscosity.

Figure 3 shows the measured shear stress plotted against the shear rate for the suspensions of η₀ = 0.0482 Pa∙s. Although good linear relationships were obtained for the whole shear rates, the shown data are limited for the low share rate region in order to clarify the intercepts. The lines did not pass through the origin but had the positive intercepts on the shear stress axis when ϕ exceeds 0.45. The same plots are shown in Fig. 4 for η₀ = 0.324 Pa∙s. The behavior of the lines is similar to that shown in Fig. 3 and the criterion was ϕ = 0.40. Therefore, the suspensions having the positive intercepts were considered as a Bingham fluid in this study and the value of the intercept was defined as the yield stress. Sukenaga³⁾ used a more viscous liquid (η₀ = 0.5−3.0 Pa∙s) and reported that the yield stress was achieved when the solid fraction exceeded 0.3.

Fig. 3.

The relation between the shear stress and the shear rate (η₀=0.0482 Pa∙s).

Fig. 4.

The relation between the shear stress and the shear rate (η₀=0.324 Pa∙s).

Figure 5 shows the effect of the solid fraction on the yield stress. The data for η₀ = 0.5 Pa∙s measured by Sukenaga³⁾ are also plotted in the figure. The yield stress seems to increase with the solid fraction and the liquid viscosity in this study; however, the effect of the liquid viscosity on the yield stress is not clear from the comparison with Sukenaga’s data.³⁾ Although the measurement in the low shear rate region is important for the determination of the yield stress, the number of measurements was limited by the capability of the equipment in this study. Measurements, more precise in nature, using advanced equipment will be needed for further investigations.

Fig. 5.

The effect of the solid fraction on the yield stress.

4.2. Observation of the Flow Discharged from a Dam

Figure 6 shows the snapshots of the flow with various solid fractions 0.2 s after the gate opening, where η₀ = 0.0482 Pa∙s. The flow is a free-jet up to ϕ = 0.3 then changes to wall flow. The velocity of the flow decreases with the solid fraction

Fig. 6.

Snapshots of the flow with various solid fractions 0.2 s after the gate opening (η₀=0.0482 Pa∙s).

Figure 7 shows the snapshots of the flow with various solid fractions 0.5 s after the gate opening, where η₀ = 0.324 Pa∙s. The flow is free jet only for ϕ = 0; wall flow is observed in the other experiments. The velocity of the flow also decreases with the solid fraction.

Fig. 7.

Snapshots of the flow with various solid fractions 0.5 s after the gate opening (η₀=0.324 Pa∙s).

In order to characterize the fluid flow discharged from a dam, a dimensional analysis was conducted and the following dimensionless number N was derived:   

$$N=\frac{\rho L\sqrt{gH}}{\eta }$$(9)

where L is the width of the dam, H is the depth of the slag, and g is the acceleration of gravity. Although the effect of the width of the dam was not measured in the experiment, the value of L may affect the behavior of the falling jet by changing its ligulate shape. However, the confirmation of Eq. (9) by the experiments should be the future work.

Figure 8 shows the relationship between tanθ and N, where θ is the free jet angle illustrated in the figure. The value of θ was determined from the snapshot when the top of the jet reached the bottom of the vessel. If N is smaller than the critical value (N = 65), tanθ becomes zero, which signifies a change from free jet to wall flow.

Fig. 8.

Relationship between tanθ and the dimensionless number, N.

The travelling time of the simulated slag was defined as the time required for the top of the slag flow to reach the bottom of the vessel. Figure 9 shows the effect of the solid fraction on the travelling time. The Newtonian fluids are shown as open marks and the Bingham fluids as solid marks. The travelling time increases with the solid fraction and its effect is significant for the Bingham fluids. Since the flow is in the wall flow region, the existence of the yield stress and high viscosity reduces the falling velocity.

Fig. 9.

Effect of the solid fraction on the travelling time of the simulated slag.

4.3. SPH Simulation of the Slag Flow

The developed SPH simulation program was applied to the experimental system. The initial configuration of the particles is shown in Fig. 10. The number of fluid particles was 102000 and a particle distance of r = 1.0 mm was used. Since the calculation cost was significantly increased by the implicit scheme, a graphic processing unit (NVIDIA Tesla K40C) with Core i7 5820 K(3.30 GHz) was used. The calculation time was reduced by about five times comparing with the conventional calculation using Core i7-4790 K (4.00 GHz).

Fig. 10.

Image of the initial configuration in the SPH simulation.

Figure 11 shows the comparison of the simulated image with the observations for two different Bingham fluids. The velocity magnitude is also shown in the figure, where the velocity unit is cms⁻¹. The simulation image of the flow 0.5 s after the gate opening replicates the picture observed at 0.7 s for ϕ = 0.3 and η₀ = 0.324 Pa∙s in the experiment. When the solid fraction increases to ϕ = 0.5, the simulation image at 0.5 s replicates the picture observed at 2.0 s. This discrepancy in time is caused primarily by opening of the sliding gate. In the Bingham fluid region, the fluid was pulled upward with the moving gate; relaxation was needed to start the falling flow, which became significant with the increase in the solid fraction. This problem will be solved by developing a suitable model for the interaction between aluminum plate and suspension. The second reason for this discrepancy may be the attractive interaction between the simulated slag and the acrylic resin wall. The introduction of a suitable interaction model in the simulation will mitigate this effect. The third reason may be the choice of the stress growth exponent m in Eq. (4). Although the increase in m enhances the accuracy of the approximation, the significant increase in the calculation cost would be caused. The further examination must be needed in the future.

Fig. 11.

The velocity distribution in the simulated slag and the comparison of the simulated images with the experimental observations.

Since the simulation program replicated the cold model experiments, the flows of the suspension with high solid fractions in the experiments were confirmed as a Bingham flow. The simulation for the actual multi-phase slag was tested based on the data for 45CaO-45SiO₂-10K₂O slag at 1643 K reported by Saito.²⁾ The viscosity was 2.25 Pa∙s and the yield stress was 2.3 Pa. The density was estimated to be 26000 kgm⁻³ by using the available data for CaO–SiO₂ system.¹²⁾ Figure 12 shows the calculated image of the slag flow. The simulation developed in this study is expected to be applied not only to refining reactors, but also to the new technology for recovering sensible heat from those slags where the partially solidified molten slag is supplied to twin-roll type solidification equipment.¹³⁾ If the simulation can replicate the actual flows, the multi-phase slag flows in those processes will be concluded as a Bingham flow.

Fig. 12.

The simulated image and velocity power distribution of the multi-phase 45CaO-45SiO₂-10K₂O slag at 1643 K.

5. Conclusions

The simulated multi-phase slags were prepared by making slurry from silicone oil and spherical polyethylene beads. The viscosity of the slurry was measured by a rotating viscometer and its flow, discharged from a dam, observed by a high-speed video system. A three-dimensional SPH simulation program for Bingham fluid was developed and its calculations were compared with the experimental observations. The results obtained are as follows:

(1) The viscosity of simulated slag is described by Einstein–Roscoe’s equation.

(2) The criterion for the free jet and wall flow is obtained by dimensional analysis.

(3) The travelling time of the flow increases dramatically with the solid fraction of simulated slag in the Bingham fluid region.

(4) The simulation replicates the observed flow in the cold model experiment and the flows of the suspension with high solid fractions in the experiments were confirmed as a Bingham flow.

References

• 1)   F.  Tsukihashi: Tetsu-to-Hagané, 95 (2009), 187.
• 2)   N.  Saito,  S.  Yoshimura,  S.  Haruki,  Y.  Yamaoka,  S.  Sukenaga and  K.  Nakashima: Tetsu-to-Hagané, 95 (2009), 282.
• 3)   S.  Sukenaga,  S.  Haruki,  Y.  Yamaoka,  N.  Saito and  K.  Nakashima: Tetsu-to-Hagané, 95 (2009), 807.
• 4)   M.  Nakano and  K.  Ito: ISIJ Int., 56 (2016), 1297.
• 5)   M.  Nakano and  K.  Ito: ISIJ Int., 56 (2016), 1537.
• 6)   L. B.  Lucy: Astron. J., 82 (1977), 1013.
• 7)   K.  Nagai and  S.  Ushijima: Annu. J. Hydraul. Eng., 54 (2010), 1183.
• 8)   T. C.  Papanastasiou: J. Rheol., 31 (1987), 385.
• 9)   Y.  Fukuzawa,  H.  Toyama,  K.  Shibata and  S.  Koshizuka: Trans. Jpn. Soc. Comput. Eng. Sci., 2014 (2014), paper No. 20140007.
• 10)   J.  Kim and  P.  Moin: J. Comput. Phys., 59 (1985), 308.
• 11)   R.  Roscoe: Br. J. Appl. Phys., 3 (1952), 267.
• 12)  Slag Atlas, 2nd ed., Verlag Stahleisen GmbH, VDEh, Düsseldorf, (1995), 332.
• 13)   H.  Tobo,  Y.  Ta,  M.  Kuwayama,  Y.  Hagio,  K.  Yabuta,  H.  Tozawa,  T.  Tanaka,  K.  Morita,  H.  Matsuura and  F.  Tsukihashi: Tetsu-to-Hagané, 99 (2013), 683.