2. A unified formulation for the tundish and continuous casting processes

The tundish and continuous casting operation units are connected by metal transfer systems to account for the smooth operation and strict control of both. However, a common formulation is possible based on transport phenomena principles. In this section we present a turbulent flow coupled with heat and mass transfer for interconnected processes. The tundish is modelled as a reactor including the metal, slag and inclusions flows, while the refractories and internal protective devices are considered. A multiphase formulation is considered: (a) liquid metal, (b) liquid slag, (c) solidified metal, (d) solidified slag, (e) particle inclusions and (f) refractory:

$$\frac{\partial(\rho u\_i)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} \left(\rho u\_j u\_i\right) = \frac{\partial}{\partial \mathbf{x}\_j} \left(\left(\mu + \mu\_t\right) \frac{\partial u\_i}{\partial \mathbf{x}\_j}\right) - \frac{\partial}{\partial \mathbf{x}\_j} \left(\tau\_{ij} + \mathbf{C}\_{ij} + L\_{ij}\right) - \frac{\partial P}{\partial \mathbf{x}\_i} - \frac{\left(1 - f\_s\right)}{K\_{u\_i}} u\_i + \rho \mathbf{g}\_{ui} \tag{1}$$

$$\frac{\partial(\rho T)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_{j}} \left(\rho u\_{j} T\right) = \frac{\partial}{\partial \mathbf{x}\_{j}} \left( \left(\frac{k}{\mathbb{C}p} + \frac{k\_{t}}{\mathbb{C}p\_{t}}\right) \frac{\partial T}{\partial \mathbf{x}\_{j}}\right) - \frac{\partial}{\partial \mathbf{x}\_{j}} \left(\theta + \mathbb{C} + L\right) - \frac{\partial}{\partial t} \left(\rho \Delta H f\_{s}\right) \tag{2}$$

$$\frac{\partial(\rho)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_{\dot{\jmath}}} \left(\rho u\_{\dot{\jmath}}\right) = 0 \tag{3}$$

τij ¼ μ<sup>t</sup>

in Holman [23]:

their specific constants.

depending on the materials used [24].

CC l 

CMn l 

provides Scheil's equation [24].

the emissivity.

The local liquid concentration of each species is given by

<sup>P</sup> � ½ � <sup>r</sup><sup>C</sup> C, old

<sup>P</sup> � ½ � <sup>r</sup><sup>C</sup> Mn, old

<sup>P</sup> <sup>þ</sup> <sup>β</sup>Mnr<sup>S</sup> <sup>1</sup> � gnþ<sup>1</sup>

<sup>P</sup> <sup>þ</sup> <sup>β</sup><sup>C</sup>r<sup>S</sup> <sup>1</sup> � gnþ<sup>1</sup>

<sup>P</sup> <sup>¼</sup> ½ � <sup>r</sup><sup>C</sup> <sup>C</sup>

rl gnþ<sup>1</sup>

<sup>P</sup> <sup>¼</sup> ½ � <sup>r</sup><sup>C</sup> Mn

operational conditions or temperature monitoring data.

k ∂T

rl gnþ<sup>1</sup> Lij <sup>þ</sup> Cij <sup>¼</sup> <sup>Δ</sup><sup>k</sup>

Sij <sup>¼</sup> <sup>1</sup> 2

∂ui ∂xj þ ∂uj ∂xi 

12

∂ui ∂xj þ ∂uj ∂xi

In the solid phase, thermal conductivity was assumed as a function of temperature, according

where ψ and γ are constants for a specific metal alloy and the refractories considered for each layer and formed solidified shell. For the liquid phases, a similar relationship is assumed with

The specific heat for the solid and the liquid phases is obtained directly from ThermoCalc calculation using TCFE5 database, while for the refractories, specific relations are used

<sup>P</sup> <sup>þ</sup> <sup>r</sup>lgold

P kC

<sup>P</sup> <sup>þ</sup> <sup>r</sup>lgold

P kMn

Heat fluxes on the water-cooled surfaces and on the radiation zones are given by

<sup>∂</sup><sup>x</sup> <sup>¼</sup> heffð Þþ Tsur � Te <sup>σ</sup>rε<sup>r</sup> <sup>T</sup><sup>4</sup>

where heff is the effective heat transfer coefficient provided by Eq. (17), Tsur is the surface temperature, Te is the environment temperature, σ<sup>r</sup> is the Stefan-Boltzmann constant and ε<sup>r</sup> is

The segregation parameter β can vary as 0 ≤ β ≤ 1. Assuming β=1 means the lever rule, and β=0,

The heat flux boundary conditions for the continuous caster machine are estimated depending on the region and operational conditions. In the mould and in the foot roll, the cooling water flow is specified at the four faces, internal and external large faces and right and left narrow faces, while at the other zones, heat fluxes were imposed only at two faces, the internal and the external large faces. For both processes the initial conditions are specified by the measured

<sup>P</sup> <sup>þ</sup> <sup>β</sup><sup>C</sup>r<sup>S</sup> <sup>1</sup> � gold

<sup>0</sup> <sup>þ</sup> <sup>1</sup> � <sup>β</sup><sup>C</sup> rSk

<sup>P</sup> <sup>þ</sup> <sup>β</sup>Mnr<sup>S</sup> <sup>1</sup> � gold

<sup>0</sup> <sup>þ</sup> <sup>1</sup> � <sup>β</sup>Mn rSk

CMn

sur � <sup>T</sup><sup>4</sup> e

CC

P k

> C <sup>0</sup> gold

P k

> Mn <sup>0</sup> gold

C 0

l old P

(14)

<sup>P</sup> � gnþ<sup>1</sup> P

> l old P

<sup>P</sup> � gnþ<sup>1</sup> P (15)

(16)

Mn 0

þ 2 3

∂ui ∂xk þ ∂uj ∂xk

Numerical Study of Turbulent Flows and Heat Transfer in Coupled Industrial-Scale Tundish of a Continuous…

kδij (10)

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293

(11)

(12)

k ¼ ψ � 0:01γT (13)

$$\frac{\partial}{\partial t} \begin{pmatrix} \rho \ C^i \end{pmatrix} \ + \quad \frac{\partial \left( \rho \ u\_j \ C^i \right)}{\partial \mathbf{x}\_j} = \frac{\partial}{\partial \mathbf{x}\_j} \left( \left( D^i + \frac{\mu\_t}{\rho} \right) \frac{\partial \left( C^i \right)}{\partial \mathbf{x}\_j} \right) \tag{4}$$

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} (\rho u\_\dagger k) = \frac{\partial}{\partial \mathbf{x}\_j} \left( \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_j} \right) + 2\mu\_t \mathbf{S}\_{\vec{\eta}} \mathbf{S}\_{\vec{\eta}} - \rho \varepsilon + \beta \mathbf{g}\_i \frac{\mu\_t}{\mathbf{Pr}} \frac{\partial T}{\partial \mathbf{x}\_i} \tag{5}$$

$$\frac{\partial(\rho\varepsilon)}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_{\dot{\jmath}}} \left(\rho\mu\_{\dot{\jmath}}\varepsilon\right) = \frac{\partial}{\partial \mathbf{x}\_{\dot{\jmath}}} \left(\left(\mu + \frac{\mu\_t}{\sigma\_\varepsilon}\right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_{\dot{\jmath}}}\right) + \mathbb{C}\_{1\epsilon} \frac{\varepsilon}{k} \left(2\mu\_t \mathbb{S}\_{\ddot{\jmath}} \mathbb{S}\_{\dddot{\jmath}} + \mathbb{C}\_{3\epsilon} \beta \mathcal{g}\_{\dot{i}} \frac{\mu\_t}{\mathbf{Pr}\,\mathbf{d}\mathbf{x}\_{\dot{\jmath}}}\right) - \mathbb{C}\_{2\epsilon} \rho \frac{\varepsilon^2}{k} \tag{6}$$

where gu is the gravity acceleration component at the velocity component direction, depending on the number of species, which can be written as

$$\mathcal{g}\_u = \,^u \mathcal{g}\_0 \sum\_{\mathcal{C}\_r \text{Mn}} \left[ \beta\_\mathcal{S}^i \left( \mathbf{C}\_l^i - \mathbf{C}\_{l,0}^i \right) + \beta\_T^i (T - T\_0) \right] \tag{7}$$

Viscosity was treated as effective viscosity [21] in the following form:

$$
\mu\_{\it eff} = \frac{\overline{\sigma}}{\Im \overline{\varepsilon}} + \mu\_t \tag{8}
$$

$$
\mu\_t = \left(\Delta\right)^2 \sqrt{\mathcal{S}\_{\vec{\eta}} \mathcal{S}\_{\vec{\eta}}} \tag{9}
$$

where σ is the material mean stress and ε is the effective deformation rate presented by Zienkiewicz [22] and Δ is a sub-grid scale for the isotropic turbulence filtering. All the variables used in the formulation are taken as the filtered values. The constants c<sup>1</sup><sup>ε</sup> ¼ 1:44, c<sup>2</sup><sup>ε</sup> ¼ 1:92, c<sup>3</sup><sup>ε</sup> ¼ 0:09 and σε ¼ 1:30 are related with turbulent kinetic energy (k) and its dissipation rate (ε):

Numerical Study of Turbulent Flows and Heat Transfer in Coupled Industrial-Scale Tundish of a Continuous… http://dx.doi.org/10.5772/intechopen.75935 293

$$
\pi\_{i\dot{\jmath}} = \mu\_t \left( \frac{\partial u\_i}{\partial \mathbf{x}\_{\dot{\jmath}}} + \frac{\partial u\_{\dot{\jmath}}}{\partial \mathbf{x}\_i} \right) + \frac{2}{3} k \delta\_{\dot{\jmath}} \tag{10}
$$

$$L\_{i\dagger} + C\_{i\dagger} = \frac{\Delta\_k}{12} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_k} + \frac{\partial u\_\dagger}{\partial \mathbf{x}\_k} \right) \tag{11}$$

$$\mathcal{S}\_{i\dot{\jmath}} = \frac{1}{2} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_{\dot{\jmath}}} + \frac{\partial u\_{\dot{\jmath}}}{\partial \mathbf{x}\_{i}} \right) \tag{12}$$

In the solid phase, thermal conductivity was assumed as a function of temperature, according in Holman [23]:

$$k = \psi - 0.01\gamma T\tag{13}$$

where ψ and γ are constants for a specific metal alloy and the refractories considered for each layer and formed solidified shell. For the liquid phases, a similar relationship is assumed with their specific constants.

The specific heat for the solid and the liquid phases is obtained directly from ThermoCalc calculation using TCFE5 database, while for the refractories, specific relations are used depending on the materials used [24].

The local liquid concentration of each species is given by

2. A unified formulation for the tundish and continuous casting processes

The tundish and continuous casting operation units are connected by metal transfer systems to account for the smooth operation and strict control of both. However, a common formulation is possible based on transport phenomena principles. In this section we present a turbulent flow coupled with heat and mass transfer for interconnected processes. The tundish is modelled as a reactor including the metal, slag and inclusions flows, while the refractories and internal protective devices are considered. A multiphase formulation is considered: (a) liquid metal, (b) liquid slag, (c) solidified metal, (d) solidified slag, (e) particle inclusions and (f) refractory:

∂ð Þ rui ∂t þ

∂ ∂xj

∂ð Þ rT ∂t þ

> ∂ð Þ rk ∂t þ

∂ ∂xj

∂ð Þ rε ∂t þ

pation rate (ε):

rujui � � <sup>¼</sup> <sup>∂</sup>

292 Numerical Simulations in Engineering and Science

∂ ∂xj ∂xj

<sup>r</sup>ujT � � <sup>¼</sup> <sup>∂</sup>

<sup>∂</sup><sup>t</sup> <sup>r</sup>C<sup>i</sup> � � <sup>þ</sup>

<sup>r</sup>ujk � � <sup>¼</sup> <sup>∂</sup>

∂xj

gu<sup>¼</sup> ug<sup>0</sup>

on the number of species, which can be written as

∂xj

<sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> σe � � ∂ε

> X C, Mn

Viscosity was treated as effective viscosity [21] in the following form:

� �

∂

∂ ∂xj

<sup>r</sup>uj<sup>ε</sup> � � <sup>¼</sup> <sup>∂</sup>

μ þ μ<sup>t</sup> � � <sup>∂</sup>ui

∂xj

� �

∂xj

k Cp þ

∂ð Þ r ∂t þ

∂ r ujCi � � ∂xj

> <sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> σk � � ∂k

> > ∂xj

βi <sup>S</sup> Ci <sup>l</sup> � <sup>C</sup><sup>i</sup> l,0 � �

<sup>μ</sup>eff <sup>¼</sup> <sup>σ</sup> 3ε

<sup>μ</sup><sup>t</sup> <sup>¼</sup> ð Þ <sup>Δ</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffi

where σ is the material mean stress and ε is the effective deformation rate presented by Zienkiewicz [22] and Δ is a sub-grid scale for the isotropic turbulence filtering. All the variables used in the formulation are taken as the filtered values. The constants c<sup>1</sup><sup>ε</sup> ¼ 1:44, c<sup>2</sup><sup>ε</sup> ¼ 1:92, c<sup>3</sup><sup>ε</sup> ¼ 0:09 and σε ¼ 1:30 are related with turbulent kinetic energy (k) and its dissi-

� �

� ∂ ∂xj

kt Cpt � � ∂T

> ∂ ∂xj ruj

> > ¼ ∂ ∂xj

> > > ∂xj

þ C1<sup>e</sup> ε <sup>k</sup> <sup>2</sup>μ<sup>t</sup>

where gu is the gravity acceleration component at the velocity component direction, depending

� �

∂xj

τij þ Cij þ Lij � � � <sup>∂</sup><sup>P</sup>

> � ∂ ∂xj

<sup>D</sup><sup>i</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> r � � ∂ Ci � �

þ 2μ<sup>t</sup>

<sup>þ</sup> <sup>β</sup><sup>i</sup>

h i

SijSij q

!

∂xi

ð Þ� <sup>θ</sup> <sup>þ</sup> <sup>C</sup> <sup>þ</sup> <sup>L</sup> <sup>∂</sup>

∂xj

SijSij � rε þ βgi

� �

SijSij þ C3eβgi

<sup>T</sup>ð Þ T � T<sup>0</sup>

� � <sup>¼</sup> <sup>0</sup> (3)

� <sup>1</sup> � <sup>f</sup> <sup>s</sup> � � Kui

<sup>∂</sup><sup>t</sup> <sup>r</sup>ΔHf <sup>s</sup>

μt Pr ∂T ∂xi

� C2er

ε2 <sup>k</sup> (6)

μt Pr ∂T ∂xi

þ μ<sup>t</sup> (8)

ui þ rgui (1)

(4)

(5)

(7)

(9)

� � (2)

$$\left[\mathbf{C}\_{l}^{\mathbf{C}}\right]\_{P} = \frac{[\rho\mathbf{C}]\_{P}^{\mathbf{C}} - [\rho\mathbf{C}]\_{P}^{\mathbf{C},old} + \left[\rho\_{l}\mathbf{g}\_{P}^{old} + \beta^{\mathbf{C}}\rho\_{S}(1 - \mathbf{g}\_{P}^{old})k\_{0}^{\mathbf{C}}\right] \left[\mathbf{C}\_{l}^{\mathbf{C}}\right]\_{P}^{old}}{\rho\_{l}\mathbf{g}\_{P}^{n+1} + \beta^{\mathbf{C}}\rho\_{S}(1 - \mathbf{g}\_{P}^{n+1})k\_{0}^{\mathbf{C}} + (1 - \beta^{\mathbf{C}})\rho\_{S}k\_{0}^{\mathbf{C}}(\mathbf{g}\_{P}^{old} - \mathbf{g}\_{P}^{n+1})} \tag{14}$$

$$\left[\mathbf{C}\_{l}^{\text{Mn}}\right]\_{P} = \frac{[\rho\mathbf{C}]\_{P}^{\text{Mn}} - [\rho\mathbf{C}]\_{P}^{\text{Mn,old}} + \left[\rho\_{l}\mathbf{g}\_{P}^{\text{old}} + \beta^{\text{Mn}}\rho\_{S}\left(1 - \mathbf{g}\_{P}^{\text{old}}\right)\mathbf{k}\_{0}^{\text{Mn}}\right] \left[\mathbf{C}\_{l}^{\text{Mn}}\right]\_{P}^{\text{old}}}{\rho\_{l}\mathbf{g}\_{P}^{\text{n+1}} + \beta^{\text{Mn}}\rho\_{S}\left(1 - \mathbf{g}\_{P}^{\text{n+1}}\right)\mathbf{k}\_{0}^{\text{Mn}} + \left(1 - \beta^{\text{Mn}}\right)\rho\_{S}\mathbf{k}\_{0}^{\text{Mn}}\left(\mathbf{g}\_{P}^{\text{old}} - \mathbf{g}\_{P}^{\text{n+1}}\right)} \tag{15}$$

The segregation parameter β can vary as 0 ≤ β ≤ 1. Assuming β=1 means the lever rule, and β=0, provides Scheil's equation [24].

The heat flux boundary conditions for the continuous caster machine are estimated depending on the region and operational conditions. In the mould and in the foot roll, the cooling water flow is specified at the four faces, internal and external large faces and right and left narrow faces, while at the other zones, heat fluxes were imposed only at two faces, the internal and the external large faces. For both processes the initial conditions are specified by the measured operational conditions or temperature monitoring data.

Heat fluxes on the water-cooled surfaces and on the radiation zones are given by

$$k\frac{\partial T}{\partial \mathbf{x}} = h\_{\text{eff}}(T\_{sur} - T\_e) + \sigma\_r \varepsilon\_r \left(T\_{sur}^4 - T\_e^4\right) \tag{16}$$

where heff is the effective heat transfer coefficient provided by Eq. (17), Tsur is the surface temperature, Te is the environment temperature, σ<sup>r</sup> is the Stefan-Boltzmann constant and ε<sup>r</sup> is the emissivity.

The heat transfer coefficient in the sprays zones (foot roll, bender and secondary cooling zone) was obtained by the water cooling enthalpy balance, providing

$$h\_{\rm eff} = \frac{m\_w c\_p \Delta T}{A(T\_{sur} - T\_e)}\tag{17}$$

where mw is the water flow, cp is the water-specific heat, A is the cross-sectional area and ΔT is the water temperature difference given as a setup parameter for the cooling system.

The mould region was modelled by using the steel residence time in the mould to calculate the effective heat transfer coefficient. This coefficient regards the effect of thermal resistance due to air gap formation:

$$h\_{mol} = 1004.6 \exp\left(0.02 \, t\_m\right) \tag{18}$$

where tm is the steel residence time, calculated by means of the cast velocity setup (Vc) and the mould height (Y) as

$$t\_m = \frac{Y}{V\_c} \tag{19}$$

One additional restriction for the continuous casting vein was imposed by using 20% of the total volumes on the oscillating mould region to account for the accuracy of the solution in this region due to the strong gradients developed within the oscillating mould with solidifying shell with strong heat release and solute redistribution. Details of the mesh generated and the

Figure 3. Physical and computational domains indicating the zones of the continuous casting slab and subdomains.

Numerical Study of Turbulent Flows and Heat Transfer in Coupled Industrial-Scale Tundish of a Continuous…

The turbulent quantities at the inlet and outlet are calculated based on the averaged velocities

Uav <sup>¼</sup> <sup>Q</sup>

Eqs. (20)–(22) are applied depending on the geometry of the valves and feeding systems. The averaged values for the temperature and compositions are either set using the measured values or, in the case of transfer system, the values calculated in the previous connected

The thermophysical properties of the liquids (steel and slags) and solids formed during the solidification process are determined by using computational thermodynamics database. The solid barriers such as refractories and inhibitors are included by using their tabled thermophysical property data furnished by the suppliers. By using the thermodynamics database, a typical steel is modelled using their pseudo-binary diagrams. Figure 4 shows the temperatures and phase composition dependency for the whole system and specific regions of the diagram. These data are continuously accessed for local predictions of the thermophysical

<sup>ε</sup>av <sup>¼</sup> <sup>2</sup>

<sup>A</sup> (20)

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295

kav <sup>¼</sup> <sup>0</sup>:01ð Þ Uav <sup>2</sup> (21)

<sup>D</sup> ð Þ kav <sup>2</sup>=<sup>3</sup> (22)

subdomains assumed in the simulations are presented in Figure 3.

for both processes, as follows:

properties during the transient calculation.

domains.

The inlet and outlet boundary conditions are specified using mass flow, compositions and temperatures. The wall standard log law is used for modelling the liquid to walls and barrier interfaces in both domains of the tundish and continuous casting vein. Figure 2 shows the geometry and computational domain of the tundish with the refractories and internal barriers. The numerical mesh used was obtained by using continuous refinement using 20% of the total volume increment of each calculation for a standard operational conditions assuming averaged error less than 1% on the temperature and velocity fields. Same procedure was used to obtain the suitable mesh distribution along the continuous casting vein. The final mesh total volumes in the tundish domain were 201,300 and for the continuous casting vein were 288,000.

Figure 2. Physical and computational domains including the refractory layers and internal features of the tundish (60 ton).

Numerical Study of Turbulent Flows and Heat Transfer in Coupled Industrial-Scale Tundish of a Continuous… http://dx.doi.org/10.5772/intechopen.75935 295

The heat transfer coefficient in the sprays zones (foot roll, bender and secondary cooling zone)

heff <sup>¼</sup> mwcpΔ<sup>T</sup> A Tð Þ sur � Te

where mw is the water flow, cp is the water-specific heat, A is the cross-sectional area and ΔT is

The mould region was modelled by using the steel residence time in the mould to calculate the effective heat transfer coefficient. This coefficient regards the effect of thermal resistance due to

where tm is the steel residence time, calculated by means of the cast velocity setup (Vc) and the

tm <sup>¼</sup> <sup>Y</sup> Vc

The inlet and outlet boundary conditions are specified using mass flow, compositions and temperatures. The wall standard log law is used for modelling the liquid to walls and barrier interfaces in both domains of the tundish and continuous casting vein. Figure 2 shows the geometry and computational domain of the tundish with the refractories and internal barriers. The numerical mesh used was obtained by using continuous refinement using 20% of the total volume increment of each calculation for a standard operational conditions assuming averaged error less than 1% on the temperature and velocity fields. Same procedure was used to obtain the suitable mesh distribution along the continuous casting vein. The final mesh total volumes in the tundish domain were 201,300 and for the continuous casting vein were 288,000.

Figure 2. Physical and computational domains including the refractory layers and internal features of the tundish

hmold ¼ 1004:6 exp 0ð Þ :02tm (18)

the water temperature difference given as a setup parameter for the cooling system.

(17)

(19)

was obtained by the water cooling enthalpy balance, providing

air gap formation:

294 Numerical Simulations in Engineering and Science

mould height (Y) as

(60 ton).

Figure 3. Physical and computational domains indicating the zones of the continuous casting slab and subdomains.

One additional restriction for the continuous casting vein was imposed by using 20% of the total volumes on the oscillating mould region to account for the accuracy of the solution in this region due to the strong gradients developed within the oscillating mould with solidifying shell with strong heat release and solute redistribution. Details of the mesh generated and the subdomains assumed in the simulations are presented in Figure 3.

The turbulent quantities at the inlet and outlet are calculated based on the averaged velocities for both processes, as follows:

$$\mathcal{U}\_{\text{av}} = \frac{\mathcal{Q}}{A} \tag{20}$$

$$k\_{av} = 0.01(\mathcal{U}\_{av})^2\tag{21}$$

$$
\varepsilon\_{av} = \frac{2}{D} (k\_{av})^{2/3} \tag{22}
$$

Eqs. (20)–(22) are applied depending on the geometry of the valves and feeding systems. The averaged values for the temperature and compositions are either set using the measured values or, in the case of transfer system, the values calculated in the previous connected domains.

The thermophysical properties of the liquids (steel and slags) and solids formed during the solidification process are determined by using computational thermodynamics database. The solid barriers such as refractories and inhibitors are included by using their tabled thermophysical property data furnished by the suppliers. By using the thermodynamics database, a typical steel is modelled using their pseudo-binary diagrams. Figure 4 shows the temperatures and phase composition dependency for the whole system and specific regions of the diagram. These data are continuously accessed for local predictions of the thermophysical properties during the transient calculation.

Figure 4. Pseudo-binary phase diagram of Fe-C-Mn-P-S as a function of carbon content (a) phase diagram, (b) and (c) magnification of high temperature range closed to 0, 15 wt% C, which is close to the steel used in this study.

Figure 5 shows the density and heat of phase transformation during temperature evolution. These quantities are accessed to estimate the heat capacity and latent heat released during the solidification and flowing paths. Figure 6 shows the solid fraction during the solidification path considering the local conditions predicted by computational thermodynamics. With these parameters and physical properties, all the information need for the coupled calculation of the tundish and continuous casting operation are closed.

3. Numerical features

used during the calculations accessed from the thermodynamic database.

computational modelling.

Momentum, mass, energy and species equations were discretizated by using the finite volume method (FVM) applied for general coordinate system [25, 26], where the integration is taken

Figure 6. Solid fraction of all solids as a function of temperature during the solidification process within the steel slab

Figure 5. Thermophysical properties inside mushy zone: (a) density and (b) heat and latent heat for the steel used in this

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Numerical Study of Turbulent Flows and Heat Transfer in Coupled Industrial-Scale Tundish of a Continuous… http://dx.doi.org/10.5772/intechopen.75935 297

Figure 5. Thermophysical properties inside mushy zone: (a) density and (b) heat and latent heat for the steel used in this computational modelling.

Figure 6. Solid fraction of all solids as a function of temperature during the solidification process within the steel slab used during the calculations accessed from the thermodynamic database.

## 3. Numerical features

Figure 5 shows the density and heat of phase transformation during temperature evolution. These quantities are accessed to estimate the heat capacity and latent heat released during the solidification and flowing paths. Figure 6 shows the solid fraction during the solidification path considering the local conditions predicted by computational thermodynamics. With these parameters and physical properties, all the information need for the coupled calculation of the

Figure 4. Pseudo-binary phase diagram of Fe-C-Mn-P-S as a function of carbon content (a) phase diagram, (b) and (c)

magnification of high temperature range closed to 0, 15 wt% C, which is close to the steel used in this study.

tundish and continuous casting operation are closed.

296 Numerical Simulations in Engineering and Science

Momentum, mass, energy and species equations were discretizated by using the finite volume method (FVM) applied for general coordinate system [25, 26], where the integration is taken over a typical control volume. The final product of this operation is a set of algebraic equations. Coefficients are obtained by the so-called power law scheme, according to Patankar [26]. The SIMPLE algorithm is used to iteratively determine the velocity components and pressure linked equations. The numerical solution of the set of algebraic equations demands large computational effort. A line-by-line solver based on the tridiagonal matrix algorithm (TDMA) was used to solve the system of algebraic equations. The Alternate Direction Implicit (ADI) iterative procedure was applied within a common solver for all discretized equations. The iterative solution was obtained for each time step in a fully implicit scheme [25, 26]. The convergence criteria were used for all variables admitting a maximum local error less than 1% for all variables simultaneously.

The initial step of the tundish feeling presents strong turbulence features and plays important role on the stable flowing development and security of the whole operation. Figure 7 shows the flowing pattern (t= 3 s) for a thin slab operation while the slab extraction is off. As can be observed, the inhibitor apparatus is important to avoid splashing and protect the refractories. Figure 8 shows the conditions where the stable flow rates are achieved with the liquid level of the tundish nearly constant. The flow pattern indicates that a complex turbulent flow is

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In order to assure the coupled model formulation and the simultaneous solution in the parallel platform simulation, a confrontation with measured temperature profile measured in the industrial machine was performed. Figure 9 showed a comparison of the model predictions

As can be observed, a close agreement with the industrial operation measurements for the temperature is reached. The measurements and calculations were compared for the stable casting operation, and the measured values were obtained using infrared pyrometer, and the

As can be observed also, the values obtained with the serial and parallel versions are virtually the same. A complete view of the solid portion of the continuous casting domain is shown in Figure 10 for stable flowing state. A thin skin formed in the oscillating mould region and continuous growing along the bending and cooling zones is observed. A recalescence and final cooling regions are observed. These regions are critical for the process due to the possibility of crack and defect susceptibility depending on the cooling rates and inclusions dragged and

for the serial, parallel and the pyrometer measurement at the industrial machine.

plotted values are the average of five runs with intervals of 5 min.

observed and the liquid flow promotes strong mixing.

formed during the casting development [22–24].

Figure 7. Fluid flow pattern during initial stage of the tundish feeling period.
