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

Abstract

Nozzle clogging in continuous casting of steel originates by the adherence of alumina particles and other oxides, precipitated during the liquid steel deoxidation, on the refractory material’s surface. Hence, these particles’ nucleation and growth rates in supersaturated melts are analyzed considering, specifically, the role of the interfacial tensions between alumina, silica, and other oxides and the liquid metal. Weak steel deoxidizers like silicon do not need high supersaturations favoring high nucleation rates, giving particles’ narrow size distributions thanks to fast diffusion and Ostwald-ripening coagulation. Strong deoxidizers, like aluminum, need high supersaturation levels leading to broad size distributions. Besides, the morphology of these particles depends on the nucleation and growth mechanisms. The adhesion forces among the deoxidation particles, forming clusters, depending on the morphology and the oxide’s chemistry. The stability of the nozzle’s clog, adhered to the nozzle’s wall, depends on the interface tensions between the melt and the nozzle’s refractory surface and between the melt and the inclusion. The results obtained here help set up basic recommendations in steel refining and materials specifications of casting nozzles.

1. Introduction

Clogging of mold nozzles, also called submerged entry nozzles (SEN), disrupts the steel casting process, affecting the caster’s productivity. The nozzle clogging produces inconsistent flow and temperature variations, steel level fluctuations in the mold, impairment of steel quality, and the steel casting’s abrupt interruption. Clogging starts when solid compounds, mainly steel skull and non-metallic inclusions, are non-uniformly deposited at the inner nozzle wall, at some typical preferential zones characterized for neighboring dead flow conditions [1, 2, 3, 4, 5]. These inclusions have as primary sources: (1) The reaction between the dissolved oxygen with the deoxidizers [6, 7, 8, 9]; (2) re-oxidation in the tundish or the nozzle [10, 11]; and (3) the entrainment of slag or refractory particles [11, 12, 13, 14]. Researchers who have worked on the determination of inclusions sources and clogging recognize that the deposited inclusions at the nozzle wall are mainly alumina inclusions [7, 8, 9, 15, 16]. Steel re-oxidation occurs due to possible air aspiration under the flow control valves (slide gate) to maintain the entry flow to the molds [17, 18, 19]. Besides, regardless of the refractory nozzle composition (alumina-graphite, zirconia, and magnesia), the steel melt infiltrates the refractory and removes the protective surface [11, 20], allowing the entrapment of refractory particles and inclusion attachment at the nozzle wall.

The non-metallic inclusions come from the steelmaking process; several researchers have focused on studying the variables that induce the inclusion deposition at the inner nozzle wall producing the clogging phenomena [1, 11, 12, 13, 14, 16, 21, 22]. Steel chemistry and, in particular, steel grades containing titanium, like Ti-SULC, (Ti Stabilized Ultra-Low Carbon Steels), steels enhance the nozzle clogging due to the surface tension properties of this element in liquid steel [23, 24, 25]. The wetting of inclusions, rich in Ti oxide, assists in the clustering and compaction of particles. When the ratio Ti/Al is above a threshold, dictated by thermodynamics, the wettability of complex oxides of Ti and Al works intensifying the nozzle’s clogging under the presence of oxygen. This series of papers provides an insight into the clogging phenomena while casting these steel grades.

In the present chapter, the authors deal with the physical–chemical aspects of the nozzle-clogging problem by inclusions originated through the deoxidation reactions of steel. Hence, to understand the fundamentals of the problem first, these particle’s nucleation and growth are considered using non-equilibrium thermodynamics principles. Second, the influence of the steel refining processes on the deoxidation particle morphology and the relation with their further growth through aggregation and clustering mechanisms is under scrutiny. The analysis continues with studying the thermodynamics related to the oxide particle’s adherence to the refractory. Finally, a dynamic analysis lets the establishment of balance among drag, buoyancy, adherence, and lift forces leading to a detachment criterion for an inclusion forming part of a first layer of the clogging. After this work, conclusions and recommendations are provided.

2. Non-equilibrium thermodynamics

2.1 Nucleation and growth rates of oxide inclusions

The nucleation of a foreign phase in an originally homogeneous solution, in the present case an oxide particle in a liquid solution of iron, is driven by the free energy of the reaction.

$$\mathit{x}\left[\mathit{M}\right]+\mathit{y}\left[\mathit{O}\right\}={\mathit{M}}_{\mathit{x}}{\mathit{O}}_{\mathit{y}}$$E1

$$∆{\mathit{G}}_{\mathit{m}}=∆{\mathit{G}}_{\mathit{m}}^{\mathrm{0}}+\mathit{\text{RTln}}\frac{{\mathit{a}}_{{\mathit{M}}_{\mathit{x}}{\mathit{O}}_{\mathit{y}}}}{{\mathit{h}}_{\mathit{M}}^{\mathit{x}}{\mathit{h}}_{\mathit{O}}^{\mathit{y}}}=∆{\mathit{G}}_{\mathit{m}}^{\mathrm{0}}+\mathit{\text{RTlnQ}}$$E2

where Q is the activity quotient, in the thermodynamic equilibrium $\mathit{∆}{\mathit{G}}_{\mathit{m}}\mathit{=}\mathrm{0}$ and the Eq. (2) changes to,

$$∆{\mathit{G}}_{\mathit{m}}^{\mathrm{0}}=-\mathit{\text{RTln}}{\mathit{K}}_{\mathit{E}}$$E3

where K_E is the equilibrium constant, which depends only on the temperature. Combining Eqs. (2) and (3), dividing the result between the molar volume of the oxide results in,

$$\begin{array}{cc}∆{\mathit{G}}_{\mathit{V}}& =\mathit{\text{RTln}}\\ \frac{\mathit{Q}}{{\mathit{K}}_{\mathit{E}}}& =-\mathit{\text{RTln}}\frac{{\mathit{S}}_{\mathrm{0}}}{{\mathit{V}}_{\mathrm{0}}}\end{array}$$E4

and S₀ is the supersaturation ratio given by,

$${\mathit{S}}_{\mathrm{0}}\mathit{=}\frac{\mathit{Q}}{{\mathit{K}}_{\mathit{E}}}\mathit{=}\frac{{\mathit{K}}_{\mathit{MO}}}{{\mathit{K}}_{\mathit{E}}}\mathit{=}\frac{\left[{\mathit{h}}_{\mathit{M}}^{\mathit{x}}{\mathit{h}}_{\mathit{O}}^{\mathit{y}}\right]}{{\left[{\mathit{h}}_{\mathit{M}}^{\mathit{x}}{\mathit{h}}_{\mathit{O}}^{\mathit{y}}\right]}_{\mathit{E}}}\mathit{=}\frac{{\left[\%\mathit{M}\right]}^{\mathit{x}}{\left[\%\mathit{O}\right]}^{\mathit{y}}}{{\left\{{\left[\%\mathit{M}\right]}^{\mathit{x}}{\left[\%\mathit{O}\right]}^{\mathit{y}}\right\}}_{\mathit{E}}}$$E5

K_MO is the solubility product of the oxide. The free energy balance for the nucleation of an oxide precipitated in the homogeneous melt includes the volumetric free energy and the surface energy opposing to the stabilization and growth of the nucleus,

$$∆\mathit{G}=\mathrm{4}\mathit{\pi }{\mathit{r}}^{\mathrm{2}}{\mathit{\sigma }}_{\mathit{PL}}+\left(\frac{\mathrm{4}}{\mathrm{3}}\right)\mathit{\pi }{\mathit{r}}^{\mathrm{3}}∆{\mathit{G}}_{\mathit{V}}$$E6

Making zero the derivative of the free energy (Table 1 [26, 27] reports the molar Gibbs free energy of deoxidation reactions of iron melts) allows the calculation of the critical radius for the onwards growth of the nucleus, obtaining,

Reaction	log Keq. (at 1873 K)
Al2O3 (s) = 2 Al + 3 O	−55.76 (= 11.80–62790/T)
ZrO2 (s) = Zr + 2 O	−36.24 (= 21.76–57000/T)
MgO (s) = Mg + O	−32.85 (= 12.45–38050/T)
SiO2 (s) = Si + 2 O	−19.39 (= 11.58–30400/T)
MnO (s) = Mn + O	−5.55 (= 6.70–15050/T)
CaO (s) = Ca + O	−42.76 (= 7.77–33700/T)

Table 1.

Equilibrium constants [26, 27].

$${\mathit{r}}_{\mathit{c}}=\frac{-\mathrm{2}{\mathit{\sigma }}_{\mathit{PL}}}{∆{\mathit{G}}_{\mathit{V}}}=\frac{\mathrm{2}{\mathit{\sigma }}_{\mathit{PL}}{\mathit{V}}_{\mathit{O}}}{\mathit{\text{RTln}}{\mathit{S}}_{\mathrm{0}}}$$E7

The free energy required for the nucleation is obtained by substituting this radius in Eq. (6),

$$∆\mathit{G}=\frac{\mathrm{16}\mathit{\pi }{\mathit{\sigma }}_{\mathit{PL}}^{\mathrm{3}}}{\mathrm{3}∆{\mathit{G}}_{\mathit{V}}^{\mathrm{2}}}=\frac{\mathrm{16}\mathit{\pi }{\mathit{\sigma }}_{\mathit{PL}}^{\mathrm{3}}{\mathit{V}}_{\mathit{O}}^{\mathrm{2}}}{\mathrm{3}{\left(\mathit{\text{RTln}}{\mathit{S}}_{\mathrm{0}}\right)}^{\mathrm{2}}}$$E8

The nucleation rate is [26, 28],

$$\mathit{I}=\mathit{exp}\left[\frac{\mathrm{16}\mathit{\pi }{\mathit{\sigma }}_{\mathit{PL}}^{\mathrm{3}}{\mathit{V}}_{\mathit{O}}^{\mathrm{2}}}{\mathrm{3}{\mathit{k}}_{\mathit{B}}{\mathit{R}}^{\mathrm{2}}{\mathit{T}}^{\mathrm{3}}{\left(\mathit{ln}{\mathit{S}}_{\mathit{O}}\right)}^{\mathrm{2}}}\right]\mathit{exp}\left(\frac{-∆\mathit{G}}{{\mathit{k}}_{\mathit{B}}\mathit{T}}\right)=\mathit{\text{Aexp}}\left[\frac{-\mathrm{16}\mathit{\pi }{\mathit{\sigma }}_{\mathit{PL}}^{\mathrm{3}}{\mathit{V}}_{\mathit{O}}^{\mathrm{2}}}{\mathrm{3}{\mathit{k}}_{\mathit{B}}{\mathit{R}}^{\mathrm{2}}{\mathit{T}}^{\mathrm{3}}{\left(\mathit{ln}{\mathit{S}}_{\mathrm{0}}\right)}^{\mathrm{2}}}\right]$$E9

$$\mathit{A}=\mathit{exp}\left[\frac{\mathrm{16}\mathit{\pi }{\mathit{\sigma }}_{\mathit{PL}}^{\mathrm{3}}{\mathit{V}}_{\mathit{O}}^{\mathrm{2}}}{\mathrm{3}{\mathit{k}}_{\mathit{B}}{\mathit{R}}^{\mathrm{2}}{\mathit{T}}^{\mathrm{3}}{\left(\mathit{ln}{\mathit{S}}_{\mathit{O}}\right)}^{\mathrm{2}}}\right]$$E10

where k_B = 1.38X10⁻²³ J K⁻¹, is the Boltzmann constant, A = 10³² m⁻³ s⁻¹ is the frequency factor, V_O is the oxide’s molar volume, ${\mathit{\sigma }}_{\mathit{PL}}$ is the interfacial tension between liquid iron and an oxide particle, R is the gas constant, and T is temperature. The critical supersaturation ${\mathit{S}}_{\mathit{O}}^{*}$ is the minimum one to nucleate one nucleus m⁻³ s¹. Hence, making I = 1 m⁻³ s⁻¹ in Eq. (9) implies ${\mathit{S}}_{\mathit{O}}^{*}={\mathit{S}}_{\mathrm{0}}$, and gives,

$${\mathit{S}}_{\mathrm{0}}=\mathit{exp}\left[\frac{{\mathit{V}}_{\mathrm{0}}}{\mathit{RT}}\sqrt{\frac{\mathrm{16}\mathit{\pi }{\mathit{\sigma }}_{\mathit{PL}}^{\mathrm{3}}}{\mathrm{3}{\mathit{k}}_{\mathit{B}}\mathit{\text{TlnA}}}}\right]$$E11

The supersaturation of iron melts in contact with different oxides was measured through electrochemical methods [29]. The critical supersaturation, ${\mathit{S}}_{\mathit{O}}^{*}$, depends on the oxide’s nature and surface tension, as seen in the precedent equation. The experimental results provide the inclusions sizes’ statistical dispersion directly in the sense that the larger surface tensions, contributing to the opposing free surface energy to the nucleation, lead to larger size dispersions [26]. Table 2 shows the experimental results of supersaturation experiments reported in Ref. [26] using different metal deoxidizers. The second and third columns are the molar volume and the interfacial tension between the oxide and the melt, the fourth and fifth, the experimental and calculated critical supersaturations through Eq. (11). The sixth and seventh columns include the experimental and corrected supersaturations, respectively. Revision of these data shows that the magnitudes of the calculated critical supersaturations are larger than the experimental critical supersaturations. The other observations are the small magnitudes of the experimental supersaturation. The large differences behind the critical supersaturations are due to the experimental data’s nature as they correspond to the metal bulk property. On the other hand, the other magnitude (the corrected supersaturations) is theoretically related to the nuclei’s curvature, as is schematized in Figure 1. The curvature raises the gradients of concentration of all solutes, M (Al, Si, Mn, Mg, Ti, Zr, Ca) and O in the nuclei’s tip and, consequently, the critical supersaturations are larger in the proximities of the tip than in the metal bulk. The curvature, given by ${\mathit{r}}_{\mathit{C}}^{\mathit{-}\mathrm{1}}$ is calculated through Eq. (7) and matching its magnitude by considering an embryo formed by two or three pairs of M-O dimers, (consulting for that purpose the atomic radius of the involved elements of the corresponding oxides in the Periodic Table).

OXIDE	Vo [m^3. Mol^−1]	${\sigma }_{\mathit{PL}}$[J. m^2]	S^*O (exp.)	S^*O (cal.)	SO	S^*O (exp.) corr	SO corr.
MgO	1.10E-05	1.8	8.4	280	3.6 ∼ 7.2e4	280.224	1801440
ZrO2	1.01E-05	1.63	4.5	85	48 ∼ 68	84.825	1093.3
Al2O3	8.60E-06	2.11	14.7	529	37 ∼ 60	530.229	1803.5
CaO	1.65E-05	1.17	3.4	83
CaO-Al2O3	1.47E-05	1.3	3.6	92	8.5E+04	90.936	2147100
SiO2	1.13E-05	1.24	1	27
MnO-SiO2	1.36E-05	1	1.7	18	1.5	18.598	16.41

Table 2.

Experimental, calculated and corrected critical supersaturation degree for precipitation of oxides [26].

Figure 1.

Supersaturations of alumina close and far away from surfaces with finite and infinite curvature radius.

The correction of the experimental supersaturations is possible through the Gibbs–Thomson’s Equation which gives the ratio between both types of supersaturations as,

$${\left[{\mathit{S}}_{\mathrm{0},\mathit{\text{corr}}}\right]}_{\mathit{r}=\mathit{r}}={\left[{\mathit{S}}_{\mathit{O},\mathit{exp}}\right]}_{\mathit{r}=\infty }\mathit{exp}\left[\frac{{\mathit{V}}_{\mathrm{0}}}{\mathit{RT}}\frac{\mathrm{2}\mathit{\sigma }}{{\mathit{r}}_{\mathit{c}}}\right]$$E12

The nucleation process consumes short times of the order of microseconds [26, 29, 30], as shown schematically by Figure 2. The fluctuations of concentrations in the liquid structure, after the de-oxidant addition, require a critical supersaturation reached in the point (a) of this figure. Once reached the critical supersaturation, the embryo starts with a group of M-O dimers that develop into metastable structures becoming into a nuclei, and as the thermodynamic and kinetic conditions permit it, evolves in alumina with time, as seen in Figure 3. After stabilizing the nuclei, the local supersaturation decreases by diffusion process [31], due to the local consumption of M and O, to the point (b) in Figure 2 where the nucleation kinetics overlaps with the diffusion process. The supersaturation continues decreasing and the diffusion overlaps with the particle growth through the Ostwald ripening process summarized by the following expression [32],

Figure 2.

Evolution of supersaturations with time during nucleation and initial growths by diffusion and Ostwald ripening mechanisms.

Figure 3.

Nucleation of an alumina lattice from the union of Al-O dimers.

$${\acute{\mathit{r}}}^{\mathrm{3}}-{\acute{\mathit{r}}}_{\mathrm{0}}^{\mathrm{3}}={\mathit{\alpha k}}_{\mathit{d}}\mathit{t}$$E13

$${\mathit{k}}_{\mathit{d}}=\frac{\mathrm{2}\mathit{\sigma }{\mathit{D}}_{\mathit{O}}{\mathit{V}}_{\mathit{O}}{\mathit{C}}_{\mathit{O}}}{\mathit{RT}\left({\mathit{C}}_{\mathit{P}}-{\mathit{C}}_{\mathit{O}}\right)}$$E14

where C_O and C_P are the concentrations of dissolved oxygen and oxygen content in the oxide, respectively, and D_O is the diffusion coefficient of oxygen. Further growth phenomena of the particles, in industrial vessels, includes Stokes collisions, collisions among particles driven by turbulent flows deriving in aggregates of particles and clusters. Grown particles are easily floated out thanks to the bottom stirring with argon of steel ladles which carries these particles through their contact with gas bubbles, and melt convection making them contact the slag facilitating their absorption in this phase.

Figure 4a and b show the effect of the supersaturation on the nucleation kinetics on the nucleation rates of different oxides recalculated from reference [26]. There are two important differences, the first is that the nucleation rates are considerably lower than those reported in Ref. [26] and the second is the larger supersaturations required to precipitate the MgO shown in Figure 4b. Another important feature is the small supersaturations required for the precipitation of silica and manganese silicate. Therefore, those particles requiring small or relatively small supersaturations yield the largest nucleation rates meaning, physically, the fast generation of million of nuclei distributed inside the reaction and diffusion boundaries. The main obstacle of a particle to born is the creation of a new surface preceding the formation of a small volume in a hosting matrix, as is seen in Figure 3, with a different structure. Those particles requiring larger supersaturations would, eventually, nucleate smaller populations of particles with a broader size distribution once their first step of development ends. The precipitates characterized by high and low supersaturations are schematized in Figure 5a and b, respectively, showing the period for each one of them. The first case will yield numerous small particles with limited growth as the supersaturation decays providing a narrow size distribution. The longer times lead to larger diffusion and growth time scales yielding broader size distributions in the second case. The time scales change to exponentially larger ones when the precipitates of solid-state transformations take place. For example, steel aging by nitrogen diffusion consumes long times as the diffusion coefficients of this interstitial element are very small in ferrite or austenite phases at 300°C [32, 33] compared with its diffusion coefficient at 1600°C [34], as is schematized in Figure 5c and d. Hence, the interfacial tension between the melt and the nuclei governs the nucleation rate, as seen in Figure 6. Alumina has one of the lowest nucleation rates as its interfacial tension is large, and its crystals will have a broad spectrum of sizes after the nucleation and the diffusion end.

Figure 4.

Effect of supersaturation ratios on nucleation rate at 1600°C, (a) various metal deoxidizers, (b) with magnesium.

Figure 5.

Schematics of nucleation rate evolution with time. (a) High nucleation rates at low supersaturation in liquid state, (b) high nucleation rates in solid state, (c) low nucleation rates at high supersaturation in liquid state, (d) low nucleation rates in solid state.

Figure 6.

Nucleation rate as a function of surface tension between the particle and the melt at 1600°C.

2.2 Alumina morphology

Alumina may acquire a wide diversity of morphologies depending on the concentrations of oxygen and the deoxidant. Accordingly, the initial supersaturation ratio influences the morphology of alumina. Figure 7 shows a scheme of the relation between oxygen concentrations and the deoxidant with the particle morphology [35]. In oxidized melts, the inclusions are rounded spheroids, as the oxygen activity decreases the surface roughness develops reaching the stage of dendritic precipitation. With further deoxidation the morphology changes to faceted, disks and crystalline rhomboids.

Figure 7.

Evolution of the growth shapes of oxide inclusions as a function of the local oxygen activity (solid line) and deoxidizer activity (dashed line) according to Steinmetz [35].

The heterogeneous nucleation of alumina particle also yields characteristic morphologies. Other foreign particles catalyze alumina’s nucleation in the melt, such as those of metastable iron oxide precipitated in the steelmaking furnace and manganese silicates in the refining ladle [36]. A catalyzed nucleation process means that the foreign particle decreases the supersaturation required by the precipitation process of the particle through a decrease of the free energy according to [32],

$$∆{\mathit{G}}_{\mathit{het}}=\left\{\frac{-\mathrm{4}}{\mathrm{3}}\mathit{\pi }{\mathit{r}}^{\mathrm{3}}∆{\mathit{G}}_{\mathit{v}}+\mathrm{4}\mathit{\pi }{\mathit{r}}^{\mathrm{2}}{\mathit{\sigma }}_{\mathit{PL}}\right\}\mathit{S}\left(\mathit{\theta }\right)$$E15

where

$$\mathit{S}\left(\mathit{\theta }\right)=\frac{\left(\mathrm{2}+\mathit{\text{cosθ}}\right){\left(\mathrm{1}-\mathit{\text{cosθ}}\right)}^{\mathrm{2}}}{\mathrm{4}}$$E16

Note that except for factor S(θ) this expression is the same as that obtained for homogeneous nucleation, Eq. (6). S(θ) has a numerical value ≤1 dependent only on θ, i.e. contact angle and the nucleus’s shape. A catalyzed alumina particle will have a morphology depending on the oxide’s nature over which it nucleates as shown in Figure 8 [36]. If the foreign o catalyzer particle is a silicate, already existent given its high nucleation rate, a chemical reaction intervenes,

Figure 8.

Heterogeneous nucleation of alumina on iron oxide and silicate particles [36].

$$\mathrm{2}\left[\mathit{Al}\right]+\left(\mathit{MnO}\bullet {\mathit{SiO}}_{\mathrm{2}}\right)=\left({\mathit{Al}}_{\mathrm{2}}{\mathit{O}}_{\mathrm{3}}\right)+\left[\mathit{Mn}\right]+\left[\mathit{Si}\right]$$E17

if the particle is iron oxide, then

$$\mathrm{2}\left[\mathit{Al}\right]+\mathrm{3}\left(\mathit{FeO}\right)=\left({\mathit{Al}}_{\mathrm{2}}{\mathit{O}}_{\mathrm{3}}\right)+\mathrm{3}\left[\mathit{Fe}\right]$$E18

The interfacial tension between the alumina and silicate particles is reduced due to the chemical reaction, a layer of alumina, (it is initially nucleated and the alumina layer grows by the reaction), surrounds the particle of silicate making sluggish the diffusion of aluminum through the alumina layer to continue the reaction. During the process, some particles of Mn and Si precipitate as products of the reaction (17). No dendrites were observed and there is the presence of needle like clusters with disk type terminations indicating a growth under small supersaturations, once the oxygen content decreases. Iron oxide suffers a rapid reduction by aluminum, reaction (18) and the product is an alumina particle without other phases. Other alumina particles nucleate heterogeneously on the original alumina particles and yield dendritic morphologies.

2.3 Further growth of inclusions

After concluding the nucleation and growth by diffusion and Ostwald ripening, the inclusions continue their growth through direct collision. In a Stokes flow regime, the probability for collision among inclusions of sizes R₁ and R₂ is [37],

$${\mathit{w}}_{\mathit{s}}=\frac{\mathrm{2}}{\mathrm{9}}\mathit{g}\frac{∆\mathit{\rho }}{\mathit{\mu }}\left|{\mathit{R}}_{\mathrm{1}}-{\mathit{R}}_{\mathrm{2}}\right|{\left({\mathit{R}}_{\mathrm{1}}+{\mathit{R}}_{\mathrm{2}}\right)}^{\mathrm{3}}$$E19

and the probability of collision under the action of turbulent eddies is expressed by the Saffman’s Equation as [38],

$$\mathit{w}=\mathrm{1.3}\sqrt{\mathit{\pi }}{\left({\mathit{R}}_{\mathrm{1}}+{\mathit{R}}_{\mathrm{2}}\right)}^{\mathrm{3}}\sqrt{\mathit{\epsilon }/\mathit{\nu }}=\mathrm{7.2}\left|{\mathit{R}}_{\mathrm{1}}-{\mathit{R}}_{\mathrm{2}}\right|{\left({\mathit{R}}_{\mathrm{1}}+{\mathit{R}}_{\mathrm{2}}\right)}^{\mathrm{3}}\left\{\mathit{\text{Noexp}}\left(-\mathit{\alpha }{\mathit{R}}_{\mathrm{1}}\right)\mathit{\text{Noexp}}\left(-\mathit{\alpha }{\mathit{R}}_{\mathrm{2}}\right)\right\}/\mathrm{2}$$E20

In Stokes regime, the highest collision probability is observed when the inclusions have large size differences as seen in Figures 9 and 10 indicates that collisions of silicate inclusions smaller than ten μm, have low probabilities for collisions under turbulent flow conditions and those with large dimensions have higher collision probabilities. Figure 11 shows the corresponding collision probabilities for alumina inclusions and particles as small as three μm yield the highest probabilities for a collision. Therefore, alumina inclusions grow from microscopic inclusions to large aggregates and clusters by collisions among small particles forming aggregates [40]. Roughly speaking, in the turbulence regions, in a bottom stirred ladle, the ratio between Stokes and Safmann’s regimes is approximately 6X10⁴. Thereby, deoxidation during steel tapping in bottom stirred ladles is the most indicated step to deoxidize and grow inclusions by turbulent collisions. This is particularly applicable to the growth of alumina particles. Indeed, vigorously stirred melts at tapping times, lead to cleaner steel heats [41]. In the secondary refining of steel, where the turbulence levels are considerably smaller, this ratio decreases and is the highest in the argon-plume regions, while in the top layer of the bath, the Stokes regime dominates.

Figure 9.

Comparison of the collision probability (Stoke’s model) with the radius of the inclusions. Miki Y, Kitaoka H, Sakuraya T, Fujii T. mechanism for separating inclusions from molten steel stirred with a rotating electro-magnetic field. ISIJ Int. 1992;32:142–149. DOI: 10.2355/isijinternational.32.142. Reproduced with permission [39].

Figure 10.

Comparison of the collision probability (Saffman’s model) with the radius of the inclusions. Miki Y, Kitaoka H, Sakuraya T, Fujii T. mechanism for separating inclusions from molten steel stirred with a rotating electro-magnetic field. ISIJ Int. 1992;32:142–149. DOI: 10.2355/isijinternational.32.142. Reproduced with permission [39].

Figure 11.

Prediction of the collision probability (Saffman’s model) for alumina inclusions. Miki Y, Kitaoka H, Sakuraya T, Fujii T. mechanism for separating inclusions from molten steel stirred with a rotating electro-magnetic field. ISIJ Int. 1992;32:142–149. DOI: 10.2355/isijinternational.32.142. Reproduced with permission [39].

2.4 Bond strength among particles

Another important aspect of inclusions growth is the stability of aggregates and clusters, forming large particles that float out of the bath faster as larger are their sizes. Strong bond strengths are desirable as once the particles form aggregates or clusters, their integrity must prevail, avoiding the generation of smaller particles, by breaking processes due to turbulence, which may bring on floatation slowness. The variety of bonds among inclusions with a wide spectrum of morphologies is simplified into three basic cases, sphere-sphere, sphere-plate and plate-plate, see Figure 12. According to thermodynamic calculations of surface tensions, the plate-plate geometry yields the largest bond strength, as suggested by Figure 13 [42]. The bond strength, eventually, will increase by the thickenings of the neck formed by the union of two particles of alumina through diffusion processes according to,

Figure 12.

Geometries of the gas cavity of different contact types [42].

Figure 13.

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

$$\frac{{\mathit{x}}^{\mathrm{5}}}{{\mathit{R}}^{\mathrm{2}}}=\frac{{\mathit{K}}_{\mathrm{1}}{\mathit{\sigma }}_{\mathit{PP}}{\mathit{V}}_{\mathrm{0}}}{\mathit{RT}}{\mathit{D}}_{\mathit{V}}\mathit{t}$$E21

In this equation, D_V = $\mathrm{1.3}\mathit{X}\mathrm{10}\mathit{exp}\left(\frac{\mathit{-}\mathrm{110000}}{\mathit{RT}}\right)$ [43], ${\mathit{\sigma }}_{\mathit{pp}}\mathit{=}\mathrm{1}\frac{\mathit{J}}{{\mathit{m}}^{\mathrm{2}}}$ is the interfacial tension 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].

Figure 14.

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

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],

Figure 15.

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

$${\mathit{\sigma }}_{\mathit{SL}}={\mathit{\sigma }}_{\mathit{SV}}-{\mathit{\sigma }}_{\mathit{LV}}\mathit{cos}{\mathit{\theta }}_{\mathit{Y}}$$E22

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],

$$\mathit{cos}{\mathit{\theta }}_{\mathit{W}}=\mathit{\text{rcos}}{\mathit{\theta }}_{\mathit{Y}}$$E23

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],

$${\mathit{\theta }}_{\mathit{CB}}=\frac{{\sum }_{\mathit{i}=\mathrm{1}}^{\mathit{n}}{\mathit{f}}_{\mathit{n}}\left({\mathit{\sigma }}_{\mathit{i},\mathit{SV}}-{\mathit{\sigma }}_{\mathit{i},\mathit{SL}}\right)}{{\mathit{\sigma }}_{\mathit{LV}}}$$E24

Where the limits i and n, in Eq. (24), corresponding to the stable phases forming part of the refractory material and f_i is the surface fraction of phase i. The following conditions establish the wettability conditions: when ${\mathit{\sigma }}_{\mathit{SV}}\mathit{>}{\mathit{\sigma }}_{\mathit{SL}}\mathit{,}{\mathrm{0}}^{\mathrm{0}}\mathit{<}{\mathit{\theta }}_{\mathit{CB}}\mathit{<}{\mathrm{90}}^{\mathrm{0}}\mathit{,}$ the liquid wets the solid. When ${\mathit{\sigma }}_{\mathit{SL}}\mathit{>}{\mathit{\sigma }}_{\mathit{SV}}\mathit{,}{\mathrm{90}}^{\mathrm{0}}\mathit{<}{\mathit{\theta }}_{\mathit{CB}}\mathit{<}{\mathrm{180}}^{\mathrm{0}}$, 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,

$${\mathit{W}}_{\mathit{ad}}={\mathit{\sigma }}_{\mathit{SV}}+{\mathit{\sigma }}_{\mathit{LV}}-{\mathit{\sigma }}_{\mathit{SL}}={\mathit{\sigma }}_{\mathit{LV}}\left(\mathrm{1}+\mathit{cos}{\mathit{\theta }}_{\mathit{CB}}\right)$$E25

A specific case of changes of surface tension of liquid steel is the case of ultra-low 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 f_S and f_V respectively, where ${\mathit{f}}_{\mathit{S}}\mathit{+}{\mathit{f}}_{\mathit{V}}\mathit{=}\mathrm{1}$. Substituting in the CB Equation for the solid–liquid fraction ${\mathit{f}}_{\mathrm{1}}\mathit{=}{\mathit{f}}_{\mathit{S}}$ and $\mathit{cos}{\mathit{\theta }}_{\mathit{CB}\mathit{.}\mathrm{1}}\mathit{=}\mathit{cos}{\mathit{\theta }}_{\mathit{CB}\mathit{,}\mathrm{2}}$ and the gas pocket fraction f₂ = f_V and $\mathit{cos}{\mathit{\theta }}_{\mathit{CB}\mathit{,}\mathrm{2}}\mathit{=}\mathit{-}\mathrm{1}$ because the fraction is completely dry (no-wetting) and combining the roughness ratio factor r with the CB Equation, we get

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].

Figure 17.

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].

$$\mathit{cos}{\mathit{\theta }}_{\mathit{app}}=\mathit{r}{\mathit{f}}_{\mathit{S}}\mathit{cos}{\mathit{\theta }}_{\mathit{CB}}+{\mathit{f}}_{\mathit{S}}-\mathrm{1}$$E26

The examination of Eq. (26) indicates that if the fraction f_S approaches 0, by increasing the asperities of the surface, there will be a condition of perfect non-wettability. This trend is shown in Figure 17b. On the contrary, if f_S 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 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:

1. The first is to use CaO as a surface cover, which would be wetted it by liquifying the alumina inclusions and decreasing the contact angle between the melt and the calcium aluminate inclusion.

2. Use a conventional AG material with a special treatment leading to smooth surfaces according to the results presented in Figure 17.

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

1. 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 adhesion of the inclusion to the wall.

2. 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.

3. 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.

4. 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 remain adhered to it.

Figure 18.

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

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

Figure 19.

Adhesion of alumina clusters and aggregates contained in the liquid flow on the nozzle surface.

3.1 Adherence force

When a particle approaches the solid surface or to another particle, both unwetted by the liquid, the formation of a cavity between them becomes thermodynamically favorable, see Figure 20. This phenomenon is because the replacement of a particle-liquid interface by a particle-vapor interface leads to a negative change of the Gibbs free energy according to Young’s Equation. Once the cavity is formed, an attractive force of adhesions develops, F_A, as expressed in the following Equation [50],

Figure 20.

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

$${\mathit{F}}_{\mathit{A}}=\mathrm{2}\mathit{\pi }{\mathit{\sigma }}_{\mathit{LV}}\mathit{l}+\mathit{\pi }{\mathit{l}}^{\mathrm{2}}∆\mathit{P}$$E27

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

$$∆\mathit{P}={\mathit{\sigma }}_{\mathit{LV}}\left(\frac{\mathrm{1}}{\mathit{r}}-\frac{\mathrm{1}}{\mathit{l}}\right)$$E28

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 25X10⁻⁶ N [51], large enough to keep fixed the particle on the refractory surface.

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]:

Figure 21.

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

Drag force

$$F_d=6\mathit{\pi \mu \gamma }\frac{d_p^2}{4}f$$E29

Adhesion force

$$F_a=\frac{3}{4}\mathit{\pi \sigma }{d}_p$$E30

Buoyancy force

$$F_b=\left({\rho }_L-{\rho }_p\right)g\frac{1}{6}\pi d_p^3$$E31

Lift force

$$F_L=9.22\frac{\mathit{\gamma \mu }{d}_p^2}{4}\frac{\gamma d_p^2}{4v}$$E32

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

$$R_v=\frac{F_L}{F_a}$$E33

Ratio for the sliding:

$$R_s=\frac{F_d+F_b}{k_s\left(F_a-F_L\right)}$$E34

Ratio for the tangential lift-off:

$$R_t=\frac{\left(1.4r_p\right)F_d+a{F}_L+r_p{F}_b}{a{F}_a}$$E35

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 25X10⁻⁶ 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.

Parameters for the calculation of ratios [53]
Shear rate, $\gamma$	1 x 10^5 s^−1	Ratio of forces
Poisson’s ratio, $v_1$	0.27	$R_v=5.12x{10}^{-8}$
Poisson’s ratio, $v_2$	0.28
Poisson’s ratio, $E_1$	8.01x10^10
Poisson’s ratio, $E_2$	2.15x10^11
Surface energy, $\sigma$	0.001 J.m^−2	$R_s=$0.064
Coefficient of friction, $k_s$	0.3
$f$	1.7009
Viscosity, ${\mu }_l$	0.006 Pa.s
Liquid density, ${\rho }_l$	7100 kg/m^3	$R_t=7.60$
Particle density, ${\rho }_p$	3600 kg/m^3
Particle diameter, $d_p$	10 x 10^−6 m
Gravity, $g$	9.8 m/ s^2

Table 3.

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

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 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.

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.

Nomenclature

CO	Oxygen concentration in the melt
CP	Oxygen concentration in the oxide
dp	Particle diameter
DO	Diffusion coefficient of oxygen in the melt
DV	Coefficient diffusion of vacancies
f	Correction factor
F	Force
Fs, Fv	Surface fraction of solid and vapor phase, respectively
g	Gravity constant
hi	Henryan activity of i
I	Nucleation rate
kB	Boltzmann’s constant
l	Principal radii
No	Constant
P	Pressure
r	Inclusion radii
r0	Initial radii of the inclusion
rc	Critical nuclei radii
R	Gas constant and particle radii
R1	Particle 1 radii
R2	Particle 2 radii
S0	Supersaturation
t	Time
T	Temperature
VO	Oxide molar volume
w	Collision probability of two particles in turbulent flow regime
Wad	Work of adhesion
Ws	Probability for collisions for two particles in Stokes’s regime
x	Neck radii
Greek letters
α	Accommodation factor
γ	Shear rate
ε	Dissipation rate of kinetic energy
θ	Contact angle
μ	Dynamic liquid viscosity
ν	Kinematic liquid viscosity
ρ	Density
σ	Surface tension
Sub indexes
a	Adhesion
app	Apparent
b	Buoyancy
c	critical
CB	Cassie-Baxter
d	Drag
het	Heterogeneous
L	Lift
LV	Liquid–Vapor
M	Metal
O	Oxide
PL	Particle-Liquid
SL	Solid–Liquid
SV	Solid-Vapor
v	Molar volume
w	Wenzel
y	Young