*3.1. Supersonic Coherent Jet Modeling*

The supersonic coherent jet is modeled with the assumption that the jet flow is conducted in a steady, compressible, non-isothermal process. The corresponding Navier-Stokes equations were solved with the modification of the *k* − ε turbulence model, which aims to improve the prediction accuracy of the jet potential core length and oxygen delivery rate and provide correct numerical conditions

to estimate the subsequent jet penetration cavity shape and simulate the decarburization process. The governing equations solved are listed below.

The continuity conservation equation can be expressed by:

$$\nabla \cdot \left(\rho \overrightarrow{\upsilon}\right) = 0,\tag{1}$$

The momentum conservation equation is represented as:

$$\nabla \cdot \left( \rho \stackrel{\textstyle \cdot}{\vec{v}} \stackrel{\textstyle \cdot}{\vec{v}} \right) = -\nabla p + \nabla \cdot \left( \stackrel{\textstyle \cdot}{\vec{\tau}} \right) + \rho \stackrel{\textstyle \cdot}{\vec{g}} + \stackrel{\textstyle \cdot}{F} \tag{2}$$

where ρ is the fluid density; <sup>→</sup> *<sup>v</sup>* is the velocity vector; *<sup>p</sup>* is the static pressure; <sup>=</sup> τ is the stress tensor; <sup>→</sup> *g* is the acceleration of gravity and <sup>→</sup> *F* is the external body force.

The energy conservation equation can be written as:

$$\nabla \cdot \left[ \overrightarrow{\boldsymbol{v}} \left( \rho \boldsymbol{E} + \boldsymbol{p} \right) \right] = \nabla \cdot \left| \left( \boldsymbol{k} + \frac{\boldsymbol{c}\_{p} \mu\_{t}}{Pr\_{t}} \right) \nabla T - \sum\_{j} \boldsymbol{h}\_{j} \overrightarrow{\boldsymbol{J}}\_{j} + \left( \overrightarrow{\boldsymbol{\tau}}\_{eff} \cdot \overrightarrow{\boldsymbol{\upsilon}} \right) \right| + \boldsymbol{S}\_{\boldsymbol{h} \boldsymbol{\nu}} \tag{3}$$

where *E* is the total energy related to the sensible enthalpy *h*; *k* is the thermal conductivity; *cp* is the specific heat; μ*<sup>t</sup>* is the turbulent viscosity; *Prt* is the turbulent Prandtl number whose default value is 0.85 for the *k* − ε turbulence model. For the free shear flow with high heat transfer simulation, the appropriate turbulent Prandtl number should be set as 0.5 according to the suggestions by Wilcox [14] and Alam [3]. However, the shrouding combustion flame around the primary supersonic oxygen jet prevents the entrainment of ambient gas into the center jet, which further impacts the generation of the free shear layer. Thus, 0.85 was still adopted for the turbulent Prandtl number to estimate the turbulent thermal conductivity in the current model. <sup>→</sup> *J <sup>j</sup>* is the diffusion flux of substance *j* and *Sh* is the volumetric heat sources including the heat of chemical reaction during the simulation.

Considering the supersonic state of the primary oxygen jet, the flow turbulence can be resolved through a time-averaged velocity scalar. In the present study, the averaged Reynolds stresses term was determined using the modified *k* − ε turbulence model originally proposed by Launder and Spalding [15]. The governing equations of the turbulent kinetic energy *k* and turbulence dissipation rate ε can be expressed by:

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho u\_i \mathbf{k}) = -\rho \overline{u\_i u\_j} \frac{\partial u\_j}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_i} \left(\mu + \frac{\mu\_l}{\sigma\_k} \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_l}\right) - \rho \varepsilon - \rho \varepsilon \mathbf{M}\_\tau \tag{4}$$

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho \mathbf{u}\_i \boldsymbol{\varepsilon}) = -\mathbf{C}\_{\varepsilon 1} \rho \overline{\mathbf{u}\_i \boldsymbol{u}\_j} \frac{\varepsilon}{k} \frac{\partial \mathbf{u}\_j}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left( \mu + \frac{\mu\_t}{\sigma\_\varepsilon} \frac{\partial k}{\partial \mathbf{x}\_j} \right) - \mathbf{C}\_{\varepsilon 2} \rho \frac{k^2}{\varepsilon} \mathbf{y} \tag{5}$$

where μ is the molecular viscosity; σ*<sup>k</sup>* and σε are the turbulent Prandtl number for *k* and ε, whose values are 1.0 and 1.3, respectively; *M*τ is the turbulent Mach number that can be defined as:

$$M\_{\pi} = \frac{\sqrt{2k}}{a},\tag{6}$$

where *a* is the acoustic velocity; *C*ε<sup>1</sup> and *C*ε<sup>2</sup> are constants whose values are 1.44 and 1.92, respectively.

The turbulent viscosity μ*<sup>t</sup>* used in Equations (4) and (5) is defined by:

$$
\mu\_t = \mathbb{C}\_{\mu} \rho \frac{k^2}{\varepsilon},
\tag{7}
$$

where *C*<sup>μ</sup> is a constant value originally equal to 0.09 for the standard *k* − ε turbulence model. In order to consider the influence of the entrained ambient gas that is reduced by the shrouding combustion flame, *C*μ was modified according to the formula proposed by Alam et al. [3]. The original value of *C*μ was divided by a variable *CT* to include the effects of the local total temperature gradient in estimating the turbulent viscosity, thereby further reducing the mixed growth rate of the shear layer to accurately simulate the jet potential core length [16]. The modified *CT* can be expressed as:

$$\mathbf{C}\_{\mu} = \frac{0.09}{\mathbf{C}\_{T}} \tag{8}$$

and

$$C\_T = 1 + \frac{C\_1 T\_\mathcal{g}^m}{1 + C\_2 f(M\_\mathcal{r})},\tag{9}$$

where *C*1, *C*<sup>2</sup> and *m* is constantly equal to 1.2, 1.0 and 0.6, respectively; *Tg* is the normalized local total temperature gradient, which can be calculated by:

$$T\_{\mathcal{S}} = \frac{k^{\frac{3}{2}} |\nabla T\_t|}{\varepsilon |T\_t|},\tag{10}$$

where *Tt* is the local total temperature of the flow field; *f*(*M*τ) is a function that further considers the influence of turbulent Mach number, which can be estimated by:

$$f(M\_{\mathbb{T}}) = \left(M\_{\mathbb{T}}^{\cdot 2} - M\_{\mathbb{T}0}{}^{2}\right)H(M\_{\mathbb{T}} - M\_{\mathbb{T}0})\_{\prime} \tag{11}$$

where *H*(*x*) is the Heaviside function; *M*τ<sup>0</sup> is a constant equal to 0.1 [17]. All aforementioned modifications of the standard *k* − ε turbulence model are incorporated into the CFD-solver Fluent through the user-defined function (UDF) code based on C language and compiled in the CFD solver for the simulation.

In order to capture the shrouding combustion flame, the species transport model with the eddy dissipation concept (EDC) [18] was employed to simulate the 28-step natural gas-oxygen combustion reactions. The Discrete Ordinates (DO) radiation model with Weighted-Sum-of-Gray-Gases Model (WSGGM) [19] was adopted to model the radiation heat transfer phenomenon for the combustion.

The numerical simulation domain of the supersonic coherent jet is shown in Figure 2, which contains 3 million computational cells totally. Total computational time is around 15 h if using 80 cores in the High Performance Computing (HPC) cluster to obtain the converged results. The simulation domain is a cylindrical-shaped vessel originating from the exit of the converging-diverging nozzle where the nozzle structure is ignored. The dimension of the vessel is much larger than the burner, which can be used to simulate the supersonic coherent jet behavior in the open space. Therefore, except for the wall where the nozzle exit is located, the other walls of the vessel are set as outlets. More detail on burner operating conditions and other information are mentioned in another published paper [20].

**Figure 2.** 3D computational domain of the supersonic coherent jet modeling.

#### *3.2. Jet Penetration Cavity Estimation*

The present study utilized a novel method to consider the jet penetration in the liquid steel bath so that the direct simulations of interaction between multi phases can be avoided. The basic idea is to calculate a theoretical interface to represent the jet penetration cavity inside the liquid steel bath and this cavity will be estimated based on the characteristics of the coherent jet reaching the bath surface and used as the physical boundary of the computational domain for subsequent decarburization simulations. The shape of the cavity interface is assumed to be a revolution paraboloid according to Memoli et al. [11], which is more precise for the coherent jet with high momentum, as its penetration depth is greater than the radius of its cross-section. The three-dimensional mathematical expression of a revolution paraboloid in Cartesian coordinate can be written as:

$$z = \frac{x^2 + y^2}{c},\tag{12}$$

where *c* is the constant need to be defined by a given volume of the jet penetration cavity and the penetration depth.

The volume of the jet penetration cavity *V* can be determined by calculating liquid steel replaced by the gas flow based on the impulsive balance at the cavity interface if ignoring the impact of the liquid steel surface tension [21]. The expression of the jet cavity volume can be written as:

$$V = \frac{\pi \rho\_{\rangle} v\_{\rangle}^2 d\_{\rangle}^2}{4g\rho\_{\mathfrak{s}}},\tag{13}$$

where ρ*<sup>j</sup>* and ρ*<sup>s</sup>* are the density of primary oxygen jet and liquid steel, respectively; *vj* and *dj* are the primary oxygen jet velocity and diameter when at bath surface, which can be determined through the supersonic coherent jet modeling of a given distance from the nozzle exit to the bath.

Jet penetration depth *D* refers to an empirical formula derived by Ishikawa et al. [22], which describes the penetration depth created by the turbulent jet. For the supersonic coherent jet, the constants in the formula need to be modified accordingly. The empirical formula shows the relationship between the jet penetration depth of a single-hole or multi-holes nozzle and the burner operating conditions, which can be expressed as:

$$D = \gamma\_{h\_0} e^{-\frac{\sigma\_1 L}{\gamma \hbar\_0 c^{\rm as} \theta}} \tag{14}$$

$$\gamma\_{\hbar\_0} = \sigma\_2(\frac{\dot{V}}{\sqrt{3}nd}),\tag{15}$$

where *L* is the axial distance between the nozzle exit to the bath surface; θ is the angle of the jet inclination; . *V* is the volume flow rate of primary oxygen jet; *n* is the number of the nozzle and equal to 1 for the current study; *d* is the nozzle exit diameter for primary oxygen jet; σ<sup>1</sup> and σ<sup>2</sup> are two constants originally equal to 1.77 and 1.67, respectively and those two parameters are determined through experiments for a specific type of coherent jet used in the present study.

The actual refining process has the slag layer covering the liquid steel bath to protect the arc and reduce heat radiation loss. The coherent jet needs to pass through the slag layer before reaching the liquid steel bath. During this period, the jet will lose some of its momentum. Therefore, the jet penetration depth should be shorter than the one without the slag layer. In the current model, the slag layer is assumed to be converted equivalently to a corresponding liquid steel layer to include its effect on the jet penetration depth. The equivalent slag layer height *hs* can be estimated by:

$$h\_{\rm s} = \frac{\rho\_{\rm sl}}{\rho\_{\rm s}} h\_{\rm sl\prime} \tag{16}$$

where ρ*sl* and *hsl* are the values for slag layer density and slag layer height, respectively. The actual jet penetration depth *Dact* reads as:

$$D\_{\rm act} = D - h\_s. \tag{17}$$

Once the constant *c* is determined by solving Equations (13) to (17), the theoretical parabolic jet cavity interface can be defined and included as the physical boundary for the computational domain of the bottom section of the EAF for the decarburization simulation. This eliminates the need to include the consideration of supersonic jets and its interaction with the liquid surface in the decarburization simulation. The estimation of the three-dimensional jet penetration cavity based on actual burner operating conditions is illustrated in Figure 3 and the computational domain with five jet penetration cavities established according to the actual burner arrangement provided by industry is given in Figure 4. This computational domain is going to be used in subsequent decarburization simulations.

**Figure 3.** Sketch of 3D jet penetration cavity estimation.

**Figure 4.** 3D computational domain with jet penetration cavities for decarburization modeling.

Notice that when the supersonic coherent jet impinges on the liquid steel bath forming the jet penetration cavity, the exchange of energy and substance occurs intensively between the gas phase and liquid phase. Therefore, the jet penetration cavity surface, as the physical boundary of the computational domain, needs to establish appropriate boundary conditions to consider the energy and substance transfer during the jet impingement. In the present study, both jet momentum transfer and delivery of the oxygen were considered. Based on the energy balance on the cavity surface, the jet momentum transferred to the liquid steel bath *Ps*,*avg* can be expressed as:

$$P\_{s, \text{avg}} = a \rho\_{\text{O}\_2} v\_{\text{O}\_2} ^2 A = \frac{a \rho\_{\text{O}\_2} ^2 A}{\rho\_{\text{s}}} \left[ \frac{1}{\Delta z} \int\_{z\_2}^{z\_1} v\_{\text{O}\_2}(z) dz \right]^2,\tag{18}$$

where α is the transferable percentage of the jet total momentum at liquid steel bath, which is 0.06 according to the reference [23]; *vO*<sup>2</sup> is average jet velocity along cavity centerline; *A* is the cavity surface area; Δ*z* is the length of the cavity centerline, which is equal to *z*<sup>1</sup> − *z*2.

The amount of oxygen delivered to the liquid steel *mO*2, *avg* through the jet cavity can be estimated by calculating the average oxygen distribution along the cavity centerline:

$$m\_{\rm O\_2, avg} = \frac{1}{\Delta z} \int\_{z\_2}^{z\_1} m\_{\rm O\_2}(z) dz. \tag{19}$$
