Calculation of Physical Properties for Use in Models of Continuous Casting Process-Part 1: Mould Slags

Abstract

Physical properties of both steels and mould slags are needed as input data for the mathematical modelling of the continuous casting process. Routines for calculating the properties of mould slags and for estimating steel properties have been developed and are described in Parts 1 and 2, respectively. Many mould powders, with differing compositions, are used in casting practice and their properties vary significantly. Reliable models have been developed to calculate these property values as a function of temperature from their chemical composition since this is available on a routine basis. Models have been developed to calculate the following properties: heat capacities, enthalpies, thermal expansion coefficient, density, viscosity, thermal conductivity and surface tension. Solid mould slags can exist as glassy or crystalline phases or as mixtures of the two (i.e. slag films) and the properties for the various phases can vary considerably; methods have been developed to calculate property values for these various states. The software used to calculate the properties is available via the link (i) http://www.mxif.manchester.ac.uk/resources/software (ii) https://sites.google.com/site/shyamkaragadde/software/thermophysical-properties.

1. Introduction

Recent mathematical models^1,2,3) have shown that that they are capable of providing (i) accurate predictions of both the heat flux and the powder consumption (a measure of the lubrication supplied to the shell) and (ii) valuable insights into the mechanisms underlying problems and defects (e.g. slag entrapment and oscillation mark formation). These models are being further developed to identify the causes of defects. However, models require accurate values for the physical properties of both the steel and mould slag (as input data). Properties of both mould slag and steel vary considerably with composition and a large number of steels and mould powders are in use. Consequently, a model of the continuous casting process requires a large amount of property data for the steel and mould powder used in the casting. The most satisfactory solution is to develop routines to calculate the required properties from their chemical composition, since the latter is available on a routine basis. The relationships developed to estimate the physical properties of mould slags from their chemical composition are described here in Part 1 and those used for steels are described in Part 2.The two models are linked to calculate the metal/slag interfacial tension.

Mould powders play an important role in the continuous casting process. They are added to the top of the mould and as they heat up the following events occur, sequentially (i) carbonates decompose to form CO₂(g) (ii) Carbon particles in the powder start to oxidise (iii) the oxide components sinter and (iv) then melt when the carbon is consumed, and molten globules mix to form a molten slag pool. Slag from the molten pool then infiltrates into the channel between the steel shell and the mould to form a slag film (Fig. 1); this consists of (i) a liquid layer (ca. 0.1 mm thick) and (ii) a solid layer (ca. 1–2 mm thick). The heat flux between shell and mould is determined, principally, by the thickness of this solid layer (and the amount of crystalline phase present) and the lubrication supplied to the shell is determined by the thickness of the liquid layer. Lubrication involves the following properties: viscosity (η), density (ρ) surface tension (γ) and break (or solidification) temperature (T_br). The heat flux involves the following properties of the mould slag: thermal conductivity (k), T_br, the fraction of crystalline phase formed in the slag film (f_crys) and the optical properties of the mould slag.

Fig. 1.

Schematic drawings showings showing the various layers formed in the mould for (a) the meniscus area and (b) half-section of the mould.^1,3)

The objective of this study was to develop routines to provide reliable values of the thermo-physical properties of mould slags from their chemical compositions, for input into the model of the continuous casting process. The other aim was to make these routines available as Excel software; open access is provided at (i) http://www.mxif.manchester.ac.uk/resources/software (ii) https://sites.google.com/site/shyamkaragadde/software/thermophysical-properties.

1.1. Effect of Structure on Slag Properties

The properties of slags are principally determined by the structure of the slag. In silicate slags the basic building block is the Si–O tetrahedron, in which one Si⁴⁺ cation is surrounded by 4 O⁻ ions. Furthermore, in pure silica, each of these O⁻ ions bonds to Si⁴⁺ ions in other tetrahedra to form a 3-dim. structure (these bonds are known as bridging O s (BOs)). When a cation, such as Na⁺, is introduced into SiO₂, the Na⁺ breaks the bridging O bond and forms a non-bridging O (NBO); this results in de-polymerisation of the slag. In highly-depolymerised slags some O ions are not bonded to Si ions, these are referred to as free oxygens (FOs).

In alumino- silicates, Al³⁺ ions are readily introduced into the Si⁴⁺ chain or ring but a Na⁺ or 0.5 Ca²⁺ is needed for charge-balancing (i.e. to form NaAl⁴⁺); cations on charge-balancing duties are not available for network-breaking. Thus the introduction of Al³⁺ ions result in an increase in the polymerisation of the slag, however, the Al–O bond is weaker than the Si–O bond.

Some properties, such as viscosity, electrical resistivity, diffusion coefficient and thermal conductivity are very dependent upon the degree of polymerisation. Several parameters have been used to represent the degree of polymerisation. In this paper the parameter, Q, is used; it can be calculated from Eqs. (1) and (2) where MO=CaO, MnO etc. and M₂O=Na₂O etc. Summation index N_MO etc. represents the number of different oxides. Other properties, such as density, are affected by the structure but to a much smaller extent than that in the above properties.   

$$\mathrm{N}\mathrm{B}\mathrm{O}/\mathrm{T}=\frac{2\left(\sum _1^{N_{MO}}{\text{X}}_{\text{MO}}+\sum _1^{N_{M2O}}{\text{X}}_{\text{M2O}}-\sum _1^{N_{M2O3}}{\text{X}}_{\text{M2O3}}\right)}{\left(\sum _1^{N_{MO2}}{\text{X}}_{\text{MO2}}-2\sum _1^{N_{M2O3}}{\text{X}}_{\text{M2O3}}\right)}$$(1)

  

$$\mathrm{Q}=4-\mathrm{N}\mathrm{B}\mathrm{O}/\mathrm{T}$$(2)

The O⁻ ions bonded to cations tend to be arranged into an octahedral array (i.e. they have 6- fold coordination) but this varies with cation size.⁴⁾ In general, the effect of different cations on the physical properties is much smaller than the effect of polymerisation. Nevertheless, the nature of the cations affects both the structure and the physical properties. The parameter usually used to represent the effect of cations is the field strength (=z/r² where z and r are the charge and radius of the cation). Increasing field strength has been reported to bring about (i) a wider spread of polymerised silicate anions (i.e. a wider spread of Qⁿ species) (ii) a decrease in coordination number which affects the concentrations of BOs and NBOs (iii) increasing disorder in the melt or glass and (iv) a lower probability that they will carry out charge- balancing duties (which are usually carried out in the hierarchy K⁺>Na⁺>Li⁺ ≈Ba²⁺ etc). Cations affect properties (i) directly or (ii) indirectly through their effect on the structure; two examples of direct effects are (a) the electrical conductivities (κ) and diffusion coefficients (D) are affected by the number of available cations and for X_Na2O=X_CaO there are twice as many Na⁺ ions as Ca²⁺ cations (so κ _{Na2O slag}>κ _{CaO slag} and D_{Na2O slag}>D_{CaO slag}) and (b) the mixed alkali effect occurs where some properties for slags are significantly affected by cations with differing size (e.g. K⁺ and Li⁺).

Increasing temperature has a similar effect to that of increasing the number of cations, i.e. it reduces the degree of polymerisation; thus increasing temperature reduces viscosity and increases electrical conductivity and diffusion rates.

1.2. Crystalline, Glassy and Liquid Phases

In the solid state, slags can exist as crystalline or glassy phases or as a mixture of the two phases (e.g. slag film). In crystalline solids, the ions occupy regular positions and the amount of disorder (or configurational entropy) is low. In contrast, in glasses the ions are more disordered and the configurational entropy is much higher than that in crystals.

The formation of glasses is promoted by (i) slags with high Q (degree of polymerisation) values (i.e. high SiO₂ contents) and (ii) fast cooling rates. When a frozen glass is heated, it transforms at the glass transition temperature (T_g) into a supercooled liquid (scl). This transition is accompanied by step-increases in both C_p and thermal expansion coefficient (α) and results in a concomitant increase in configurational entropy.

When a liquid slag is cooled, if crystals are precipitated, there is a sharp increase in viscosity at a certain temperature, often denoted as the break temperature (T_br). In contrast, if a scl is formed the viscosity increases smoothly with decreasing temperature.

2. Model Details

2.1. Composition, Modelling Parameters

The mould slag compositions covered by the model include conventional F- containing powders and F- free powders but do not include, at this time, the mould slags (based on calcium aluminates) now being used to cast high- Al, high- Mn steels. Furthermore, B₂O₃ additions have been reported to both reduce and increase slag viscosities of mould slags by various investigators; for this reason the effect of B₂O₃ on viscosity has been excluded.

As –received, casting powders contain both free carbon and carbonate; these constituents are lost on ignition (LOI) in the process of forming a casting slag. The fraction of powder forming slag (f*) can be calculated from the chemical composition using either Eqs. (3) or (4)   

$$\mathrm{f}*=\left(100-\%\mathrm{L}\mathrm{O}\mathrm{I}\right)/100$$(3)

  

$$\mathrm{f}*=(100-\%{\mathrm{C}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}-\left(44/12\right)\left(\%{\mathrm{C}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-\%{\mathrm{C}}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\right)$$(4)

The composition of the casting slag can then be calculated by (i) multiplying by (1/f*) and normalising the composition. The software (covering these models) has the option to take into account typical pick-up values for Al₂O₃, FeO and MnO during casting. If this option is used the slag composition is re-normalised.

Mole fractions (X) are then calculated from these normalised compositions using Eq. (5) (where i refers to the constituents e.g. CaO, SiO₂ etc.); the %F present is converted into X_CaF2 and the Ca²⁺ associated with the F^- ions is deducted from X_CaO.   

$${\mathrm{X}}_{\mathrm{i}}=\left({\mathrm{X}}_{\mathrm{i}}/{\mathrm{M}}_{\mathrm{i}}\right)/\mathrm{\Sigma }\left({\mathrm{X}}_{\mathrm{i}}/{\mathrm{M}}_{\mathrm{i}}\right)$$(5)

The mean molecular weight (M) is calculated using Eq. (6)   

$$\mathrm{M}=\mathrm{\Sigma }\left({\mathrm{X}}_{\mathrm{i}}{\mathrm{M}}_{\mathrm{i}}\right)$$(6)

The parameter (NBO/T) is used as a measure of the de-polymerisation of the slag. It is calculated from Eq. (1) where Σ X_MO=X_CaO+X_MgO+X_FeO etc. and Σ X_M2O=X_Na2O+X_K2O etc. The parameter, Q is a measure of the polymerisation of the slag and is calculated from (NBO/T) using Eq. (2). It should be noted that both X_CaF2 and X_TiO2 are ignored in Eq. (5) (on the basis of structural evidence) and the calculated Q value is that of the remaining slag (Q^rem).

The fraction of crystalline phase in the slag film (f_crys) was calculated using the relation proposed by Li et al.⁵⁾ where (NBO/T)* indicates that the parameter differs from that calculated with Eq. (1), in that (i) CaF₂ is included in network breakers and (ii) when %MgO and %MnO are >7% and >4%, respectively, they are included in the denominator of Eq. (1).   

$${\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}}=-2.84+1.41\left(\mathrm{N}\mathrm{B}\mathrm{O}/\mathrm{T}\right)*$$(7)

  

$${\%}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}}=100.{\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}}$$(8)

2.2. Temperatures (T_g, T_crit, T_liq and T_br)

These temperatures are calculated from chemical composition by the software and the temperature scale in the program is adjusted automatically.

The glass transition temperature (T_g) is the temperature where, on heating, a frozen glass transforms into a super-cooled liquid (scl). This transition is accompanied by step-increases in C_p and thermal expansion coefficient (TEC) (i.e. 3α<T_g ≈ α>T_g). The viscosity of a glassy slag is taken to have a value of 10^13.4 (d Pas) at T_g (i.e. log₁₀ η (d Pas)=13.4)).

Numerical analysis of a database of T_g values (840–910 K) obtained from C_p and TEC data⁶⁾ and chemical composition data for 23 mould slags was carried out in this study. Inspection of T_g values for mould slags indicates that values of mould slags tend to have similar values (T_g=870±30 K). The best fit results are shown in Eq. (9) and are used in the software. The uncertainty in the calculation is ca. ±20 K.   

$$\begin{array}{l}{\mathrm{T}}_{\mathrm{g}}\left(\mathrm{K}\right)=906-330.5{\mathrm{X}}_{\mathrm{S}\mathrm{i}\mathrm{O}2}+190{\mathrm{X}}_{\mathrm{C}\mathrm{a}\mathrm{O}}+440{\mathrm{X}}_{\mathrm{A}\mathrm{l}2\mathrm{O}3}-\\ 449{\mathrm{X}}_{\mathrm{N}2\mathrm{O}+\mathrm{K}2\mathrm{O}}-11{\mathrm{X}}_{\mathrm{M}\mathrm{g}\mathrm{O}}+154{\mathrm{X}}_{\mathrm{C}\mathrm{a}\mathrm{F}2}-309{\mathrm{X}}_{\mathrm{M}\mathrm{n}\mathrm{O}}-1\hspace{0.17em}\hspace{0.17em}391{\mathrm{X}}_{\mathrm{F}\mathrm{e}\mathrm{O}}\end{array}$$(9)

The critical (or deformation) temperature (T_crit) is the temperature, above which, the thermal conductivity, decreases rapidly with increasing temperature (this can be seen in Fig. 6). Thermal conductivity- viscosity plots reveal that T_crit occurs when log₁₀ η _Tcrit (dPas)=6 i.e. between the softening and flow temperatures.⁷⁾ Values of the thermal conductivity as a function of temperature for a number of mould slags indicate that T_crit occurs at 1040±10 K.^8,9)

Fig. 6.

Thermal conductivity of mould slags as a function of temperature; Glassy phase:=faint line=k_THW⁷⁾ and ●,○ k_LP values for solid and liquid calculated from 10⁷a=4.5 and 4 m²s⁻¹,respectively;^{16,18,28,29,30,31)} partially- crystalline slags= X, + and dashed, bold line and bold line: Δ,□=partially-melted powders³²⁾ values recorded on cooling after melting.

The liquidus temperature (T_liq) is the temperature where the slag becomes fully molten. A numerical analysis of T_liq values (from DSC experiments) and composition data gave Eq. (10); the calculated values are subject to uncertainties of ±30 K.   

$$\begin{array}{l}{\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}}\left(\mathrm{K}\right)=1\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}464-+11.4\%\mathrm{S}\mathrm{i}\mathrm{O}2-11\%\mathrm{C}\mathrm{a}\mathrm{O}+4.2\%\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\\ +5.7\%\mathrm{M}\mathrm{g}\mathrm{O}-10.1\%\mathrm{N}\mathrm{a}2\mathrm{O}-15.8\%{\mathrm{K}}_2\mathrm{O}+10\%\mathrm{L}{\mathrm{i}}_2\mathrm{O}\\ +1.9\%\mathrm{F}+8.3\%\mathrm{F}\mathrm{e}\mathrm{O}+11.6\%\mathrm{M}\mathrm{n}\mathrm{O}\end{array}$$(10)

More recently, numerical analysis was carried out on a database of T_liq and chemical composition data for 23 mould slags⁶⁾ and the Eq. (11) was obtained and is preferred. Values are subject to uncertainties of ±25 K.   

$$\begin{array}{l}{\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}}\left(\mathrm{K}\right)=1\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}473-1.518\%\mathrm{S}\mathrm{i}{\mathrm{O}}_2+2.59\%\mathrm{C}\mathrm{a}\mathrm{O}+1.56\%\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\\ -17.1\%\mathrm{M}\mathrm{g}\mathrm{O}-9.06\%\mathrm{N}{\mathrm{a}}_2\mathrm{O}-6.0\%{\mathrm{K}}_2\mathrm{O}+18\%\mathrm{L}{\mathrm{i}}_2\mathrm{O}\\ +4.8\%\mathrm{F}-9.87\%\mathrm{F}\mathrm{e}\mathrm{O}-2.12\%\mathrm{M}\mathrm{n}\mathrm{O}\end{array}$$(11)

The solidification temperature (T_sol) is the temperature where solids are first precipitated on cooling. It is frequently represented by the break temperature (T_br) which is the temperature below which the viscosity shows a sharp increase in viscosity during cooling. The break temperature decreases with increasing cooling rate and often T_br is cited for a cooling rate of 10 Kmin⁻¹. The thickness of the solid slag layer (Fig. 1) is partially- determined by the break temperature. The following equation was obtained from numerical analysis of T_br and chemical composition data;¹⁰⁾ these values are subject to uncertainties of ±30 K.   

$$\begin{array}{l}{\mathrm{T}}_{\mathrm{b}\mathrm{r}}\left(\mathrm{K}\right)=1\hspace{0.17em}\hspace{0.17em}393-3.3\%\mathrm{S}\mathrm{i}{\mathrm{O}}_2+8.65\%\mathrm{C}\mathrm{a}\mathrm{O}-8.45\%\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\\ -17.1\%\mathrm{M}\mathrm{g}\mathrm{O}-3.2\%\mathrm{N}{\mathrm{a}}_2\mathrm{O}-2.2\%{\mathrm{K}}_2\mathrm{O}-6.6\%\mathrm{L}{\mathrm{i}}_2\mathrm{O}\\ -6.47\%\mathrm{F}-18.4\%\mathrm{F}\mathrm{e}\mathrm{O}-3.2\%\mathrm{M}\mathrm{n}\mathrm{O}\end{array}$$(12)

In general, super-cooling will ensure that T_br<T_liq. However, the calculation of both T_br and T_liq involve an uncertainty of ±25–30 K, so cases where T_liq<T_br do occur occasionally; in these cases a warning is issued on the Collected Worksheet.

2.3. Heat Capacity (C_p) Enthalpy (H_T−H₂₉₈)

The C_p- T relationships are different for different phases. Consequently, the mould slags are first sorted into individual phases i.e. (i) crystalline (b) glass (iii) slag film and (iv) liquid. The C_p and enthalpy of mould slags are little affected by the slag structure and it has been shown^11,6) that they can be calculated using routines based on partial molar quantities   

$$\text{P}=\sum _{i=1}^N\left(X_i{P}_i\right)$$(13)

where P is a property, or coefficient, X is the mole fraction and i=1,2,3,…N denotes the different components (e.g. CaO, SiO₂ etc.).The values of C_p and (H_T−H₂₉₈) are calculated in units of JK⁻¹mol⁻¹ and kJK⁻¹ kg⁻¹, respectively; values were converted from mol⁻¹ to kg⁻¹ by multiplying by (1000/M) where M, the average molecular weight of the slag.

The temperature dependence of C_p for crystalline solids is usually expressed in the form:   

$${\mathrm{C}}_{\mathrm{p}}=\mathrm{a}*+\mathrm{b}*\mathrm{T}-\mathrm{c}*/{\mathrm{T}}^2.$$(14)

where a*, b* and c* are constants (given in Table 1). In the model the parameters “a*” “b*” and “c*” are treated as partial molar quantities:¹⁰⁾ e.g.   

$$a^{*}=\sum _{i=1}^N\left(X_i{a}^{*}{\hspace{0.17em}}_i\right)$$(15)

Table 1. The coefficients used in the calculation of C_p for crystalline phase C_p (JK⁻¹mol⁻¹)=a*+b*T-c*/T²; C_p for liquids (JK⁻¹mol⁻¹): and ΔS^fus: (JK⁻¹mol⁻¹).

	SiO2	CaO	Al2O3	MgO	Na2O	K2O	Li2O	FeO	MnO	CaF2	ZrO2	TiO2	B2O3
a*	56.0	48.8	115	42.7	65.7	65.7	65.7	49	46.4	59.8	69.6	75.2
103b*	15.4	4.52	11.8	7.45	22.6	22.6	22.6	8.37	8.1	30.45	7.53	1.17
10−5c*	14.4	6.53	35.1	6.2	0	0	0	2.8	3.88	−1.97	14.1	18.2
Cp(l)	87	90.8	146.4	90.4	92.1	74.1	96.2	76.6	79.9	96.2	113	111.7
ΔSfus	4.6	24.7	50.2	24.7	33.9	33.9	33.9	9.6	18.8	18	25.1	26.8

The parameters b* and c* are calculated in a similar manner to a*.

The enthalpy (H_T−H₂₉₈) is given by:   

$$\begin{array}{l}\left({\mathrm{H}}_{\mathrm{T}}-{\mathrm{H}}_{298}\right)=\underset{298}{\overset{\mathrm{T}}{\int }}{C}_{p}dT=\mathrm{a}*\left(\mathrm{T}-298\right)+0.5\mathrm{b}*\left({\mathrm{T}}^2-{298}^2\right)\\ +\left(\mathrm{c}*/\mathrm{T}\right)-\left(\mathrm{c}*/298\right)\end{array}$$(16)

The enthalpy of fusion, ΔH^fus, is calculated by assuming the entropy of fusion (ΔS^fus) can also be calculated from partial molar terms.   

$$\mathrm{\Delta }{S}^{fus}=\sum _{i=1}^N\left(X_i\hspace{0.17em}\mathrm{\Delta }{S}_i{\hspace{0.17em}}^{fus}\right)$$(17)

  

$$\mathrm{\Delta }{\mathrm{H}}^{\mathrm{f}\mathrm{u}\mathrm{s}}={\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}}\mathrm{\Delta }{\mathrm{S}}^{\mathrm{f}\mathrm{u}\mathrm{s}}$$(18)

The C_p for the liquid phase is also calculated from partial molar terms¹¹⁾ using Eq. (19) and the enthalpy for the liquid is determined from Eq. (20).   

$$C_p=\sum _{i=1}^N\left(X_i\hspace{0.17em}{C}_{pi}\right)$$(19)

  

$$\left({\mathrm{H}}_{\mathrm{T}}-{\mathrm{H}}_{298}\right)={\left({\mathrm{H}}_{\mathrm{T}\mathrm{l}\mathrm{i}\mathrm{q}}-{\mathrm{H}}_{298}\right)}_{\mathrm{s}\mathrm{o}\mathrm{l}}+\mathrm{\Delta }{\mathrm{H}}^{\mathrm{f}\mathrm{u}\mathrm{s}}+{\mathrm{C}}_{\mathrm{p}\mathrm{\hspace{1em}}}{\mathrm{\hspace{0.17em}}}_{\mathrm{l}\mathrm{i}\mathrm{q}}\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}}\right)$$(20)

The C_p for the glass phase is close to that of the crystalline phase for temperatures below T_g. At T_g the frozen glass transforms to a super-cooled liquid (scl); this is accompanied by a step- increase in C_p. Measured values of C_p for the scl (C_{p scl}) were between 1400 and 1500 JK⁻¹kg⁻¹.^6,12) These values are in close agreement with the value calculated for the liquid.

Consequently it was assumed that C_{p scl}=C_{p liq} for the range between T_g and T_liq. Values of the enthalpy for the scl in the temperature range between T_g and T_liq are given in Eq. (21).   

$${\left({\mathrm{H}}_{\mathrm{T}}-{\mathrm{H}}_{298}\right)}_{scl}=\left({\mathrm{H}}_{\mathrm{T}\mathrm{g}}-{\mathrm{H}}_{298}\right)+{\mathrm{C}}_{\mathrm{p}\mathrm{\hspace{0.17em} }scl}\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{g}}\right)$$(21)

It should be noted that:

(i) Crystallisation occurs around 50–100 K above T_g (this is an exothermic reaction and results in an apparent decrease in C_p in DSC measurements); thus the state of the sample remains largely undefined in this temperature range.

(ii) When (H_Tliq−H₂₉₈) is calculated for both crystalline and scl it was found that (H_Tliq−H₂₉₈)_cryst>(H_Tliq−H₂₉₈)_scl which suggests that ΔH^fus ≠ 0 for the scl; high temperature C_p measurements are needed to resolve this anomaly.

The slag film consists of a mixture of glass and crystalline phases; the C_p and enthalpy for a slag film is calculated using the rule of mixtures (Eqs. (22) and (23), respectively).   

$${\mathrm{C}}_{\mathrm{p}\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{m}}={\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t}}{\mathrm{C}}_{\mathrm{p}}{\hspace{0.17em}}_{\hspace{1em}\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t}}+\left(1-{\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t}}\right){\mathrm{C}}_{\mathrm{p}\mathrm{g}\mathrm{l}}$$(22)

  

$$\begin{array}{l}{\left({\mathrm{H}}_{\mathrm{T}}-{\mathrm{H}}_{298}\right)}_{\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{m}}\\ ={\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t}}{\left({\mathrm{H}}_{\mathrm{T}}-{\mathrm{H}}_{298}\right)}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t}}+\left(1-{\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t}}\right){\left({\mathrm{H}}_{\mathrm{T}}-{\mathrm{H}}_{298}\right)}_{\mathrm{g}\mathrm{l}}.\end{array}$$(23)

There are only a few measurements of the C_p and (H_T−H₂₉₈) available for mould slags^6,12) and these are restricted to temperatures below 1000 K. there are no reported measurements for the liquid. Thus the model is based on a generic data for oxides and fluorides. Although, the predicted C_p and (H_T−H₂₉₈) values are within ±2% of the experimental values for T<1000 K, the uncertainty is probably ±5% for T>1000 K. It should also be noted that when a glass is heated to a temperature of ca. (T_g +80 K) the sample will partially crystallise and the exothermic enthalpy released causes a sudden apparent decrease in C_p values measured by DSC; this is not a true effect and the values calculated by the model will not show this effect.

2.4. Density (ρ) Thermal Expansion Coefficient (α)

Thermal expansion coefficients reported for 10 glassy samples over the temperature (298−T_g) range between 9 and 11×10⁻⁶ K⁻¹ and for one sintered mould flux (298–1100 K).⁶⁾ These measurements indicated that α decreased slightly with increasing Q (10⁶ α=15.2–2.14 Q) but the scatter was such (R²=0.41) that a constant value of α=10×10⁻⁶ K⁻¹ has been preferred in the calculations. There are few data available for partially-crystallised samples of mould slag; one sample⁶⁾ exhibited a value of α(298–1100 K)=10×10⁻⁶ K⁻¹; this value was adopted.

The thermal expansion coefficient data for binary silicates in the liquid indicate that it is dependent upon both Q and (z/r²).¹³⁾ However, there are few reported ρ- T data for liquid mould slags and individual ρ values are subject to uncertainty. Consequently, thermal expansion values for the liquid phase were obtained with a generic model for oxide slags¹¹⁾ containing CaF₂.

The densities of silicate and alumino-silicate slags exhibit only a small dependence on the slag structure. Reasonable estimates can be obtained using partial molar volumes (V) for the various slag constituents but special procedures are needed for SiO₂ and Al₂O₃.¹¹⁾

There are few reported data for the density of mould slags. Values have been reported for the glassy phase^14,15,16) the liquid state^17,18) and for some slag films.¹²⁾

The density of the liquid mould slag is calculated from partial molar volumes (V=M/ρ where M=molecular weight) as shown in Eqs. (24), (25), (26) (using the data given in Table 2) but special treatment is given to SiO₂ (V_SiO2=19.55 +7.966X _SiO2) and Al₂O₃ (V_Al2O3= 28.31+32X_Al2O3 +31.45 (X_Al2O3)²) to account for the effect of structure on the molar volume and hence the density (ρ).¹¹⁾ These routines were calculated from studies of density values for a wide range of slag compositions.¹¹⁾   

$${\mathrm{V}}_{1\hspace{0.17em}\hspace{0.17em}773}=\sum _{i=1}^N\left(X_i\hspace{0.17em}{V}_i\right)$$(24)

  

$${\mathrm{V}}_{\mathrm{T}}={\mathrm{V}}_{1\hspace{0.17em}\hspace{0.17em}773}+0.01\left(\mathrm{T}-1\hspace{0.17em}\hspace{0.17em}773\mathrm{K}\right)$$(25)

  

$${\mathrm{V}}_{\mathrm{T}}=\mathrm{M}/{\rho }_{\mathrm{T}}$$(26)

Table 2. Partial molar volumes (V₁₇₇₃) used in the calculation of the density of the liquid; a denotes for V_SiO2=19.55 +7.966X _SiO2) b for Al₂O₃ (V_Al2O3=28.31+32X_Al2O3 +31.45 (X_Al2O3)²) and V₂₉₈ for the solid; c for SiO₂, V_SiO2=23.76+3.5X _SiO2.; V₁₈₇₃ and (dV/dT) in the 4^th and fifth row are from Stebbins²¹⁾; * indicates calculated here.

	SiO2	CaO	Al2O3	MgO	Na2O	K2O	Li2O	FeO	MnO	CaF2	ZrO2	TiO2	B2O3
V1773(l)[11]	a	20.7	b	16.1	33	51.8	16	15.8	15.6	31.3		24	10
V298	c	14.4	40.4	12.5	20.2	33.5	11	16.5	17	28.5	–	–	–
V1673 *[21]	26.75	16.9	37.7	11.95	29.5	48.5	17.7*	13.7	13.3*	31*		24.25	27*
103dV/dT[21]	−0.42	3.97	1.02	2.22	7.4	2	1.5*	4.15	4*	5.1*		0.8	4*

Stebbins et al.¹⁹⁾ have also reported values of V₁₈₇₃ and dV/dT based on regressions of V-T and (dV/dT) data for most of the components shown in Table 2. These data were used to calculate both V_{1873 K} and (dV/dT). Values for the missing components in Table 2 were derived here from ρ-T and V-T data.

The recommended values of V_T were calculated using the V₁₇₇₃ value and Eq. (27) and are preferred to values calculated by Eq. (25). The values calculated using the Stebbins value for V₁₈₇₃ are given on the Density Worksheet. Although V_T values in Table 2 vary, the density values calculated with the Stebbins parameters lie within 1% of those calculated by Eq. (27).   

$${\mathrm{V}}_{\mathrm{T}}=\mathrm{\Sigma }\mathrm{\hspace{0.17em}}{\mathrm{X}}_{\mathrm{i}}{\mathrm{V}}_{1\hspace{0.17em}\hspace{0.17em}773}+\mathrm{\Sigma }\hspace{0.17em}{\mathrm{X}}_{\mathrm{i}}(\mathrm{d}\mathrm{V}/\mathrm{d}\mathrm{T})(\mathrm{T}-1\hspace{0.17em}\hspace{0.17em}773\mathrm{\hspace{0.17em} }\hspace{0.17em}\mathrm{K})$$(27)

A similar approach to that used for the liquid is taken for solid phase with different values for the molar volume (Table 2) and (V_SiO2=23.76+3.5X _SiO2) and (V_Al2O3=40.4). However, when a glassy sample crystallises it is accompanied by shrinkage (since ρ_crys>ρ_gl). This results in porosity in the sample; the porosity levels were not determined for the density measurements on slag films¹²⁾ but would be expected to lower the density values.

There are very few data reported for mould slags. The glassy phase has a lower density than the crystalline phase but there are no experimental data for mould slags to validate this statement. Thus, it has been assumed that ρ_gl=ρ_crys. for temperatures between T_g and T_liq; the density of the super-cooled liquid was calculated by assuming that (i) there was no density change associated with the (scl → liquid) transition (ii) ρ_scl is a linear function over this temperature range (i.e. ρ_T=ρ_Tg+(T−T_g)(ρ_Tliq−ρ_Tg)/(T_liq−T_g).

Values of ρ_T for solids (both the glassy and sintered samples) were calculated using a value of α=10⁻⁵ K⁻¹.Normally, the calculations produce a density decrease at T_liq but the uncertainty in the calculations of ρ_T for both solid and liquid phases can lead (with certain compositions) to an apparent increase in density at T_liq; this is not a true effect and is caused by the combined uncertainties. In such circumstances a warning is given on the Collected Worksheet.

There are few reported density data for mould slags. Nevertheless, the estimated values for the density for temperatures below 1000 K, the uncertainties are probably 2–3% and are within ±5% for T>1000 K. Data are needed for the density change associated with glass→ crystalline phase change.

2.5. Viscosity (η)

The principal factor affecting the viscosities of slags is the degree of polymerisation present in the slag; this can be clearly seen in Fig. 2 where both ln η_{1900 K} and the parameter, B_η exhibit considerable sensitivity to the parameter, Q (which is a measure of the polymerisation). The scatter of individual points in Fig. 2(a) is due to the effect of different cations (with viscosity increasing with increasing cation size or decreasing field strength (z/r²)).

Fig. 2.

(a) Viscosity (ln η_{1900 K}) and (b) the parameter, B _η, as functions of Q; note the upper curve in (b) is for MO-silicates and the lower curve is for M₂O-silicates.⁷⁾

The introduction of Al³⁺ into the Si⁴⁺ structure for slags with Q>2.8 results in a small linear increase in viscosity with increasing Q for alumina-silicates, in contrast with the dramatic increase in viscosity with increasing Q for silicates shown in Fig. 2(a).

Viscosity measurements on mould slags have been reported by several workers.^{20,21,22,23,24,25)} Experimental uncertainties can be up to 25%; McCauley²¹⁾ cites six sets of viscosity measurements on the same slag and the reported viscosities were found to vary by ±25%. Another source of uncertainty is that few workers cite chemical analysis on the post-measurement sample and fluorine and sodium losses can be significant during the experiment.

The new model to calculate the viscosities of mould slags is based on an earlier model proposed for slags in general⁷⁾ but which has required certain modifications in applying it to mould slags. In the original model, ln(η_{1900 K}) and the activation energy term, B_η, are represented as functions of Q as double exponential relations (Fig. 2). The small deviations from the curve due to cation effects can be accommodated in terms of the field strength for the cations (i.e. mean z/r ²).The activation energy term, B_η can also represented as a function of Q by two different double- exponential equations for MO-silicates and M₂O-silicates (Fig. 2(b) and Eqs. (28) and (29)). In mould slags, the charge balancing duties are carried out by K⁺ and Na⁺ ions, and this coupled with the fact that X_MO>>X_M2O implies that the upper curve in Fig. 2(b) and Eq. (28) is more appropriate for casting slags. The viscosities at temperatures other than the reference temperature can be calculated from the Eq. (30).   

$$\begin{array}{l}\mathrm{M}\mathrm{O}-\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}:{\mathrm{B}}_{\eta }=795\hspace{0.17em}\hspace{0.17em}830.8+8.476\\ \times {10}^{-6}\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{Q}/0.2611\right)+795\hspace{0.17em}\hspace{0.17em}841.4\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{Q}/211\hspace{0.17em}\hspace{0.17em}587.4\right)\end{array}$$(28)

  

$$\begin{array}{l}{\mathrm{M}}_2\mathrm{O}-\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}:{\mathrm{B}}_{\eta }=-24.6+6.122\\ \times {10}^{-13}\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{Q}/0.1258\right)+28.465\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{Q}/7.727\right)\end{array}$$(29)

  

$$\mathrm{l}\mathrm{n}{\eta }_{\mathrm{T}}=\mathrm{l}\mathrm{n}{\eta }_{1\hspace{0.17em}\hspace{0.17em}900}+\left({\mathrm{B}}_{\eta }/\mathrm{T}\right)-\left({\mathrm{B}}_{\eta }/1\hspace{0.17em}\hspace{0.17em}900\right)$$(30)

The above model must be modified to apply it to mould slags since:

(i) the reference temperature, 1573 K, is far removed from 1900 K; this would have little effect on B_η but would have a dramatic effect on the ln η term since viscosity is sensitive to temperature changes.

(ii) both B_η and ln η_{1573 K} are affected by both Q and the CaF₂ content, simultaneously.

Consequently, it is necessary to separate the effects of CaF₂ and Q. In the case of B_η (little affected by temperature) a value for a CaF₂-free slag can be obtained from Eq. (28); this is denoted as a solid curve in Fig. 3(a). It can be seen that B_η (i) increases with increasing Q and (ii) departs from the curve with deviations increasing with increasing CaF₂.Since B_η is dependent upon two factors (Q and X_CaF2), the various slags were then sorted into individual groups according to their X_CaF2.values (e.g. (0–0.05); (0.05–0.1); etc.) and allocated a mean X_CaF2 value (e.g. 0.025; 0.075, respectively). The departure of experimental values (B_{η expt}) from the curve (i.e. from Eq. (28), calculated using the remaining slag composition, B_{η rem} Eq. (28)) is denoted ΔB_η, (defined in Eq. (31)) and was then plotted as a function of X_CaF2 for specific groups of similar Q (e.g.{1.5±0.25}; {2±0.25} etc.); the results are shown in Fig. 3(b).   

$$\mathrm{\Delta }{\mathrm{B}}_{\eta }={\mathrm{B}}_{\eta \mathrm{r}\mathrm{e}\mathrm{m}}{\hspace{0.17em}}_{\mathrm{\hspace{0.17em} }\mathrm{E}\mathrm{q}28}-{\mathrm{B}}_{\eta \mathrm{e}\mathrm{x}\mathrm{p}\mathrm{t}}$$(31)

  

$${\mathrm{B}}_{\eta }=\left({\mathrm{B}}_{\eta }{\hspace{0.17em}}_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{E}\mathrm{q}28}-\mathrm{\Delta }{\mathrm{B}}_{\eta }\right)={\mathrm{B}}_{\eta }{\hspace{0.17em}}_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{E}\mathrm{q}28}-32.7{\mathrm{X}}_{\mathrm{C}\mathrm{a}\mathrm{F}2}$$(32)

Fig. 3.

(a) The parameter, B_η, as a function of Q for slags containing varying X_CaF2: for following X_CaF2 ranges: line=0; o=0–0. 05; ◊=0.05–0.1; Δ=0.1–0.15; ♦=0.15–0.2; ■=0.2–0.25; ●=>0.25 and (b) ΔB_η as a function of X_CaF2.

The same approach can not be adopted to treat the ln η_{1573 K} data because of the large differences in ln η values for 1900 K and 1573 K. Consequently, it is difficult to calculate a value of ln η_{1573, CaF2}=0 for CaF₂-free slags, since both Q and X_CaF2 affect the viscosity simultaneously. The experimental data were first sorted into the same groups of similar X_CaF2 values and these values were then plotted as function of Q (Fig. 4(a)) and equations were derived for ln η_{1573 K} as a function of Q. A linear best fit of the data (ln η_{1573 K}=3.073+1.93Q) was then derived since the relationship is close to linear in the range Q=1.5 to 3.Then the database was divided into groups of similar Q values and plotted against the mole fraction of CaF₂ (Fig. 4(b)). The mean gradient of these plots (d ln η₁₅₇₃/dX_CaF2)=−9 was determined from these plots. Values of ln η_{1573, CaF2=0} were then calculated using Eq. (33); the values of ln η_{1573, CaF2=0} were then plotted against Q (Fig. 5) and the best fit curve for ln η_{1573, CaF2=0} is given in Eq. (34). It can be seen that the mean scatter in ln η₁₅₇₃ values is about 0.3.which corresponds to an uncertainty of ca. ±25% in η. Attempts will be made in the future to reduce this uncertainty from a best fit regression of ln η₁₅₇₃ data as functions of Q and X_CaF2. The calculated viscosity values are compared with values calculated by the Riboud²⁶⁾ and Iida models²⁷⁾ in the Viscosity worksheet and mean values for the three models were also calculated.   

$$\mathrm{l}\mathrm{n}{\eta }_{1\hspace{0.17em}\hspace{0.17em}573,}{\hspace{0.17em}}_{\mathrm{C}\mathrm{a}\mathrm{F}2=0}=\mathrm{l}{\mathrm{n}}_{\eta 1\hspace{0.17em}\hspace{0.17em}573}-9{\mathrm{X}}_{\mathrm{C}\mathrm{a}\mathrm{F}2}$$(33)

  

$$\mathrm{l}\mathrm{n}{\eta }_{1\hspace{0.17em}\hspace{0.17em}573\mathrm{K},}{\hspace{0.17em}}_{\mathrm{r}\mathrm{e}\mathrm{f}}=0.1227\mathrm{e}\mathrm{x}\mathrm{p}\left(1.1454\mathrm{Q}\right)$$(34)

Fig. 4.

The parameter, ln η_{1573 K} as functions of (a) Q; X_CaF2=(0.025±0.025)=O,+; black line (0.075±0.025)=♦ and red line,; (0.125±0.025)=▲ and green line;(0.175±0.025)=♦; (0.225 ±0.025)=■; >0.25=● and (b) mole fraction CaF₂; O=(Q=1.5); ■=(Q=2); ▲=(Q=2.5); ♦=(Q=3).

Fig. 5.

Values of ln η_{1573, CaF2=0} for various mould slags as a function of the parameter Q; ◊=Lanyi;²⁰⁾ ■, x=McCauley²¹⁾ ●=Persson²²⁾ ♦ Elahipanah.²³⁾

Although viscosity measurements for supercooled liquids are available for minerals and glasses, measurements on slags are rare for metallurgical slags and the authors were unable to obtain any experimental data for the scl state of mould slags. However, it is possible to calculate values for these slags in the scl range since log₁₀ η (d Pas) has values of 13.4 and 6, respectively, at T_g and T_crit (=1040 ±10 K). These data can be coupled with calculated values for the liquid range at T_liq, 1573 and 1673 K. Attempts were made to obtain a best fit for these data using a polynomial relation (log₁₀ η(d Pas)=a+bT+cT²+dT³ where a, b, c and d are constants). Unfortunately, the resulting value for log₁₀ η at T_g departed significantly from log₁₀ η (dPas)=13.4 in some cases. Consequently, values of log₁₀ η (d Pas) were obtained from linear relationships between T_g and T_crit and between T_crit and T_liq; these values should not be regarded as accurate values for the scl but do provide the reader with approximate values which are unavailable in the literature.

The performance of the new model was tested by calculating the viscosity at 1573 K and comparing it with the experimental values for ca. 30 mould powders.^{20,21,22,23,24,25)} The slags were randomly selected to give a range of η₁₅₇₃ values (0.5–12 dPas) but slags containing >1% B₂O₃, >1% ZrO₂ and >3%FeO were excluded. The deviation (Δ=100 (η_calc-η_exp)/η_exp) was calculated and the mean deviation determined (Δ_mean=[ Δ₁²+ Δ₂²+ Δ₃²+...]^0.5). Then the Δ_mean values were compared with those for the Riboud²⁶⁾ and Iida²⁷⁾ models. It was found that performances of the new model, the Riboud and Iida models were similar (Δ_mean=35%; 35% and 43%, respectively). It was found that the mean η₁₅₇₃ value for the three studies was closer to the experimental values (Δ_mean=27%). It should be noted the experimental uncertainties associated with both the viscosity measurements and changes in chemical composition during the measurement sequence are ca. ±25% for most of the reported data.

2.6. Thermal Conductivity (k) and Thermal Diffusivity (a)

The thermal conductivity of the slag film is important in controlling the heat transfer from the shell. Thermal conductivities of liquid slags are difficult to measure since it is difficult to eliminate contributions from convection. This is usually achieved by using transient methods where the experiment is completed before convection is initiated. The two principal, transient techniques are the laser pulse (LP) and the Transient Hot wire (THW) methods.

However, glassy and liquid slags are also semi-transparent to infra-red radiation and heat transfer can occur simultaneously by both lattice conductivity (k_lat) and radiation conductivity (k_R).Most investigators try to determine the values of k_lat and k_R, individually and then combine them to give a total value. However, recent measurements of k_lat on mould slags using the LP and THW methods reveal that:

(i) Values of k_LP^{16,18,28,29,30,31)} and k_THW^{8,9,35,36,37)} are in good agreement up to 1000 K (Fig. 6) but deviate markedly at higher temperatures.

(ii) The k_THW values for mould slags drop markedly for temperatures >1040 K, in contrast to k_LP values which continue to rise (Fig. 6); this contradictory behaviour has been attributed variously, to electrical leakage in the THW measurements or to k_R contributions to k_LP in the LP measurements.

The source of the discrepancy between k_LP and k_THW is unresolved at the present time. Here it has been assumed that k_LP measurements contain contributions from k_R and the k_THW measurements have been adopted.

The thermal conductivities of slags are also affected by (i) the degree of crystallinity developed in the slag (i.e. f_crys) since k_crys ≈ 2 k_glass due to the higher packing density in the crystalline phase than in the glass and (ii) porosity which tends to reduce the thermal conductivity. Note crystallisation is accompanied by shrinkage and results in the formation of fine pores in slag films.

Plots of k_THW for glassy, M₂O–CaO–SiO₂ slags versus the calculated viscosities indicated that the sharp drop in k_THW occurred at the temperature where η=10⁶ d Pas;^7,13) thus T_crit occurs midway between the softening temperature (where the sample can no longer support its own weight) and the flow temperature.^7,13) The collapse in k_THW occurs for the scl phase (i.e. in glassy samples). This strongly suggests that the magnitude of the thermal conductivity is linked to the rigidity of the silicate lattice.^7,13) Measured k_THW values ranged from 1.05–1.09 Wm⁻¹K⁻¹ at 298 K for glass samples;^8,9) so a value of k₂₉₈=1.07±0.02 Wm⁻¹K⁻¹ was adopted for all glassy mould fluxes.

The following values were also reported for the glassy phase, T_crit=1040 ±10 K and k_Tcrit=1.65±0.05 Wm⁻¹K⁻¹.⁷⁾ Measurements of k_THW for 15 mould slags indicated that k_{1040 K}= 1.65±0.05 Wm⁻¹K⁻¹;^8,9) thus k_T values between 295 K and 1040 K can be calculated using Eq. (35). Note that if k₂₉₅>1.65 Wm⁻¹K⁻¹ the temperature coefficient (dk/dT) will be negative and if k₂₉₅<1.65 Wm⁻¹K⁻¹, (dk/dT) will be positive   

$$\begin{array}{l}\left(295-1\hspace{0.17em}\hspace{0.17em}040\mathrm{K}\right):\\ {\mathrm{k}}_{\mathrm{T}}\left(\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)=1.07+0.00078\left(\mathrm{T}-295\mathrm{K}\right)\end{array}$$(35)

A large number of thermal diffusivity (a_LP) values for casting slags in the glassy, crystalline and liquid states have been measured^{16,28,29,30,31)} with the LP method; values of k_LP were derived using Eq. (36).   

$${\mathrm{k}}_{\mathrm{L}\mathrm{P}}={\mathrm{a}}_{\mathrm{L}\mathrm{P}}./\rho \hspace{0.17em}\hspace{0.17em}{\mathrm{C}}_{\mathrm{p}}$$(36)

The model determines the thermal conductivity at three temperatures, namely, 298 K, T_crit (=1040 K) and T_liq; the k-T curves between (298 K and T_crit) and also (T_crit and T_liq) are assumed to be linear. There are insufficient data available for the temperature coefficient (dk/dT) of the liquid slag at the present time to permit calculation of values for liquid mould slags.

The super-cooled liquid phase is formed at temperatures above T_g (> ca. 870 K) but the sharp drop in k_THW does not occur until the temperature exceeds the critical (or deformation) temperature, T_crit. Values of k_T between T_crit and T_liq are calculated using Eq. (37).   

$$\begin{array}{l}\left(1\hspace{0.17em}\hspace{0.17em}040\mathrm{K}-{\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}}\right):\hspace{1em}{\mathrm{k}}_{\mathrm{T}}\left(\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)\\ =1.65-\left\{\left(1.65-{\mathrm{k}}_{\mathrm{T}\mathrm{l}\mathrm{i}\mathrm{q}}\right)\left({\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}}-\mathrm{T}\right)\right\}/\left({\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}}-1\hspace{0.17em}\hspace{0.17em}040\mathrm{K}\right)\end{array}$$(37)

In partially-crystalline slags (or slag films) the thermal conductivity increases with increasing crystallinity^8,28,30,33) since (k_cryst≈ 2 k_glass). Values of k₂₉₅ for partially-crystalline slags are calculated from Eq. (38)^8,9) using the f_cryst derived with Eq. (7). It should be noted that Eq. (38) may produce a slightly low k₂₉₅ value for a 100% crystalline slag (i.e. k≈ 2 Wm⁻¹K⁻¹) since crystallisation is accompanied by porosity, which lowers the conductivity of the sample. A fully crystalline sample would not be expected to show the collapse in k_T above T_crit exhibited by glasses (Fig. 6); thermal diffusivity values for a fully crystalline sample appear to remain reasonably constant with increasing temperature above 1040 K. Thus, k_T has been assumed to remain constant for crystalline samples until the sample melts. However, slag films with a reasonable amount of glassy phase would be expected to show some collapse in k_T above T_crit.   

$${\mathrm{k}}_{295}\left(\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)=1.07+0.7{\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t}}$$(38)

When the glass is heated above ca. 1000 K, crystallisation of the sample occurs; note crystallisation occurs roughly in the same temperature range as T_crit; the model takes no account of any subsequent change in f_crys. Values of k_T for partially-crystalline samples in the temperature range (1040 K -T_liq) are calculated using Eq (37).

There are few measurements of the thermal conductivity of liquid mould slags using the k_THW method,^8,32,34) but several workers^{18,28,29,30,31)} have reported values for the thermal diffusivity of liquid mould slags (a_LP=4×10⁻⁷ m²s⁻¹) using the LP method which yields values for k_LP which are about 10 x higher than k_THW values (Fig. 6). The k_THW values have been tentatively, adopted. Eq. (39) was obtained from a relation reported for slags covering a wide compositional range.⁷⁾   

$${\mathrm{k}}^{\mathrm{m}}{\hspace{0.17em}}_{\mathrm{T}\mathrm{H}\mathrm{W}}\left(\mathrm{l}\mathrm{i}\mathrm{q}\right)=0.139+3.65\times {10}^{-5}\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{Q}/0.3421\right)$$(39)

Values of k₂₉₈ have been reported for slag films taken from the mould.^9,12) Approximate values for k₂₉₅ of the slag film can be calculated by using Eq. (40) where f_cryst is calculated from Eq. (7). Note k₂₉₈=1.77 Wm⁻¹K⁻¹ is calculated with Eq. (38) for f_crys=1 and refers to a sample containing pores.   

$${\mathrm{k}}_{298}\left(\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{m}\right)=1.77\mathrm{\hspace{0.17em} }{\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}}+1.07\left(1-{\mathrm{f}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}}\right)\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$$(40)

There are two outstanding problems to be resolved concerning thermal conductivity measurements, namely (i) the differences between k_THW and k_LP for temperatures above 1040 K and (ii) the degree of porosity in slag film and partially-crystalline samples which has not yet been determined and which makes it difficult to determine the k value for a fully-dense slag. The calculated thermal conductivity values of the glass and partially-crystalline phases are probably prone to uncertainties of ca. ±5% and ±10%, respectively, for temperatures between 298 and 1040 K. However for temperatures >1040 K, it is difficult to attribute uncertainties until the differences between k_THW and k_LP are fully explained.

2.7. Surface Tension (γ) and Interfacial Tension (γ_msl)

Surface tension is a surface property and not a bulk property. The magnitude of the surface tension is determined by the composition of the surface. Components with low surface tension (denoted surfactants) tend to occupy the surface layer and hence largely determine the surface tension. Components such as CaO, MgO and Al₂O₃ have high surface tensions (Table 3) compared with the surface tension of SiO₂ and the surfactants (B₂O₃, K₂O, Na₂O and CaF₂).

Table 3. Values for slag components of surface tension (γ, mNm⁻¹), temperature coefficient (dγ/dT, mNm⁻²)⁴¹⁾ and Gibbs energy of formation for liquid oxides per O atom (−ΔG_i)⁴³⁾ used to calculate interaction parameter, φ.⁴²⁾ The last row provides equations used to calculate depression of surface tension (X₁γ₁) by surfactants.⁴¹⁾

	SiO2	CaO	Al2O3	MgO	Na2O	K2O	Li2O	FeO	MnO	CaF2	ZrO2	TiO2	B2O3
γ1773	260	625	655	635	297	160	[300]	645	530	290	400	350	110
dγ/dT	0.031	−0.094	−0.177	−0.13	−0.11	−0.11	−0.11	−0.1	−0.1	−0.07	−0.15	−0.15
−ΔGi kJmol−1	292.2	408.2	354.6	368.4	195.2	140	357	158	240.5	466	365	304
Surfactant		Na2O			K2O			CaF2			B2O3
Equation (Xiγi)surf		=−0.8−1388 Xisurf +6723 (Xisurf)2			=−0.8−1388 Xisurf +6723 (Xisurf)2			=−2.934 Xisurf +4769 (Xisurf)2			=−5.2−3.454 Xisurf +22178(Xisurf)2

The temperature dependence of surface tension (dγ/dT) is usually negative for most components but as the concentration of surfactants increases (dγ/dT) becomes less negative and eventually changes to a positive value.

Few surface tension measurements have been reported for mould slags.^{17,18,35,36,37,38,39,40)} The model is based on a previous model⁴¹⁾ in which the following assumptions were made:

(i) The various slag components were divided into, surfactants and bulk components.

(ii) The surface tension of bulk components (Fig. 7(a)) at 1773 K were calculated using partial molar values for the various components (γ=X₁γ₁+X₂γ₂+X₃γ₃) ^bulk.

Fig. 7.

Schematic diagram showing compositional dependence (X) of surface tension (γ=solid line) and X₁ γ₁ and X₂ γ₂ (=dashed lines) for a binary slag system containing (a) no surface active constituents and (b) one bulk constituent (1) and one surfactant (2).⁴¹⁾

(iii) It was found that X_iγ_i contributions for surfactants resulted in a sharp drop in surface tension until a certain point was achieved (in the range, X_i=0.1 to 0.12) and the relation (up to this point) is expressed as, (X₁γ₁=a’ +b’X₁+c’X₁²). Above this point, γ increased gradually (Fig. 7(b)) with increasing X (in range, 0.12 to 1.0 and is expressed as (X₁γ₁=d’+e’X₁).

When this model was applied to mould slags (which contain a reasonably high concentration of surfactants) it was found that it tended to produce low values for the surface tension.⁴¹⁾

Consequently, the model was modified and the following assumptions were made:

(i) That the minimum in the surfactant curve in Fig. 7(b) corresponded to the point where the slag surface was saturated with surfactants (X_surf=0.12).

(ii) That the surface is occupied preferentially by B₂O₃>K₂O>Na₂O>CaF₂ until the surface becomes saturated with surfactant at ΣX_surf=0.12; (X_iγ_i)^surf is calculated using the values for surfactants given in Table 3.

(iii) Any excess surfactant (left after ΣX_surf=0.12) is treated as a bulk component (γ=X₁^xsγ₁+X₂^xsγ₂).

(iv) The temperature dependence (dγ/dT) was calculated from Eq. (41).   

$$(\mathrm{d}\gamma /\mathrm{d}\mathrm{T})=[{\mathrm{X}}_1(\mathrm{d}{\gamma }_1/\mathrm{d}\mathrm{T})+{\mathrm{X}}_2(\mathrm{d}{\gamma }_2/\mathrm{d}\mathrm{T})]$$(41)

The surface tension is then calculated from Eqs. (42) and (43)   

$$\begin{array}{l}{\gamma }_{1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}773\mathrm{K}}\\ =[{({\mathrm{X}}_1{\gamma }_1+{\mathrm{X}}_2{\gamma }_2+{\mathrm{X}}_3{\gamma }_3)}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}+({\mathrm{X}}_1{\hspace{0.17em}}^{\mathrm{x}\mathrm{s}}{\gamma }_1+{\mathrm{X}}_2{\hspace{0.17em}}^{\mathrm{x}\mathrm{s}}{\gamma }_2)+\mathrm{\Sigma }{({\mathrm{X}}_{\mathrm{i}}{\gamma }_{\mathrm{i}})}^{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}]\end{array}$$(42)

  

$${\gamma }_{\mathrm{T}}={\gamma }_{1\hspace{0.17em}\hspace{0.17em}773\mathrm{K}}+(\mathrm{d}\gamma /\mathrm{d}\mathrm{T})\left(\mathrm{T}-1\hspace{0.17em}\hspace{0.17em}773\right)$$(43)

There is a limited amount of data to test the predictions and the reported data show significant variation. Nevertheless, the calculated values are in good agreement with the measured values.^{1,18,35,36,37,38,39,40)} The uncertainty associated with the estimations is probably ca. ±10%.

The interfacial tension (γ_msl) can be calculated from Eq. (44) where γ_m=steel surface tension, γ_sl can be derived from Eqs. (42) and (43) and the interaction coefficient, φ, can be calculated using Eqs. (45) and (46), where ΔG_i is the free energy of formation of the individual, liquid oxides^42,43) with the values being given in Table 3.⁴³⁾ Thus, φ is treated as a measure of the oxide components in the slag to supply soluble oxygen in the metal and thereby, decrease the interfacial tension, γ_msl. The value of γ_msl is largely determined by γ_m since γ_m≈ (4 to 5) γ_sl and the value of γ_m, in turn, is largely determined by the S content of the steel. The calculated values of γ_msl obtained with this procedure were reported to lie within ±100 mNm⁻¹ of the experimental values.⁴²⁾ Values of γ_msl are calculated in steel property estimation software, worksheet “surface tension”.   

$${\gamma }_{\mathrm{m}\mathrm{s}\mathrm{l}}={\gamma }_{\mathrm{m}}+{\gamma }_{\mathrm{s}\mathrm{l}}-2\phi {({\gamma }_{\mathrm{m}}{\gamma }_{\mathrm{s}\mathrm{l}})}^{0.5}.$$(44)

  

$$\phi =0.89-1.5\times {10}^{-3}\mathrm{\Delta }{\mathrm{G}}_{\mathrm{i}}{\hspace{0.17em}}^{\mathrm{D}}.$$(45)

  

$$\mathrm{\Delta }{\mathrm{G}}_{\mathrm{i}}{\hspace{0.17em}}^{\mathrm{D}}.=\hspace{0.17em}=\mathrm{\Delta }{\mathrm{G}}_{\mathrm{F}\mathrm{e}\mathrm{O}-}\mathrm{\Sigma }{\mathrm{X}}_{\mathrm{i}}\mathrm{\Delta }{\mathrm{G}}_{\mathrm{i}}.$$(46)

2.8. Emissivity (ε_TN)

Total normal emissivity measurements have been reported for mould slags;⁶⁾ values were recorded for the solid (at T>1030 K) ε_TN=0.92±0.05 and liquid, ε_TN=0.91±0.03.

3. Results and Discussion

Routines have been developed in this study to calculate the following properties of mould slags from their chemical composition for subsequent use as input data in mathematical models of the continuous casting process: T_liq, T_g, T_br; T_crit; C_p (H_T−H₂₉₈); density and α; viscosity, thermal conductivity, surface and interfacial tension and emissivity.

Although mould slags have relatively low melting points and are benign in their behaviour, there are few measurements available for some properties (e.g. C_p, enthalpy, density and surface tension). By contrast, there are a large number of reported data for viscosity and for thermal conductivity/diffusivity. However, there is high level of experimental uncertainty with viscosity measurements (±10–25%) and there is an unresolved dispute regarding the thermal conductivity values at temperatures at >1040 K derived using the THW and LP methods. It is important that this dispute be settled quickly. One further problem is that many workers fail to report post-measurement analysis of their samples and F and Na losses can be significant during a measurement campaign and affect the property values. This study aims to provide reliable property data for the mould slag for use in mathematical models of continuous casting. Consequently, it is intended to update the existing software as new data become available. The following improvements in the software are planned:

(i) The present software does not include B₂O₃ additions since various workers report B₂O₃ causes both increases and decreases in slag viscosity.

(ii) To include calcium-aluminate- based slags used for casting high Al steels.

(iii) To improve the viscosity model for the super-cooled liquid phase.

4. Conclusions

(1) Routines have been developed to calculate the thermo-physical properties of mould slags from their chemical composition for their subsequent use in mathematical models of heat and fluid flow in the continuous casting mould.

(2) The accuracies of estimated values are affected by the limitations in the experimental data.

(3) The cause of the huge discrepancies in thermal conductivities for T>1050 K must be resolved.

Acknowledgements

This work was made possible by the facilities and support provided by the EU (RFSR-PR-10005 DDT), the Research Complex at Harwell, and the EPSRC (EP/I02249X/1).

Symbols Abbreviations

a =Thermal diffusivity (m²s⁻¹)

C_p =Heat capacity (JK⁻¹mol⁻¹ or JK⁻¹kg⁻¹)

f_cryst =Fraction crystalline phase

f* =Fraction of powder forming slag

(H_T−H₂₉₈) =Enthalpy relative to 298 K (Jmol⁻¹ or Jkg⁻¹)

k =Thermal conductivity (Wm⁻¹K⁻¹)

S =Entropy (JK⁻¹kg⁻¹)

T =temperature (K)

X =Mole fraction

α =Thermal expansion coefficient(K⁻¹)

γ =Surface tension (mNm⁻¹)

γ_ms =Interfacial tension (mNm⁻¹)

η =Viscosity (dPas)

ρ =Density (kgm⁻³)

BO =Bridging Oxygens

FO =Free Oxygens

LP =Laser pulse method

NBO =Non-bridging Oxygens

NBO/T =Measure of de-polymerisation

THW =transient hot wire method

Q =measure of polymerisation

scl =super-cooled liquid

Subscripts and Supercripts

^m =Value for liquid at T_liq.

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