Fluid Flow and Mixing Phenomena in Mechanically Agitated and Gas Stirred Ladle Systems and Their Comparisons

Abstract

A sliding mesh-based LES (large eddy simulation) mathematical model has been developed to investigate fluid flow and mixing phenomena in a mechanically agitated (MA) water model ladle (D = 0.30 m) fitted with an impeller/paddle. Parallel to such, liquid velocity and 95% bulk mixing times were experimentally measured as a function of rotational speed of the impeller. These were applied to validate mathematical model predictions. On the basis of results derived from the present physical and mathematical modelling investigation and already published data on gas stirred (GS) ladle systems, a performance comparison between mechanical and gas agitated systems has been presented. It is shown that at specific stirring power, similar to those practised in ladle refining (~10⁻³ W/kg), while intensity of motion is more pronounced and mixing is considerably faster, corresponding flow establishment periods are relatively longer in the mechanically agitated system. Furthermore, while relationship between mixing time – specific stirring power in both the systems are found to be very similar e.g., τ_mix,95%,MA∞ vs. τ_mix,95%,GS∞, markedly different and contrasting dependence of mixing time on liquid depths i.e., τ_mix,95%,MA∞H^1.8 vs. τ_mix,95%,GS∞H^−1.0 in the two systems have been observed. In addition to such, dynamic similarity criterion for mechanically agitated systems has been investigated and it is demonstrated that similar to gas stirred ladle systems, a Froude based modelling criteria suffices.

1. Introduction

Rates of metallurgical processes such as melting, dissolution, dispersion, interfacial reactions etc. in refining vessels e.g., ladles are generally transport controlled and therefore, intensity of flow and turbulence tend to exert considerable influence on the associated process rates. Melts contained in furnaces, ladles etc. are therefore often vigorously stirred to expedite process rates and to this end, inert/reactive gas injection, electromagnetic stirring (EMS) as well as mechanical agitation have all been practised in the industry. Due to varied technology and equipment used in such stirring practices, flow, turbulence etc. in metal processing units and the attendant rates of various transport-controlled processes are likely to depend not only on stirring power but also on the mode of stirring.

Extensive research on turbulent flow phenomena during inert gas and electromagnetic stirring have been carried out and reported in the literature.^1,2,3,4) In contrast, equivalent efforts on mechanically agitated systems, particularly in metallurgical engineering,^5,6,7) have been less common and include such studies as, the k-ε turbulence model-based calculations of flow and mass transfer phenomena⁵⁾ as well as numerical modelling of combined inert gas and mechanical stirring processes.⁶⁾ Several topics, of interest in the mechanically agitated systems, such as measurements of flow and turbulence, modelling criteria, comparison among different stirring techniques etc. have remained mostly unattended and therefore, not reported so far. Similarly, comparative hydrodynamic performance between mechanically agitated and gas stirred ladle systems, a subject of considerable importance to steelmakers as well as researchers, has remained unexplored; neither much information on the topic is available in open literature.

To address the above, a combined computational and experimental study on fluid flow and mixing phenomena in a mechanically agitated system has been carried out. On the basis of such and embodying already published results on flow and mixing phenomena in axi-symmetric gas stirred ladles, a hydrodynamic performance comparison, between mechanical and gas agitated systems, has been carried out. These are presented, analysed and discussed in the following in detail.

2. Present Work

A cylindrical shaped PERSPEX water model ladle system (D = 0.30 m), with a simple paddle type stirrer, was fabricated to investigate fluid flow and mixing phenomena in a mechanically agitated system, both numerically and experimentally. A schematic of the model vessel is presented in Fig. 1(a) while a photograph of the same is shown in Fig. 1(b). There as seen, the impeller, connected to a motor and capable of rotating with a wide range of speed, was applied to stir water, contained in the cylindrical shaped vessel. In the present work, impeller speed in the range of 116–232 rpm and a fixed liquid depth of 0.25 m (i.e., H/D~0.8) were applied. These resulted in specific stirring power of ~10⁻³ W/kg (see later) and Reynolds number (i.e., $R{e}_{imp}=\rho N{D}_{imp}^2/\mu$) greater than 10⁴ or so, similar to those encountered in ladle metallurgy steelmaking operations.¹⁾ In the chosen range of impeller speed, negligible distortion of the free surface was encountered and entrainment of air from the surrounding into bulk liquid via the rotating impeller/paddle could be restricted, leading to a horizontal, flat free surface. The PERSPEX water model vessel was constructed to primarily generate experimental data such that the mathematical models developed in this work (see Sec. 2.1 later) can be validated and applied to predict different phenomena of interest in the mechanically agitated system over the desired range of impeller speeds.

Fig. 1.

(a) A schematic of the mechanically agitated system and (b) the corresponding water model applied to measure flow velocities and 95% bulk mixing times. (Online version in color.)

2.1. Mathematical Modelling

In a mechanically agitated system such as the one shown in Fig. 1, the liquid surrounding the rotating impeller/paddle continuously changes with reference to a fixed frame of reference, since space occupied by the impeller, during rotation, gets filled-up in a periodic manner by the adjacent liquid. To numerically simulate the effect of such a rotating object on liquid flow, turbulence and mixing phenomena, a sliding mesh-based LES (Large eddy simulation)^8,9) formalism has been adopted in the present work. Large eddy simulation has been preferred over the k-ε model since previous work¹⁰⁾ clearly indicated that the two-equation k-ε model and its derivatives are largely inadequate to reliably resolve turbulent flow phenomena in mechanically agitated flow systems.

To formulate governing equations representing flow phenomena in the mechanically agitated ladle, in addition to the routine assumptions of a Newtonian, incompressible, constant property, isothermal and non-reactive flow, the following were embodied viz.,

(i) Flow phenomena are three-dimensional and transient. Despite symmetry of the vessel, the rotation of the paddle together with (e.g., Fig. 1) LES necessitate a three-dimensional transient formulation.

(ii) The impeller is considered to be rotating with full rotational speed from the starting time, t = 0. In other words, a specified and constant rotational speed of the impeller has been assumed to be achieved instantaneously.

(iii) The ladle free surface is considered to be mobile and perfectly horizontal since deformation of the free surface, due to vortex formation around the rotating shaft, has been found to be less than 1% of total free surface area, throughout the range of rotational speed applied.

(iv) On the basis of (iii), homogeneous or single phase flow formulation has been considered most appropriate for the present problem

(v) The species transport equation applied to predict mixing time is taken to coupled one-way with the flow equations implying essentially that addition of the inert, neutral density tracer in the study of mixing, does not affect the induced flow and turbulence in the vessel and finally,

(vi) No overlying buoyant phase is present in the system.

In the sliding mesh approach,^8,9) a part of the computational mesh embodying the rotating impeller, is made to move rigidly in the flow domain with respect to the surrounding stationary mesh. This implies that all boundaries and cells of the moving zone (i.e., containing the impeller) move together without getting deformed, i.e., like a rigid body, and the rigidly moving mesh slides over the stationary mesh zone along an interface which is inherently non-conformal and is updated with time, during numerical computation. Furthermore, continuity of flow between the moving and stationary mesh, associated respectively with the impeller and the surrounding liquid, is maintained by incorporating conservative interpolation techniques at the interface. More details of these are available in ref. 8 and 9 and are therefore, not reproduced here.

For a dynamically moving control volume V, appropriate conservation statement for a general variable, ψ, with respect to a stationary frame of reference, can be written in the following generalized integral form^8,9) viz.,   

$$\frac{d}{dt}\underset{V}{\overset{ }{{\displaystyle \int }}}\rho \psi dV+\underset{\partial V}{\overset{ }{{\displaystyle \int }}}\rho \psi \left(\overrightarrow{u}-\overrightarrow{m}\right).d\overrightarrow{A}=\underset{\partial V}{\overset{ }{{\displaystyle \int }}}\mathrm{\Gamma }\nabla \psi .d\overrightarrow{A}+\underset{V}{\overset{ }{{\displaystyle \int }}}{S}_{\psi }dV$$(1)

In Eq. (1), ρ is the density of the fluid, $\overrightarrow{u}$ is flow velocity, $\overrightarrow{m}$ is velocity of the moving mesh, Γ is generalised diffusion coefficient and S_ψ is source term of ψ. In the part of flow domain having stationary mesh, modified version of Eq. (1), with $\overrightarrow{m}$ = 0 is applicable. Consequently, Eq. (1), can be conveniently adapted to both, moving and stationary, mesh regions.

In the spirit of Eq. (1), governing equations of conservation of mass and momentum, in terms of filtered velocity (see later) can be expressed in compact tensorial notation as:   

$$\frac{\partial }{\partial x_j}\left({\overline{u}}_j-m_j\right)=0$$(2)

  

$$\frac{\partial }{\partial t}\left(\rho {\overline{u}}_i\right)+\frac{\partial }{\partial x_j}\rho ({\overline{u}}_j-m_j){\overline{u}}_i=-\frac{\partial \overline{p}}{\partial x_i}+\frac{\partial }{\partial x_j}\left(\mu \frac{\partial {\overline{u}}_i}{\partial x_j}\right)-\frac{\partial \left(\rho {\tau }_{ij}\right)}{\partial x_j}$$(3)

in which, m_j, as stated earlier, represents mesh velocity associated with the moving impeller/paddle and the over barred entities represent filtered variables^a. Governing equations of filtered velocity components presented via Eq. (3), as seen, needs an appropriate closure since unknown residual stress tensor (τ_ij), arising from the filtering operation is not known a priori. The residual stress tensor appearing in Eq. (3) is modelled in LES^11,12) via an appropriate, sub-grid scale model, wherein, the residual stress is expressed as a sum of deviatoric and isotropic stresses viz.,   

$${\tau }_{ij}=\underset{\text{deviatoric}\mathrm{\hspace{0.17em} }\text{stress}}{\underbrace{{\tau }_{ij}-\frac{1}{3}{\tau }_{kk}{\delta }_{ij}}}+\underset{\text{isotropic}\mathrm{\hspace{0.17em} }\text{stress}}{\underbrace{\frac{1}{3}{\tau }_{kk}{\delta }_{ij}}}$$(4)

The isotropic residual stress, as shown above, is generally included in the filtered pressure term whereas the corresponding deviatoric component is modelled on the basis of eddy-viscosity hypothesis as:   

$${\tau }_{ij}-\frac{1}{3}{\tau }_{kk}{\delta }_{ij}=-2{\mu }_{SGS}{\overline{S}}_{ij}$$(5)

in which, μ_SGS is subgrid-scale turbulent viscosity and ${\overline{S}}_{ij}$ is strain-rate tensor of the resolved large-scale motions, defined as,   

$${\overline{S}}_{ij}=\frac{1}{2}\left(\frac{\partial {\overline{u}}_i}{\partial x_j}+\frac{\partial {\overline{u}}_j}{\partial x_i}\right)$$(6)

For modelling of subgrid-scale turbulent viscosity, μ_SGS, Wall-Adapting Local Eddy-Viscosity (WALE) model was applied in this work. Accordingly, subgrid-scale turbulent viscosity has been represented as:^13,14)   

$${\mu }_{SGS}=\rho L_{SGS}^2\frac{{\left(S_{ij}^d{S}_{ij}^d\right)}^{\frac{3}{2}}}{{\left({\overline{S}}_{ij}{\overline{S}}_{ij}\right)}^{\frac{5}{2}}+{\left(S_{ij}^d{S}_{ij}^d\right)}^{\frac{5}{4}}}$$(7)

in which, $S_{ij}^d$ is defined as:   

$$S_{ij}^d=\frac{1}{2}\left({\left({\overline{h}}_{ij}\right)}^2+{\left({\overline{h}}_{ji}\right)}^2\right)-\frac{1}{3}{\delta }_{ij}{\left({\overline{h}}_{kk}\right)}^2,\mathrm{\hspace{0.17em} }{\overline{h}}_{ij}=\frac{\partial {\overline{u}}_i}{\partial x_j}$$(8)

Furthermore, the subgrid scale mixing length, L_SGS, appearing in Eq. (7), is expressed as:   

$$L_{SGS}=min\left(\kappa d,\mathrm{\hspace{0.17em} }{C}_w{V}_c{}^{\frac{1}{3}}\right)$$(9)

In the above, κ, C_w, V_c, d and δ_ij respectively represent von Karman constant, WALE constant, volume of the computational cell, distance to the closest wall and the Kronecker delta.^11,12,13,14) A default value of 0.325 was applied for the model constant, C_w.

To predict transient concentration field following pulse addition of an inert tracer and hence to estimate mixing time, a scalar transport equation, has been solved in conjunction with the flow equations presented already. In the LES formalism, the governing equation, expressing the conservation of an inert tracer, i in the flow systems, is expressed as:   

$$\frac{\partial }{\partial t}\left({\varphi }_i\right)+\frac{\partial }{\partial x_j}\left({\overline{u}}_j-m_j\right){\varphi }_i=\frac{\partial }{\partial x_j}\left(\frac{\mu }{\rho Sc}.\frac{\partial {\varphi }_i}{\partial x_j}\right)-\frac{\partial }{\partial x_j}\left(J_{j,SGS}\right)$$(10)

in which, J_j,SGS represents the subgrid scale material flux of the species, i = $\left(-\frac{{\mu }_{SGS}}{\rho S{c}_t}.\frac{\partial {\varphi }_i}{\partial x_j}\right)$, μ_SGS is the subgrid scale viscosity, estimated via Eq. (7) and Sc_t is the turbulent Schmidt number, the latter set to its typical value of 0.7. Equations (2), (3) and (10), as represented, were applied in the rotating mesh zone (enclosing the impeller) with a finite and non-zero rotational velocity, while outside the rotating mesh zone, m_j was set to zero.

To solve the governing equations of continuity, motion and species conservation, a null initial velocity field was applied, and the rotating impeller was assigned to its desired rotational speed at time t = 0. Furthermore, tracer addition was considered over a small region of fluid, close to the free surface and near the axis of the vessel where, φ_i at time t = 0 was set to unity. Principal dimensions of the mechanically agitated system together with relevant bounding surfaces and cell zones are illustrated schematically in Fig. 2. Applicable boundary conditions are summarised in Table 1.

Fig. 2.

A schematic of the mechanically agitated system illustrating the rotating impeller and the cylindrical shaped vessel together with relevant dimensions, surfaces and cell zones.

Table 1. Boundary condition applied to model turbulent flow and mixing phenomena in a mechanically agitated ladle.

Physical boundary	Applicable boundary conditions
Ladle side and bottom walls	No slip, non-rotating wall in absolute frame together with zero mass flux of i
Impeller surface and the tiny shaft part inside enclosure (e.g., the dashed box around the impeller viz., Fig. 2)	No slip moving wall with zero rpm with respect to adjacent cell zone and zero mass flux of i
Portion of solid shaft outside enclosure	No slip moving wall rotating with impeller speed in absolute frame and zero mass flux of i
Free surface	Flat free mobile surface with zero normal velocity and zero tangential and radial velocity gradients; zero mass flux of i

The governing equations of continuity, flow and species transport were solved via ANSYS Fluent^TM Version18.0 embodying the dynamic or sliding mesh approach mentioned already. To resolve flow field around the rotating impeller accurately, an element size of 1 mm was applied in majority of the enclosure domain, containing the rotating impeller. Furthermore, in the vicinity of the moving and rotating interface, an element size of 3 mm was deployed whereas relatively larger mesh element, having a size of 4 mm, was applied in the remainder of the ladle. A total number of 510339 elements together with time step, smaller than 5 × 10⁻⁴ s, was applied in the numerical solution scheme.

In addition to mathematical modelling, measurements of flow and 95% mixing times were also carried out in the cylindrical vessel shown in Fig. 1(b). Towards these, while LDV (Laser Doppler Velocimeter) was applied to measure velocity fields, electrical conductivity measurement technique was used to monitor mixing times. These measurement techniques have been common in metallurgical flow modelling literature and relevant details have been summarised by many previous investigators.^15,16,17) Therefore, these are not described here further. In such context, it is however important to mention that all measurements were carried out under steady flow condition and due care was always taken to ensure sufficient reproducibility.

^a  It is to be noted here that while the flowing fluid has three components of motion, the impeller (and hence, m_j) on the other hand has only one finite, non-zero, i.e., the rotational component of motion.

2.2. Mathematical Model Validation

Since numerically computed flows and mixing times have been extensively applied in this work to draw various conclusions, it is therefore naturally important to demonstrate the adequacy and appropriateness of the LES-based mathematical model, presented earlier. To this end, experimentally measured (i) velocity components at different locations in the system as well as (ii) 95% bulk mixing times¹⁷⁾ were applied and assessed directly against corresponding numerical predictions. The comparisons are illustrated in Fig. 3. There, as seen, numerically predicted rotational and axial components of the flow, at various locations in the system, as well as bulk 95% mixing times, are in excellent agreement with corresponding experimental measurements. These evidently demonstrate that the sliding mesh-based, LES flow calculation procedure predicts flows and mixing times in the mechanically agitated systems with reasonable certainty.

Fig. 3.

Comparisons between numerically predicted and experimentally measured axial (ai-aii) and rotational (bi-bii) components of flows as well as 95% bulk mixing times (c) in the mechanically agitated system. (Online version in color.)

2.3. Specific Stirring Power in Mechanically Agitated and Gas Stirred Systems and Their Equivalence

In mechanically agitated system, energy is supplied to the bath via a submerged, rotating impeller/paddle, typically driven by an electrical motor. Under such condition, stirring power input to the bath can be estimated by computing the torque acting on the surface of the rotating impeller/paddle as well as the shaft (see Fig. 2). For a rotating body, power, P is defined as a dot product of torque, $\overrightarrow{T}$ and angular velocity ${\overrightarrow{\omega }}_a$ and represented as:   

$$P=\overrightarrow{T}\mathrm{\hspace{0.17em} }.\mathrm{\hspace{0.17em} }{\overrightarrow{\omega }}_a$$(11)

Therefore, specific stirring power (${\dot{\epsilon }}_{M,MA}$) input to a mechanically agitated, cylindrical shaped ladle, can be expressed as:   

$${\dot{\epsilon }}_{M,MA}=\frac{P}{M}=\frac{T_{total}\omega }{M}=\frac{T_{total}\omega }{{\rho }_L\pi R^{2}H}$$(12)

in which, T_total represents total torque along the rotational axis, needed to rotate the impeller and the shaft at the prescribed angular velocity of ω(rad/s) and M is total mass of the liquid (= πρ_LR²H) in the vessel. Since rotational speed of the impeller is known a priori (i.e., from the reading of a Tachometer), estimation of stirring power, as seen from Eq. (12) necessitates knowledge of the total torque, T_total. In the present work, the total torque and hence the input stirring power to the mechanically agitated system has been calculated numerically via the three-dimensional, transient, turbulent flow model presented earlier. To this end, the built-in, torque calculation function, available in ANSYS Fluent was applied. Representative specific input stirring power, corresponding to each specific impeller speed, was computed numerically following attainment of steady flow condition in the system.

Estimation of specific input power to a cylindrical shaped gas stirred ladle system, on the other hand, is straightforward and can be accomplished analytically incorporating key operating parameters. Thus, at gas flow rates typically practised during ladle homogenisation operation, input stirring power in a gas stirred ladle system can be conveniently calculated from the following expression.¹⁷⁾   

$${\dot{\epsilon }}_{GS}={\rho }_L\mathrm{\hspace{0.17em} }g{Q}_{G}H$$(13)

In Eq. (13), Q_G is the volumetric gas flow rate referenced to the mean operating temperature and pressure (at the mid-bath depth level) of the liquid. Since mass of liquid in the ladle is equal to πρ_LR²H, the specific input stirring power in a gas stirred system can be readily expressed as:   

$${\dot{\epsilon }}_{M,GS}=\left({\rho }_L\mathrm{\hspace{0.17em} }g{Q}_{G}H\right).\frac{1}{\pi {\rho }_L{R}^{2}H}=\frac{g{Q}_G}{\pi R^2}$$(14)

At identical input specific power in the two systems, ${\dot{\epsilon }}_{M,GS}={\dot{\epsilon }}_{M,MA}$. Therefore, by equating Eqs. (12) and (14), and embodying the numerically estimated total torque, equivalent gas flow rate in the gas stirred system, corresponding to a specific rotational speed of the impeller in the mechanically agitated system, can be readily calculated. Thus, setting T_Totalω = ρ_L gQ_GH, equivalent gas flow rate in similar capacity ladles can be deduced from:   

$$Q_{G,Eq.}=\frac{T_{Total}\omega }{{\rho }_{L}gH}$$(15)

Alternatively, for different size ladle systems, the equality expressed via Eq. (15) can be re-cast as:   

$$Q_{G,Eq.}=\left(\frac{M_{GS}}{M_{MA}}\right)\frac{T_{Total}\omega }{{\rho }_{L}gH}$$(16)

in which, M_MA and M_GS respectively represent the mass of liquid contained in the mechanically agitated and gas stirred systems.

On the basis of numerically estimated torque at different impeller speeds, corresponding equivalent gas flow rates in two different size gas stirred ladle systems (i.e., D = 0.30 m and 1.12 m respectively) were estimated via Eqs. (15) and (16) and these are summarised in Table 2. Estimates thus obtained show gas flow rates of about 1–2 lit/min in the smaller vessel and 6–18 lit/min in the bigger ladle. Such flow rates, as one would note here, correspond to specific stirring power of the order of 10⁻³ w/kg, typical of ladle homogenisation operations.

Table 2. Numerically estimated total torque at various impeller speeds, corresponding input specific stirring power in the mechanically agitated system (H = 0.25 m and D = 0.30 m) and equivalent gas flow rates in two different size gas stirred ladle systems.

Impeller speed (rpm)	Numerically calculated torque (10−3 N-m) in the mechanically agitated system [D = 0.30 m, H = 0.25 m and H/D = 0.83]	Estimated stirring power input (10−3 W/kg ) in the mechanically agitated system	Equivalent gas flow rate (10−5 m3/s) in a similar size vessels [D = 0.30 m and H = 0.25 m]19)	Equivalent gas flow rate (10−4 m3/s) in a pilot scale vessel [D = 1.12 m and L = 0.93 m and H/D = 0.83]17)
155	1.15	1.06	0.76	1.06
194	1.86	2.15	1.55	2.16
213	2.00	2.54	1.83	2.55
232	2.18	3.01	2.17	3.02

2.4. Fluid Flow Phenomena in Mechanically Agitated and Gas Stirred Ladle Systems

In the range of stirring power consider in this work, Mazumdar and co-workers,^{17,18,19,20,21,22)} have extensively investigated fluid flow and mixing phenomena in different size, axisymmetric gas stirred water model ladle systems, including the two, shown in Table 2 i.e., D = 0.30 m and D = 1.12 (with H/D~1) respectively. Several other investigators have also reported flow and mixing in similar size vessel, having similar aspect ratios, at similar specific stirring power.^15,16) These together with computational and experimental results derived from the present study have been applied to carry out a hydrodynamic performance comparison between mechanically agitated and gas stirred systems, at equivalent specific input stirring power. To this end, it is important to mention here that comparisons between the two different types of agitated systems, in the present work, has been carried out for slag less (no upper buoyant phase), cylindrical shaped ladle systems (H/D~1.0) at specific input power of ~ 10⁻³ W/kg and therefore, extrapolation of various inferences and their generalisation, beyond their range of validity, is uncertain.

Numerically predicted flow patterns in the mechanically agitated and an axi-symmetrical gas stirred ladle system (D = 0.30 m) at a specific stirring power of 3 × 10⁻³ W/kg are shown in Fig. 4. There, flow velocity distributions have been presented on the (i) horizontal plane at mid-bath depth level as well as (ii) central vertical plane of the two systems. These indicate that while axial flow predominates in the gas stirred system due to the vertically rising bubble plume, the rotating impeller in the mechanically agitated system, on the other hand, produces relatively more intense rotational flows on the horizontal plane. As expected, rotational velocity on the basal plane of the gas stirred system is practically zero^b, reflecting essentially the characteristics of an axi-symmetrical flow system. Figures 4(a) through 4(d) clearly indicate that fluid flow in the two systems, at equivalent input stirring power, is very different both qualitatively and quantitatively owing to different modes of stirring. Accordingly, varied hydrodynamic performances can be anticipated from the two systems.

Fig. 4.

Velocity fields in the mechanically agitated and axi-symmetrical gas-stirred ladle systems. (a)–(b) flow patterns in the mechanically agitated systems on horizontal plane bisecting the ladle and the central vertical plane at an impeller speed of 232 rpm; (c)–(d) flow patterns in the gas stirred system on a horizontal plane passing through mid-bath depth and the central vertical plane at a gas flow rate of 2 lit/min. (e)–(f) Schematics of the mechanically agitated and axi-symmetrical gas stirred ladle systems. (Online version in color.)

To visualise the results presented in Fig. 4 from a quantitative standpoint, average or mean speed of liquid recirculation^c in the mechanically agitated system was calculated via the LES based model at different impeller speeds. These are shown in Fig. 5(a), in which, instantaneous mean speed of bath recirculation has been plotted as a function of time for different impeller speeds. These results indicate that depending on the impeller speed, mean speed of liquid recirculation in the system, at steady state, lies in the range of 40 to 105 mm/s. At similar specific input stirring power and equivalent gas flow rates (e.g., Table 2), mean speed of liquid recirculation in the axisymmetric gas stirred system (D = 0.30 m) has been calculated substituting H = 0.25 m and R = 0.15 m in the following relationship (in SI unit):²⁰⁾   

$${\overline{U}}_{mean}=0.86Q_G{}^{0.33}{H}^{0.25}{R}^{-0.58}$$(17)

Fig. 5.

(a) Predicted instantaneous mean speed of liquid recirculation at different impeller speeds and (b) a comparison of mean speed of bath recirculation at equivalent specific stirring power in similar size (D = 0.30 m) mechanically agitated and gas stirred systems. (Online version in color.)

Where U_mean, H, R and Q_G respectively represent mean speed of liquid recirculation, depth of liquid, vessel radius and gas flow rate (corrected to mean height and temperature of the liquid). Estimates of mean speed of bath recirculation for the two systems i.e., mechanical vs. gas agitated, are compared directly in Fig. 5(b) as a function of specific stirring power. There, it is at once apparent that at equivalent specific stirring power in similar size vessels, the average speed of liquid recirculation in the mechanically agitated system is almost twice as intense as those in the gas stirred system. This is to be expected since significant amount of stirring energy is known to be dissipated in gas stirred ladles due to gas-liquid friction or bubble slippage phenomena.²¹⁾

Experimentally measured flow establishment periods in a practically similar size, axi-symmetrical gas stirred ladle (D = 0.30 m and H = 0.21 m), in the range of 1 to 3 lit/min of gas flow rates (or, specific stirring power of ~10⁻³ W/kg), was reported in this journal several years back by Mazumdar and co-workers¹⁹⁾ and this is shown in Fig. 6(a). These experimental measurements indicate that in the range of 1 to 3 lit/min of gas flow rates, flow establishment periods in the gas stirred system vary between 10 to 20 s. More specifically, at about 1 lit/min of gas flow, Fig. 6(a) is indicative of a flow establishment period of about 12.5 s or so. At practically equivalent specific stirring power (corresponding to an impeller speed of 194 rpm), Fig. 6(b), in contrast, indicates substantially longer flow establishment period of 58 s or so in the mechanically agitated system. This is consistent given that at identical specific stirring power, mean speed of bath recirculation at steady state in the gas stirred ladle is much smaller than that in the mechanically agitated (viz., see Fig. 5(b)) system.

Fig. 6.

(a) Experimentally measured flow establishment periods in an axi-symmetrical water model gas stirred ladle (D = 0.30 m and H = 0.21 m)¹⁹⁾ at three different gas flow rates and (b) numerically predicted flow establishment period in the mechanically agitated system (D = 0.30 m and H = 0.25 m) at an impeller speed of 194 rpm (equivalent gas flow rate = 0.21 × 10⁻⁴ m³/s; see Table 2). (Online version in color.)

^b  Extremely weak rotational flow (< 10⁻³ mm/s) on horizontal plane in an axi-symmetrical gas stirred systems is due to wandering of bubble plume which are known to be captured well by discrete phase model (DPM) of bubble stirred ladle systems.

^c  Defined in the present work as: ${\overline{U}}_{mean}=\frac{\int \int \int v_{res}(x,y,z)dxdydz}{Effective\mathrm{\hspace{0.17em} }volume\mathrm{\hspace{0.17em} }occuopied\mathrm{\hspace{0.17em} }by\mathrm{\hspace{0.17em} }fluid}$

2.5. Mixing Characteristics in Mechanically Agitated and Gas Stirred Ladle Systems

Mixing rates in metallurgical reactors and vessels are known to be position dependant^16,17) and hence different regions in the mechanical agitated as well as gas stirred ladle systems are expected to mix at different rates leading to different estimates of local mixing times. Region specific transient concentration profiles as measured experimentally in the mechanically agitated system are shown in Fig. 7(a). This indicates substantial variation in mixing rates as well as corresponding mixing times in different regions of the vessels. Furthermore, as seen, mixing rate is slowest in the vicinity of the free surface (location P1) and highest close to the rotating impeller, near the base of the ladle (location P3) implying essentially that mixing time registered from the probe, kept immersed at location P1 (the region to mix last in the system) can be interpreted as the bulk or reactor mixing time¹⁷⁾ for the system. In contrast, an axi-symmetrical gas stirred ladle system (D = 1.12 m), as shown in Fig. 7(b),¹⁷⁾ is known to exhibit fastest mixing near the free surface and slowest, near the base of the ladle. Such markedly different mixing characteristics, reflected from Figs. 7(a) and 7(b), essentially arise due to different flow characteristics in the systems, discussed in the context of Figs. 4(a) through 4(d) earlier.

Fig. 7.

Variations of tracer concentration with time at different locations in (a) mechanically agitated (D = 0.30 m) and (b) gas stirred ladle (D = 1.12 m; H/D = 0.83) systems¹⁷⁾ illustrating the location of the slowest mixing region in the two systems (specific stirring power ~10⁻³ W/kg). (Online version in color.)

In Fig. 8(a), experimentally measured 95% bulk mixing times^d in the mechanically agitated ladle is shown as a function of specific stirring power summarised in Table 2. There, for the sake of a comparison, corresponding experimental 95% bulk mixing times in the axisymmetric gas stirred system (D = 0.30 m) are shown in Fig. 8(b). These clearly indicate that mixing times in the mechanically agitated system, at equivalent specific stirring power, are considerably shorter than those in the gas stirred system. This is consistent with the corresponding trends of mean speed of bath recirculation presented earlier (e.g., see Fig. 5(b)) since mixing, at the fundamental level, is governed by convection and turbulent diffusion phenomena. Thus, a higher liquid turn-over rate in turn translates into faster mixing or shorter mixing times.

Fig. 8.

(a) Experimentally measured 95% bulk mixing times as a function of specific stirring power in a ladle (D = 0.30 m) during mechanical as well as gas stirring and (b) Mixing time~ gas flow rate relationship in a pilot scale ladle (D = 1.12 m and H = 0.93 m).¹⁷⁾ (Online version in color.)

Several years ago, Mazumdar and Guthrie¹⁷⁾ reported on the variation of 95% mixing time with gas flow rate (directly proportional to specific stirring power; e.g. Eq. (13)) in a pilot scale water model ladle (i.e., D = 1.12 m and H/D = 0.83)¹⁷⁾ as shown Fig. 8(b). There, exponent on gas flow rate, particularly in the range of ~10⁻³ W/kg of specific stirring power, is seen to be practically identical to those shown in Fig. 8(a). These experimental results evidently confirm that at relatively low specific stirring power, mixing time decreases with increasing specific stirring power and practically identical laws are obeyed, in mechanically agitated as well as gas stirred systems, regardless of vessel size.

The dependence of mixing time on liquid depth in the mechanically agitated system is shown in Fig. 9(a). There, it is evident that mixing time increases with increasing bath depth, following a relationship τ_mix,95%∞H^1.8, which is in direct contrast to the one applicable to gas stirred ladle systems, i.e., τ_mix,95%∞H^−0.96, as shown in Fig. 9(b).¹⁷⁾ Very similar and an inverse dependence of mixing time on liquid depth in gas stirred ladle systems has also been reported many years back by Asai and co-workers.¹⁵⁾ This is surprising since gas stirred ladles are essentially potentially energy driven²³⁾ wherein, the rate of energy supply or power input, at any given gas flow rate, is dependant on liquid depth and tends to increase directly with increasing liquid/bath depth (see Eq. (13)). In contrast to the above, mechanically agitated systems are driven by kinetic energy supplied by the rotating impeller. In such systems, stirring power is independent of bath depth. However, with any increase in liquid depth, as mass of liquid in the vessel increases, this tends to make bath stirring sluggish, retarding mixing rates in the system.

Fig. 9.

Variation of 95% mixing times as a function of liquid depth. (a) Mechanically agitated and (b) axi-symmetrical gas stirred ladle (D = 1.12 m)¹⁷⁾ systems at specific stirring power of ~ 6 × 10⁻³ W/kg. (Online version in color.)

^d  These are defined in the present context as the time beyond which the monitoring point (slowest mixing region) concentration continuously falls with a narrow ± 5% deviation band around the final homogeneous concentration value.¹⁷⁾

2.6. Dynamic Similarity Criteria in Mechanically Agitated and Gas Stirred Ladle Systems

In physical modelling of isothermal, non-reacting flow systems, similarity criteria between model and full- scale systems are dictated by geometrical and mechanical similarities. In water modelling of iron and steelmaking processes, mechanical similarity is governed by dynamic similarity or the similarity of forces, which in turn is governed by the Navier-stokes equations, represented in dimensionless form as:²⁴⁾   

$$\frac{1}{N_{Eu}}=F\left(N_{Re},N_{Fr}\right)$$(18)

The above implies that both Reynolds and Froude number equality between model and prototype must be maintained in order to ensure dynamic similarity between the two. However, as is well known, in reduced scale water modelling of iron and steelmaking processes, it is impossible to simultaneously respect both Reynolds and Froude number equality since kinematic viscosity of molten iron/steel at 1873 K and that of water at 298 K are practically identical.²⁴⁾ As a consequence, the role of one of the two dimensionless numbers is generally dispensed with, and flow phenomena characterised either in terms of Froude or Reynolds number-based criterion.

Gas stirred ladle system have always been modelled considering flow phenomena to be essentially Froude number dominated.¹⁾ This follows since gas injection induced motion is turbulent and driven essentially by the buoyancy forces afforded by the rising gas-liquid plume. In the Froude dominated, geometrically similar model and full-scale systems, the following relationship²²⁾ has been shown to hold good, viz.,   

$$\frac{{\tau }_{mix,\mathrm{\hspace{0.17em} }mod}}{{\tau }_{mix,\mathrm{\hspace{0.17em} }fs}}={\lambda }^{0.5}$$(19)

In Eq. (19), λ is the geometrical scale factor (= H_mod/H_fs) and τ_mix,mod and τ_{mix, fs} are respectively mixing times in model and full-scale systems. The validity of Eq. (19) to gas stirred ladle systems and hence, the applicability of a Froude number-based modelling criterion has been demonstrated by many researchers in the past. To this end, typically, experimentally measured mixing times from two different geometrically similar water model ladles at corresponding gas flow rates (i.e., $\frac{Q_{mod}}{Q_{fs}}$ = λ^2.5, derived from the Froude based criterion²²⁾ i.e., $\frac{U_{mod}}{U_{fs}}$ = λ^0.5) have been applied. One such set of already published results, illustrating the adequacy of Froude based scaling criterion to gas stirred ladle system, is shown in Fig. 10(a).²²⁾

Fig. 10.

(a) Experimentally measured mixing times in geometrically similar gas stirred ladle systems illustrating the adequacy of Froude basis scaling law²²⁾ and (b) Evaluation of the relative adequacy of Froude and Reynolds number-based scaling laws on the basis of numerically predicted mixing times in two different geometrically similar mechanically agitated systems. (Online version in color.)

In contrast to the above, the application of a Froude similarity criterion to a mechanically agitated system is not naturally forthcoming since flow, per se, does not occur under the influence of the gravitational force. Similarly, application of Reynolds similarity criterion cannot be justified since, flows in the mechanically agitated system is turbulent and the magnitude of the viscous force (= μND²) is significantly small (of the order of 10⁻⁵ N or so) in comparison to the inertial forces. It is readily shown that Reynolds number (i.e., Re = $\frac{\rho N{D}^2}{\mu }$) equality in mechanically agitated model and full-scale systems, containing fluids of identical kinematic viscosity, leads to a relationship of the type $\frac{N_{mod}}{N_{fs}}$ = λ⁻², in which, N is the rotational speed of the impeller. This together with kinematic similarity considerations suggest that corresponding mixing times in the model and full-scale systems, in Reynolds number dominated flow regime, are related in accordance with: $\frac{{\tau }_{mix,\mathrm{\hspace{0.17em} }mod}}{{\tau }_{mix,\mathrm{\hspace{0.17em} }fs}}$ = λ^2.0. In contrast to the above, Froude number (i.e., Fr = $\frac{N^2{D}^2}{gD}$) equivalence together with kinematic similarity entail the relationships $\frac{N_{mod}}{N_{fs}}$ = λ^−0.5, and $\frac{{\tau }_{mix,\mathrm{\hspace{0.17em} }mod}}{{\tau }_{mix,\mathrm{\hspace{0.17em} }fs}}$ = λ^0.5 between geometrically similar model and full-scale systems.

Thus, to test the relative adequacy of Reynolds vs. Froude criterion to the modelling of mechanically agitated system, 95% mixing times in two different size water model systems (D = 0.30 m and D = 1.12 m; λ = 0.268) were predicted numerically via the sliding mesh based LES calculation procedure presented earlier. Estimates of mixing time thus obtained have been assessed against both Froude as well as Reynolds based mixing laws (i.e., $\frac{{\tau }_{mix,\mathrm{\hspace{0.17em} }mod}}{{\tau }_{mix,\mathrm{\hspace{0.17em} }fs}}$ = λ^0.5 and $\frac{{\tau }_{mix,\mathrm{\hspace{0.17em} }mod}}{{\tau }_{mix,\mathrm{\hspace{0.17em} }fs}}$ = λ^2.0 respectively) and this is illustrated in Fig. 10(b). There, it is readily evident that predicted mixing times, in geometrically similar mechanically agitated systems, corroborate the Froude scaling criterion far more closely suggesting essentially that by correlating impeller speeds between the model and full-scale systems according to, $\frac{N_{mod}}{N_{fs}}$ = λ^−0.5, dynamic similarity between the two is reasonably ensured.

As a final point, input stirring power to any mechanically agitated systems, for a given rotational speed of the impeller, is expected to be a function of impeller design as the latter determines the contact surface area and hence the resultant shearing forces/equilibrium torque. On the other hand, at any given gas flow rate, location of the gas injection plug (asymmetric vs. symmetric) does not influence specific stirring power and associated flow and mixing to a significant extent. Given the simple shape of the stirrer considered in this work vis a vis the appreciable difference in flow and mixing times between the two systems, it is anticipated that stirring intensity as well as mixing efficiency, at equivalent specific stirring power, are in general much superior in mechanically agitated system.

3. Conclusions

A LES-based turbulent flow model has been developed to investigate fluid flow and mixing phenomena in a water model of a mechanically agitated system. The mathematical model has been validated against experimentally measured flow and mixing times and applied to numerically calculate stirring power in the model ladle (i.e., D = 0.30 m) at different impeller speeds. Parallel to such, already reported studies on flow phenomena and mixing in gas stirred ladle systems have been considered and these were applied to carry out a comparison between mechanically agitated and gas stirred ladle systems at equivalent specific stirring power. It is shown that general features of fluid motion in the two systems are dependent on the mode of stirring and intensity of motion in the mechanically agitated system, at equivalent specific stirring power, is considerably stronger. As a consequence of such, shorter mixing times result in the mechanically agitated systems. Although mixing time in both the systems were found to decrease with increasing stirring power, completely opposite dependence of mixing time on liquid depth were observed i.e., τ_mix,GS,95%∞H^−1.0 as opposed to τ_mix,MA,95%∞H^1.8. Finally, dynamic similarity criterion in geometrically similar, mechanically agitated water model ladle system was analysed theoretically as well as computationally and it is shown that, similar to gas stirred ladle systems, Froude based modelling criterion holds good reasonably for mechanically agitated system.

Acknowledgement

The authors gratefully acknowledge Mr. Rishikesh Misra, Doctoral student, Dept. of MSE, IIT Kanpur for providing the DPM based estimation of flow fields in an axi-symmetrical, water model, gas stirred ladle system (D = 0.30 m and H = 0.25 m at a gas flow rate of 2 lit/min or, 0.332 × 10⁻⁴ m³/s), presented as Figs. 4(c) and 4(d) in the text.

Nomenclature

C_w: WALE constant

d: distance from the closest wall (m)

D: diameter of the ladle (m)

D_imp: Diameter of impeller (m)

J_j,SGS: subgrid scale mass flux (kg/m²-s)

H: depth of liquid in the ladle (m)

H_fs: height of the ladle in full scale system (m)

H_mod: height of the ladle in model system (m)

L_SGS: subgrid-scale mixing length (m)

$\overrightarrow{m}$: mesh velocity (m/s)

M: total mass of liquid (kg)

M_GS: mass of liquid contained in the gas stirred ladle (kg)

M_MA: mass of liquid contained in the mechanically agitated ladle (kg)

N: rotational speed of the impeller (rpm)

N_Eu: Euler number

N_Fr: Froude number

N_fs: rotational speed of the impeller in full scale system (rpm)

N_mod: rotational speed of the impeller in model system (rpm)

N_Re: Reynolds number

p: filtered pressure (pa)

P: power (W)

Q,Q_G: volumetric gas flow rate (m³/s)

Q_fs: gas flow rate in full scale system (m³/s)

Q_G,Eq.: equivalent gas flow rate (m³/s)

Q_mod: gas flow rate in model system (m³/s)

R: radius of ladle (m)

Re_imp: Reynolds number of the impeller

S_ij: filtered rate of strain tensor (1/s)

S_ψ: source term of ψ (kg/s³-m)

Sc: Schmidt number

Sc_t: turbulent Schmidt number

T: time (s)

$\overrightarrow{T}$: torque vector (N-m)

T_total: total torque on impeller and shaft surfaces along rotational axis (N-m)

$\overrightarrow{u}$: flow velocity (m/s)

u: filtered velocity (m/s)

U_fs: mean speed of liquid recirculation in full scale system (m/s)

U_mod: mean speed of liquid recirculation in model system (m/s)

U_mean: mean speed of liquid recirculation (m/s)

v_res: resultant speed (m/s)

V: control volume (m³)

V_c: volume of a computational cell (m³)

x, y: cartesian coordinate (m)

Z: cartesian and cylindrical coordinate (m)

Γ: diffusion coefficient (m²/s)

δ_ij: Kronecker delta

${\dot{\epsilon }}_{GS}$: input power for gas stirred ladle system (W)

${\dot{\epsilon }}_{M,GS},{\dot{\epsilon }}_g$: specific input power for gas stirred ladle system (W/kg)

${\dot{\epsilon }}_{M,MA},{\dot{\epsilon }}_M$: specific input power for mechanically agitated ladle system (W/kg)

κ: von Karman constant

λ: Geometrical scale factor

μ: dynamic viscosity (kg/m-s)

μ_SGS: subgrid-scale turbulent viscosity (kg/m-s)

ρ: fluid density (kg/m³)

ρ_L: density of liquid in the ladle system (kg/m³)

τ_ij: residual stress tensor

τ_m,T_m: measured 95% mixing time (s)

τ_{mix, fs}: mixing times in full scale systems (s)

τ_mix,mod: mixing times in model systems (s)

τ_mix,95%,GS: 95% mixing time of gas stirred ladle system (s)

τ_mix,,95%,MA: 95% mixing time of mechanically agitated ladle system (s)

Φ_i: mass fraction of species i

Ψ: general scalar

ω: :rotational speed (rad/s)

${\overrightarrow{\omega }}_a$: angular velocity (rad/s)

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