Mass Transfer in Steelmaking
The productivity of steelmaking processes, including production and refining of liquid steel, depends on the mass transfer rates. Due to the high temperatures involved in the processes, the rate controlling step of the processes is usually a mass transfer phenomenon . In steelmaking operations, different situations of mass transfer can occur, depending on the phases involved:
Liquid-liquid mass transfer, in the case of reactions involving liquid steel and slag.
Liquid-gas mass transfer, when a gas is injected into or onto liquid steel.
Liquid-solid mass transfer, when solid particles are injected into liquid steel to promote refining reactions and alloy additions.
In all these situations, the characterization of the mass transfer coefficients and the parameters that interfere on its value is a very important task to optimize the process and improve productivity. The first step to determine the mass transfer coefficient is the modeling of the flow of the different phases to determine the mass flux based on the concentration gradients. Far from the interface, turbulent mixing ensures that the bulk concentrations remain largely uniform and spatial concentration gradients remain small. As approaching to the interface, turbulence activity is damped and the description of a species concentration C with molecular diffusivity D is governed by the advection diffusion equation: 𝜕𝐶𝜕𝑡+𝑢∙𝛻𝐶=𝛻∙𝐽 1.3.
Turbulence interactions in Multiphase Flows
The previous section described some interactions between the bulk flow and interface that may locally change the interface properties and its shape. We are going to discuss about the effects of turbulent flows near the interface region and its implications on mass transfer.
For a more detailed discussion, it is important to start by analyzing where the turbulence is produced, and two main cases may occur . The interface may have a free slip condition, which happens in channel flows when the turbulence is produced far from the interface, at the channel bottom, with no-slip condition, and brought to the interface. A second configuration is found in studies considering multiphase flows in tanks, lakes or sea, where the wind flows over the liquid-gas interface. When sufficiently high speed winds flow over a free surface, they can generate waves due to the shear imposed on the surface .
Studies like those from Lombardi et al , Banerjee et al , Lin et al  and Komori et al  verified that for the air-water cases, the turbulence characteristics at the gas side are fairly similar to those near the wall, being damped at the vicinity of the interface. The same behavior is not observed at the liquid side, where the velocity fluctuations are high near the interface .
Later, Turney et al , by investigating the turbulence structures at the liquid side of the gas-liquid interface and its effect on the mass transfer, verified that the friction velocity at the interface of the gas side should be inferior to 0.1 m/s to avoid the formation of waves on the liquid interface. In this research field, it is customary to relate the friction velocity to the wind velocity 10 m above the interface, which can be easily measured. The maximum friction velocity suggested corresponds to a wind velocity at 10 m of about 3.5 m/s. The perturbations that may occur at the liquid interface due to the wind shear are of two types; (i) small perturbations, capillary waves, or ripples, which main characteristics are low amplitude and wave length and high frequency, and (ii) large perturbations, disturbance waves, which characteristics are high amplitudes, large wave length and frequencies lower than the capillary waves. This transition from a plane surface, with small perturbations can be seen in Figure 2.1, from Turney et al . The existence of waves at the interface favors the turbulence generation and structures characteristics of this flow regime.
Mass Transfer across Liquid-Gas Interface – A brief overview
Mass transfer occurs when there is a concentration gradient. The flux is directed from the region of high concentration to the one with lower concentration. This concentration difference of the transferred species is the driven force of the mass transfer.
There are a vast number of published investigations concerning the mass transfer across the liquid-gas interface. These investigations can be numerical (     ) or experimental (    ) allowing the development of empirical models and the validation of numerical models for the mass transfer coefficient calculations at the different phases.
Turney and Banerjee   discussed the formulations for the mass transfer coefficient at the liquid-gas interface. For their widely applications, we could mention some of these formulations as the film theory of Lewis and Whitman , the penetration theory of Higbie , the renewal surface theory of Danckwerts  and the surface divergence theory of Banerjee . Each of these groups is based on a distinct hypothesis of the flow behavior close to the liquid-gas interface and they will be explained in section 2.2. All these approaches show that the mass transfer of a given chemical species depends on a transport property, the mass diffusivity, and on the hydrodynamic conditions maintaining the contact between these phases, such as the film thickness , the time that the fresh fluid packets coming from the bulk flow remain in contact with the interface , the surface renewal time  and the divergence of the interface parallel flow . These quantities are hard to be experimentally measured, forcing us to make use of empirical relations and/or much complex mathematical models to describe all the physical phenomena involved.
Numerical investigations of the mass transfer phenomenon at the liquid-gas interface vary from those that consider a tank with a liquid phase in laminar regime   (where the liquid surface is characterized by the low perturbation degree) to those that consider the liquid phase to be on a turbulent regime    (assuming that the transport mechanisms in the gas phase are not limiting factors for the mass transfer). In these investigations, the thickness of the mesh elements at the interface region is reduced in order to guarantee at least three mesh elements at the mass boundary layer . Latter, the mass transfer calculated is correlated to the different theories and the coefficients are adjusted.
Continuous Casting (CC) Water Model
In this chapter, we will describe the experimental apparatus and the techniques used to extract the data needed to analyze the flow in such models. Such techniques include the Laser Doppler Anemometry (LDA) to measure the velocity field inside the water model, the image processing, which was performed with Python®, and the Experimental Interface Tracking (EIT) method, which we developed to locally track the liquid-liquid interface.
Similitude: Non-Dimensional Parameters
Generally speaking, the physical modeling consists in building a model in laboratorial scale of one specific reactor, and simulating the process that takes place in such reactor. For the obtained results in laboratory scale to be applicable in industrial scale, it is necessary to respect some similarities between the model and the real process. We say that the model and the industrial reactor are similar when they exhibit a constant ratio between correspondent values and scales, named similarity relations or scale relations .
The similarity between the industrial process and the model may include geometric, mechanic (which is divided in static, kinematic and dynamic), thermal and chemical similitude. For the fluid dynamic study, and considering a turbulent flow that can be modeled by the Navier- Stokes equations, the dominant forces that govern the fluid are the inertia, gravity, shear and possibly the surface tension . The dimensionless numbers obtained from these forces are:
The Reynolds number (Re): the ratio between the inertial and viscous forces 𝑅𝑒=𝜌𝐿𝑢𝜇 3.1.
The Froude (Fr): the ratio between inertial and gravitational forces 𝐹𝑟=𝑢2𝑔𝐿 3.2.
The Weber (We): the ratio between inertial and surface tension forces 𝑊𝑒=𝜌𝑢2𝜎 3.3.
In this section, the experimental measurement techniques used in our multiphase system are described. Laser Doppler anemometry (LDA) was used to characterize the velocity field at the interface region. Image processing was also performed with the aid of a code built with Python language to provide insightful information about the interface displacement and wave induced motion. A methodology to locally track the interface was developed and is described at the last section of this chapter.
Laser Doppler Anemometry (LDA)
The LDA has been used in this work due to its non-intrusive principle, which allows the measurement of the velocity field without any intrusion in the fluid flow. It measures the instantaneous velocity of the fluid by detecting the frequency shift of laser light that has been scattered by small particles suspended in the flow. The LDA equipment used is a one dimensional component LDA. With this equipment, it was possible to measure the horizontal and the vertical component of the velocity field by turning the laser source in its support. The components of a LDA system are displayed in Figure 3.3: In LDA, the light is emitted from a laser source with a specific wavelength toward the measurement point. In fact, the LDA technique does not measure the velocity of the fluid itself, but the velocity of particles dispersed in the fluid, called the seeding particles. Thus, for the measured velocity to be assumed the same as the fluid velocity, these particles must have the same specific mass as fluid of interest, in such a way that the gravity forces and the buoyancy can be neglected. The seeding particles may also have a geometric form that reduces the resistance forces that may interfere in the flow characteristics. They should scatter the light sufficiently and be generated conveniently. For a more comprehensive explanation of LDA principles the reader is referred to the Dantec website .
With the LDA technique, we could measure the mean flow velocity and the root mean square (rms) of the velocity field in each measured position, p. For each point of measure, 600 seconds of measures with an average data acquisition frequency of 80Hz were taken. We believed that this measurement time would be sufficient to characterize most of the low frequencies encountered in CC model configurations. The mean velocity and the rms at each point are calculated by 𝑢̅𝑝=1𝑁Σ𝑢𝑝,𝑖𝑁𝑖=1 3.12.
Table of contents :
1.1 Steelmaking Processes and Mass Transfer
1.2 Continuous Casting
1.3 Mass Transfer in Steelmaking
2 Mass Transfer and Fluid Flow near the Interface Region
2.1 Turbulence interactions in Multiphase Flows
2.2 Mass Transfer in Multiphase Flows
2.2.1 Mass Transfer across Liquid-Gas Interface – A brief overview
2.2.2 The film theory
2.2.3 The penetration model
2.2.4 The surface renewal time model
2.2.5 The surface Divergence Model
3 Experimental Methods
3.1 Continuous Casting (CC) Water Model
3.1.1 Similitude: Non-Dimensional Parameters
3.1.2 Physical Model and Fluid Properties
3.2 Velocity Measurements
3.2.1 Laser Doppler Anemometry (LDA)
3.3 Image processing
3.3.1 Python Image Processing
3.3.2 Experimental Interface Tracking (EIT)
4 Models and Numerical Methods
4.1 The single fluid model
4.1.1 Mass Conservation
4.1.2 Momentum Conservation
4.1.3 Navier-Stokes Equation for Single Fluid Model
4.2 Turbulence modeling
4.2.1 General featuring’s
4.2.2 LES modeling for single fluid flows
4.2.3 Spatial filtering of Navier-Stokes single-fluid model
4.2.4 Filtered Navier-Stokes equations for two-phase flows
4.3 Fictitious Domain Method (FDM) for Obstacles and solid boundaries
4.4 Approximation of the turbulent single-fluid model
4.4.1 Temporal discretization
4.4.2 The incompressibility constraint – The velocity-pressure coupling
4.5 Spatial Integration
4.6 Interface Tracking Methods
4.6.1 Interface tracking with reconstruction – VOF-PLIC
4.6.2 Capillarity effects Smooth Volume of Fluid – SVOF
5 Continuous Casting – Hydrodynamic Characterization
5.1 Experimental Analysis
5.1.1 Effect of water flow rate
5.1.2 Effect of oil layer viscosity
5.1.3 Effect of oil layer thickness
5.2 Proposed configuration for mathematical modeling
5.3 Interface Characteristics
5.4 Mean Flow Description
5.4.1 Water Oil Interface (WOI) configuration
5.4.2 Water Air (WFS) Configuration
5.5 Turbulence Characterization
5.6 Turbulence implications on Mass Transfer Coefficients
5.7 Mass Transfer distribution at the liquid/liquid interface
5.8 Influence of process parameters on mass transfer coefficients
5.8.1 Casting Speed
5.8.2 Slag viscosity
6 Liquid/Liquid Mass Transfer Experiments
6.1 Liquid/Liquid Experiments
6.1.1 Mass Transfer in a Ladle Model
6.1.2 Simplified liquid/liquid mass transfer experiments
6.1.3 Simplified liquid/liquid mass transfer modeling
7 Mass Transfer on an Industrial Configuration
7.1 Continuous Casting – Industrial Configuration
7.1.1 The mold
7.1.2 The Submerged Entry Nozzle (SEN)
7.1.3 The operational conditions
7.2 CFD Model Results
7.2.1 Hydrodynamic aspects
7.2.1 Mass Transfer Coefficient
9 References .