**4. Boundary conditions of wall adhesion**

To design nozzle materials through the principles of physical-chemistry of interfaces and techniques of computer fluid dynamics, reliable boundary conditions for inclusion adherence to the refractory wall are necessary. In other words, to develop new materials is necessary to deal with a theoretical analysis including effects of surface roughness, the effect of impurities in alumina graphite materials (mainly Na2O and K2O, oxides cover) together with control of the boundary layer by the internal design of the nozzle [46] before making investments on experimental research. Specifically, this boundary condition is applicable only during the

development of the clog's first layer and not during its growth. That is to say, when the first layer of the clog stabilizes, its future growth is guaranteed by contact with other alumina particles, which will go through a sinterization process by diffusion of vacancies, as explained above, forming high strength bonds. This boundary condition is derived from a balance of forces, according to **Figure 21.** These forces obey the following expressions [52]:

Drag force

$$F\_d = \text{б}\pi\mu\gamma \frac{d\_p^2}{4}f\tag{29}$$

Different force ratios working on the particles are useful as possible boundary

*Rv* <sup>¼</sup> *FL Fa*

*Rs* <sup>¼</sup> *Fd* <sup>þ</sup> *Fb ks*ð Þ *Fa* � *FL*

The constants and variables included in these expressions are reported in **Table 3** [53]. In these calculations, instead of using Eq. (30) to estimate the adhesion force, the force suggested of 25X10�<sup>6</sup> N was employed [51]. This force was calculated by optimizing Eq. (27) and Eq. (28) and is considered a more realistic magnitude. The force ratios calculated for a ten μm particle in a liquid steel flow are in the last column of **Table 3**. Some conclusions derived from these results are a) The lift force is negligible compared with the adhesion force. b) The ratio for sliding is small compared with the effects of the adhesion force. c) Summing the momentums of drag, lift, and buoyancy forces and compared with the adhesion force's momentum yields a magnitude larger than one, meaning that the particle may be

The clogging phenomena have their roots in the deoxidation step of the liquid steel. Operational factors like the addition time of aluminum, oxygen supersaturation, temperature, and melt stirring fixe the initial conditions of sizes distributions

Poisson's ratio, *<sup>v</sup>*<sup>1</sup> 0.27 *Rv* <sup>¼</sup> <sup>5</sup>*:*12*x*10�<sup>8</sup>

Surface energy, *<sup>σ</sup>* 0.001 J.m�<sup>2</sup> *Rs* <sup>¼</sup>0.064

Liquid density, *<sup>ρ</sup><sup>l</sup>* 7100 kg/m<sup>3</sup> *Rt* <sup>¼</sup> <sup>7</sup>*:*<sup>60</sup>

�<sup>1</sup> **Ratio of forces**

 *Fd* <sup>þ</sup> *aFL* <sup>þ</sup> *rpFb aFa*

*Rt* <sup>¼</sup> <sup>1</sup>*:*4*rp*

(33)

(34)

(35)

conditions or, simply, to compare the magnitudes of these forces as follows:

*The Physical Chemistry of Steel Deoxidation and Nozzle Clogging in Continuous Casting*

Ratio for the vertical lift-off

*DOI: http://dx.doi.org/10.5772/intechopen.95369*

Ratio for the tangential lift-off:

Ratio for the sliding:

dislodged from the wall.

**Parameters for the calculation of ratios [53]**

Shear rate, *γ* 1 x 105 s

Poisson's ratio, *v*<sup>2</sup> 0.28 Poisson's ratio, *E*<sup>1</sup> 8.01x10<sup>10</sup> Poisson's ratio, *E*<sup>2</sup> 2.15x10<sup>11</sup>

Coefficient of friction, *ks* 0.3 *f* 1.7009 Viscosity, *μ<sup>l</sup>* 0.006 Pa.s

Particle density, *ρ<sup>p</sup>* 3600 kg/m<sup>3</sup> Particle diameter, *dp* 10 x 10�<sup>6</sup> m Gravity, *g* 9.8 m/ s<sup>2</sup>

*Parameters and constants to estimate the forces on a particle with a 10 μm diameter.*

**Closure**

**Table 3.**

**105**

Adhesion force

$$F\_a = \frac{3}{4}\pi\sigma d\_p\tag{30}$$

Buoyancy force

$$F\_b = \left(\rho\_L - \rho\_p\right) \lg \frac{1}{\mathsf{G}} \pi d\_p^3 \tag{31}$$

Lift force

$$F\_L = 9.22 \frac{\chi \mu d\_p^2}{4} \frac{\chi d\_p^2}{4v} \tag{32}$$

**Figure 21.** *Forces acting on an inclusion. (FC = capillary force, FB = buoyancy force, FL = lift force, FD = drag force) [51].*

*The Physical Chemistry of Steel Deoxidation and Nozzle Clogging in Continuous Casting DOI: http://dx.doi.org/10.5772/intechopen.95369*

Different force ratios working on the particles are useful as possible boundary conditions or, simply, to compare the magnitudes of these forces as follows:

Ratio for the vertical lift-off

$$R\_v = \frac{F\_L}{F\_a} \tag{33}$$

Ratio for the sliding:

development of the clog's first layer and not during its growth. That is to say, when the first layer of the clog stabilizes, its future growth is guaranteed by contact with other alumina particles, which will go through a sinterization process by diffusion of vacancies, as explained above, forming high strength bonds. This boundary condition is derived from a balance of forces, according to **Figure 21.** These forces

*Fd* ¼ 6*πμγ*

*Fa* <sup>¼</sup> <sup>3</sup>

*Fb* ¼ *ρ<sup>L</sup>* � *ρ<sup>p</sup>* 

*FL* <sup>¼</sup> <sup>9</sup>*:*<sup>22</sup> *γμd*<sup>2</sup>

*Forces acting on an inclusion. (FC = capillary force, FB = buoyancy force, FL = lift force, FD = drag force) [51].*

*d*2 *p*

> *g* 1 <sup>6</sup> *<sup>π</sup>d*<sup>3</sup>

*p* 4

*γd*<sup>2</sup> *p* 4*v*

<sup>4</sup> *<sup>f</sup>* (29)

<sup>4</sup> *πσdp* (30)

*<sup>p</sup>* (31)

(32)

obey the following expressions [52]:

*Casting Processes and Modelling of Metallic Materials*

Drag force

Adhesion force

Buoyancy force

Lift force

**Figure 21.**

**104**

$$R\_s = \frac{F\_d + F\_b}{k\_s(F\_a - F\_L)}\tag{34}$$

Ratio for the tangential lift-off:

$$R\_t = \frac{(1.4r\_p)F\_d + aF\_L + r\_p F\_b}{aF\_a} \tag{35}$$

The constants and variables included in these expressions are reported in **Table 3** [53]. In these calculations, instead of using Eq. (30) to estimate the adhesion force, the force suggested of 25X10�<sup>6</sup> N was employed [51]. This force was calculated by optimizing Eq. (27) and Eq. (28) and is considered a more realistic magnitude. The force ratios calculated for a ten μm particle in a liquid steel flow are in the last column of **Table 3**. Some conclusions derived from these results are a) The lift force is negligible compared with the adhesion force. b) The ratio for sliding is small compared with the effects of the adhesion force. c) Summing the momentums of drag, lift, and buoyancy forces and compared with the adhesion force's momentum yields a magnitude larger than one, meaning that the particle may be dislodged from the wall.

#### **Closure**

The clogging phenomena have their roots in the deoxidation step of the liquid steel. Operational factors like the addition time of aluminum, oxygen supersaturation, temperature, and melt stirring fixe the initial conditions of sizes distributions


#### **Table 3.**

*Parameters and constants to estimate the forces on a particle with a 10 μm diameter.*

and the alumina particle's population responsible of the nozzle clogging. The high level of supersaturation required by aluminum governs the initial size distribution. Once the particle is nucleated at the initial stages, the initial growth is through diffusion and Ostwald ripening mechanisms. Although not proved yet here, the published literature reports that the bond strength among alumina particles and their morphology are important on the clogging mechanism. Once inside the nozzle, the particle, taking contact with refractory, may remain adhered to it if the adhesion force is larger than the momentums originated from the lift, buoyancy and drag forces with the particle size.

**Greek letters**

**Sub indexes**

*α* Accommodation factor

*DOI: http://dx.doi.org/10.5772/intechopen.95369*

*μ* Dynamic liquid viscosity *ν* Kinematic liquid viscosity

*ε* Dissipation rate of kinetic energy

*The Physical Chemistry of Steel Deoxidation and Nozzle Clogging in Continuous Casting*

*γ* Shear rate

*ρ* Density

a Adhesion app Apparent b Buoyancy c critical CB Cassie-Baxter

d Drag

L Lift

M Metal O Oxide

**Author details**

**107**

Jafeth Rodríguez Ávila<sup>2</sup>

*θ* Contact angle

*σ* Surface tension

het Heterogeneous

LV Liquid–Vapor

PL Particle-Liquid SL Solid–Liquid SV Solid-Vapor v Molar volume w Wenzel y Young

María-Guadalupe González Solórzano<sup>1</sup>

, Rodolfo Morales-Dávila<sup>1</sup>

1 Instituto Politécnico Nacional-ESIQIE, Department of Metallurgy, Mexico

13, Ciudad Universitaria, 25350, Arteaga Coahuila, México

202, Cerro del Gato, 98160, Zacatecas, México

provided the original work is properly cited.

\*Address all correspondence to: rmorales@ipn.mx

2 Facultad de Ingeniería, Universidad Autónoma de Coahuila, Blvd. Fundadores Km

3 Instituto Politécnico Nacional-UPIIZ, Metallurgical Engineering, Blvd. del Bote

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

, Carlos Rodrigo Muñiz-Valdés<sup>2</sup> and Alfonso Nájera Bastida<sup>3</sup>

\*,

The refractory's roughness is of no help to control the clogging as those materials that are hydrophobic or hydrophilic will enhance these properties with rough surfaces. Ideally, a smooth surface approaching the Young's Law would be the ideal material to decrease clogging. Raw materials purity is of interest as some oxides are easily reduced by the carbon of the nozzle or aluminum in the melt, all working to enhance the clogging problem.

Under the present situation, this work contributes to the understanding of the surface phenomena in the areas of inclusion nucleation and growth of inclusions and the steel refining and the interaction with the refractory. It gives options for the boundary conditions applied in computational fluid dynamics simulations, all focused on designing new nozzle materials.
