**3. Adherence of alumina particles on refractory surfaces**

The adhesion of alumina inclusions to the refractory surface has its highest repercussion in the nozzle feeding with liquid steel the continuous casting slab mold. The property governing these phenomena is the contact angle between a particle and a solid phase with a smooth surface as is shown in **Figure 15**, which is a function of the surface tension and known as Young's law [47],

$$
\sigma\_{\rm SL} = \sigma\_{\rm SV} - \sigma\_{\rm LV} \cos \theta\_Y \tag{22}
$$

In actual refractory materials, there are not smooth surfaces and have certain levels of asperities and roughness. Hence, Eq. (22) suffers a modification through a roughness factor, r, to become in the Wenzel's Equation [47],

$$
\cos \theta\_W = r \cos \theta\_Y \tag{23}
$$

*θCB* ¼

*Contact angle between surfaces in a liquid in equilibrium with a vapor phase [46].*

**Figure 14.**

**Figure 15.**

**99**

P*<sup>n</sup>*

*Models for initial stages of sintering of spherical particles showing the vacancies diffusion paths [43].*

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

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

*<sup>i</sup>*¼<sup>1</sup>*<sup>f</sup> <sup>n</sup>*ð Þ *<sup>σ</sup><sup>i</sup>*,*SV* � *<sup>σ</sup><sup>i</sup>*,*SL σLV*

Where the limits i and n, in Eq. (24), corresponding to the stable phases forming part of the refractory material and fi is the surface fraction of phase i. The following

(24)

A further modification includes the consideration of the heterogeneous structural nature of industrial materials such as refractories for continuous casting and Eq. (23) becomes into the Cassie-Baxter Equation [47],

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

**Figure 14.** *Models for initial stages of sintering of spherical particles showing the vacancies diffusion paths [43].*

**Figure 15.**

In this equation, DV = 1*:*3*X*10 *exp* �<sup>110000</sup>

*Casting Processes and Modelling of Metallic Materials*

**Figure 13.**

**98**

*RT*

*Attractive force for different contact types. Zheng L, Malfliet a, Wollants P, Blanpain B, Guo M. effect of alumina morphology on the clustering of alumina inclusions in molten iron. ISIJ Int. 2016;56:926–935. DOI:*

The adhesion of alumina inclusions to the refractory surface has its highest repercussion in the nozzle feeding with liquid steel the continuous casting slab mold. The property governing these phenomena is the contact angle between a particle and a solid phase with a smooth surface as is shown in **Figure 15**, which is a

In actual refractory materials, there are not smooth surfaces and have certain levels of asperities and roughness. Hence, Eq. (22) suffers a modification through a

A further modification includes the consideration of the heterogeneous structural nature of industrial materials such as refractories for continuous casting and

between alumina particles K1 = 10–100 [43]. **Figure 14** shows the mechanism of diffusion of vacancies in the alumina lattice to form the bond [44]. The bond strength reaches the order of MPa due to the interdiffusion between alumina particles [45].

**3. Adherence of alumina particles on refractory surfaces**

*10.2355/isijinternational.ISIJINT-2015-561. Reproduced with permission [42].*

function of the surface tension and known as Young's law [47],

roughness factor, r, to become in the Wenzel's Equation [47],

Eq. (23) becomes into the Cassie-Baxter Equation [47],

[43], *<sup>σ</sup>pp* <sup>¼</sup> <sup>1</sup> *<sup>J</sup>*

*σSL* ¼ *σSV* � *σLV cos θ<sup>Y</sup>* (22)

*cos θ<sup>W</sup>* ¼ *rcosθ<sup>Y</sup>* (23)

*<sup>m</sup>*<sup>2</sup> is the interfacial tension

*Contact angle between surfaces in a liquid in equilibrium with a vapor phase [46].*

$$\theta\_{\rm CB} = \frac{\sum\_{i=1}^{n} f\_n (\sigma\_{i,SV} - \sigma\_{i,SL})}{\sigma\_{LV}} \tag{24}$$

Where the limits i and n, in Eq. (24), corresponding to the stable phases forming part of the refractory material and fi is the surface fraction of phase i. The following conditions establish the wettability conditions: when *σSV >σSL*, 0<sup>0</sup> *<θCB <* 900, the liquid wets the solid. When *σSL >σSV*, 90<sup>0</sup> *< θCB <* 1800, the solid and the liquid have a poor wettability. The work of adhesion between a particle and a substrate is derived from Young's Equation,

$$\mathcal{W}\_{ad} = \sigma\_{\rm SV} + \sigma\_{\rm LV} - \sigma\_{\rm SL} = \sigma\_{\rm LV}(\mathbf{1} + \cos \theta\_{\rm CB}) \tag{25}$$

A specific case of changes of surface tension of liquid steel is the case of ultralow carbon steels stabilized with Ti (Ti-SULC steels) where this element decreases this property as seen in **Figure 16a**, and the work of adhesion as a function of the Ti content is shown in **Figure 16b** [46]. This effect of Ti on the surface tension of Ti-SULC steels enhances the wettability between the melt and the inclusions. The wettability of hydrophilic and hydrophobic systems (such as steelmaking and casting processes) increases and decreases, respectively, with surface roughness, leaving behind the ideal behavior, indicated by Young's Equation, as the best condition. **Figure 17a** shows the effects of the surface roughness on the contact angle or Wenzel angle. Increasing the roughness ratios make a hydrophilic system more hydrophilic and a hydrophobic system in a more hydrophobic one. It can be assumed that in the actual metal-refractory contact, due to poor wettability between the two phases, a gas can be trapped in between the asperities of the surface, such that the liquid sits on a surface having a distribution of solid asperities and gas pockets (two-component surface material) and their surface fractions are fS and fV respectively, where *f <sup>S</sup>* þ *f <sup>V</sup>* ¼ 1. Substituting in the CB Equation for the solid–liquid fraction *f* <sup>1</sup> ¼ *f <sup>S</sup>* and *cos θCB:*<sup>1</sup> ¼ *cos θCB*,2 and the gas pocket fraction f2 = fV and *cos θCB*,2 ¼ �1 because the fraction is completely dry (no-wetting) and combining the roughness ratio factor r with the CB Equation, we get

$$
\cos \theta\_{app} = \mathbf{r} f\_S \cos \theta\_{CB} + f\_S - \mathbf{1} \tag{26}
$$

hydrophobic one. Decreasing the adherence of inclusions on the refractory surface requires two simultaneous conditions that must be fulfilled: a small contact angle between the melt and the inclusion and a low contact angle between the refractory and the melt. Therefore, to manipulate the second angle, there may be two ways:

*Effect of roughness and voids on the surface refractory on interfacial properties, (a) comparison between Wenzel's and Young's equation, (b) effect of solid fraction for contact points according to the Cassie-Baxter equation. Of fluid flow of liquid steel through clogged nozzles: thermodynamics analysis and flow simulation.*

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

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

*Steel. Res. Int. 2020;91. DOI: 10.1002/srin.202000049. Reproduced with permission [46].*

1.The first is to use CaO as a surface cover, which would be wetted it by

melt and the calcium aluminate inclusion.

adhesion of the inclusion to the wall.

summarized in **Figure 18(a–d)**:

**Figure 17.**

remain adhered to it.

**101**

surfaces according to the results presented in **Figure 17**.

liquifying the alumina inclusions and decreasing the contact angle between the

2.Use a conventional AG material with a special treatment leading to smooth

The changes of wettability among the refractory, the melt, and the inclusions are

a. The contact angle 1 (between the inclusion and the melt) is larger than the contact angle 2 (between the refractory and the melt). Hence, the liquid does not wet the nozzle. Therefore, the refractory rejects the metal, and there is the

b. When the contact angle 2 is larger than angle 1, the nozzle is slightly wettable by the melt and allows that this one enters between the inclusion and the wall, making a small separation between them and reducing the strength of adhesion.

c. When angle 2 decreases further, the nozzle wall will increase its wettability by the melt. Hence, the separation between the inclusion and the wall becomes larger, making inclusion separate from the nozzle wall.

d. In this case, the liquid wets the inclusion; if the inclusion is liquid, it will go with the flow, but if it is solid, the inclusion approaches the wall and will

As the liquid steel flows through the nozzle, **Figure 19**, the alumina particles are transported along, and those close to the boundary layer might, eventually, get in this region and adhere to the refractory's surface by mechanisms of fluctuating velocities [48, 49]. Therefore, in current casting systems, the adherence of an

The examination of Eq. (26) indicates that if the fraction fS approaches 0, by increasing the asperities of the surface, there will be a condition of perfect nonwettability. This trend is shown in **Figure 17b**. On the contrary, if fS approaches 1 (complete surface smoothness), the contact angle is given by Eq. (26) with r = 1. The combined effects of roughness and a solid fraction are as follows: in a hydrophilic system, the simultaneous increases of solid fraction and surface roughness make a hydrophilic system more hydrophilic. A decrease of the solid fraction with a combined increase of surface roughness makes this hydrophobic system in a more

#### **Figure 16.**

*Interfacial properties between liquid steel containing Ti and alumina particles, (a) interfacial tension, (b) adhesion work. González-Solórzano M.G., Morales R.D. Gutiérrez E, Guarneros J, Chattopadhyay K. analysis of fluid flow of liquid steel through clogged nozzles: Thermodynamics analysis and flow simulation. Steel. Res. Int. 2020;91. DOI: 10.1002/srin.202000049. Reproduced with permission [46].*

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

**Figure 17.**

conditions establish the wettability conditions: when *σSV >σSL*, 0<sup>0</sup> *<θCB <* 900, the liquid wets the solid. When *σSL >σSV*, 90<sup>0</sup> *< θCB <* 1800, the solid and the liquid have a poor wettability. The work of adhesion between a particle and a substrate is

A specific case of changes of surface tension of liquid steel is the case of ultralow carbon steels stabilized with Ti (Ti-SULC steels) where this element decreases this property as seen in **Figure 16a**, and the work of adhesion as a function of the Ti content is shown in **Figure 16b** [46]. This effect of Ti on the surface tension of Ti-SULC steels enhances the wettability between the melt and the inclusions. The wettability of hydrophilic and hydrophobic systems (such as steelmaking and casting processes) increases and decreases, respectively, with surface roughness, leaving behind the ideal behavior, indicated by Young's Equation, as the best condition. **Figure 17a** shows the effects of the surface roughness on the contact angle or Wenzel angle. Increasing the roughness ratios make a hydrophilic system more hydrophilic and a hydrophobic system in a more hydrophobic one. It can be assumed that in the actual metal-refractory contact, due to poor wettability between the two phases, a gas can be trapped in between the asperities of the surface, such that the liquid sits on a surface having a distribution of solid asperities and gas pockets (two-component surface material) and their surface fractions are fS and fV respectively, where *f <sup>S</sup>* þ *f <sup>V</sup>* ¼ 1. Substituting in the CB Equation for the solid–liquid fraction *f* <sup>1</sup> ¼ *f <sup>S</sup>* and *cos θCB:*<sup>1</sup> ¼ *cos θCB*,2 and the gas pocket fraction f2 = fV and *cos θCB*,2 ¼ �1 because the fraction is completely dry (no-wetting) and

combining the roughness ratio factor r with the CB Equation, we get

The examination of Eq. (26) indicates that if the fraction fS approaches 0, by increasing the asperities of the surface, there will be a condition of perfect nonwettability. This trend is shown in **Figure 17b**. On the contrary, if fS approaches 1 (complete surface smoothness), the contact angle is given by Eq. (26) with r = 1. The combined effects of roughness and a solid fraction are as follows: in a hydrophilic system, the simultaneous increases of solid fraction and surface roughness make a hydrophilic system more hydrophilic. A decrease of the solid fraction with a combined increase of surface roughness makes this hydrophobic system in a more

*Interfacial properties between liquid steel containing Ti and alumina particles, (a) interfacial tension, (b) adhesion work. González-Solórzano M.G., Morales R.D. Gutiérrez E, Guarneros J, Chattopadhyay K. analysis of fluid flow of liquid steel through clogged nozzles: Thermodynamics analysis and flow simulation. Steel. Res.*

*Int. 2020;91. DOI: 10.1002/srin.202000049. Reproduced with permission [46].*

*Wad* ¼ *σSV* þ *σLV* � *σSL* ¼ *σLV*ð Þ 1 þ *cos θCB* (25)

*cos θapp* ¼ *rf <sup>S</sup> cos θCB* þ *f <sup>S</sup>* � 1 (26)

derived from Young's Equation,

*Casting Processes and Modelling of Metallic Materials*

**Figure 16.**

**100**

*Effect of roughness and voids on the surface refractory on interfacial properties, (a) comparison between Wenzel's and Young's equation, (b) effect of solid fraction for contact points according to the Cassie-Baxter equation. Of fluid flow of liquid steel through clogged nozzles: thermodynamics analysis and flow simulation. Steel. Res. Int. 2020;91. DOI: 10.1002/srin.202000049. Reproduced with permission [46].*

hydrophobic one. Decreasing the adherence of inclusions on the refractory surface requires two simultaneous conditions that must be fulfilled: a small contact angle between the melt and the inclusion and a low contact angle between the refractory and the melt. Therefore, to manipulate the second angle, there may be two ways:


The changes of wettability among the refractory, the melt, and the inclusions are summarized in **Figure 18(a–d)**:


As the liquid steel flows through the nozzle, **Figure 19**, the alumina particles are transported along, and those close to the boundary layer might, eventually, get in this region and adhere to the refractory's surface by mechanisms of fluctuating velocities [48, 49]. Therefore, in current casting systems, the adherence of an

**Figure 18.** *Effects of melt-refractory and melt-inclusion contact angles on particle-refractory surface adhesion.*

alumina particle to the surface of a nozzle refractory is favored by the low wettability of the inclusion and the refractory by the melt.

*FA* ¼ 2*πσLVl* þ *πl*

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

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

Δ*P* ¼ *σLV*

inside of it. The pressure difference obeys Laplace's Eq. (45),

*Schematization of a cavity between a sphere and a plate in a non-wetting system.*

**Figure 20.**

**103**

**4. Boundary conditions of wall adhesion**

The cavity might be filled with: (1) gaseous components initially dissolved in the melt (2) gaseous components coming from the refractory (3) melt vapor, or (4) liquid phases forming due to a local rise in the oxygen concentration. In any case, there will be a pressure drop between the liquid outside of the cavity and the phase

> 1 *r* � 1 *l*

Where l and r are the principal radii of the cavity. The two parameters determine the profile of the liquid–vapor interface, which can be either optimized using the Laplace-Young Equation at constant pressure drop or approximately described with a piece of a circle with radius r. When the cavity is in equilibrium with the liquid phase, its shape is determined by thermodynamics. The estimated adhesion force between an alumina particle and the wall of an AG (alumina-graphite nozzle) is about 25X10�<sup>6</sup> N [51], large enough to keep fixed the particle on the refractory surface.

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

2

Δ*P* (27)

(28)
