# Analysis of Non-Symmetrical Heat Transfers during the Casting of Steel Billets and Slabs

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## Abstract

**:**

## 1. Introduction

## 2. Computational Representation of Steel Casting

_{p}) for steel, are functions of the temperature and chemical composition [14,15,16,17,18,22,23,24].

- The casting temperature (TCO) is the same for all the nodes. Thus, the assignment of the energy required to define the casting temperature is for all nodes.
- Only one single steel volume is in the casting plant for the simulation. In consequence, there is no heat inter-change in the longitudinal direction of the machine. Thus, the heat removal in the cast direction is negligible. This assumption simplifies the problem and reduces the calculation time; the problem is a 2D type as the longitudinal heat transfer is negligible. Therefore, the treatment of the problem uses 2D computational arrays; one for enthalpies and one for both the latest and previous calculations of enthalpy (${H}_{i,j}^{t-1}$ and ${T}_{i,j}^{t-1}$) and (${H}_{i,j}^{t}$ and ${T}_{i,j}^{t}$), reducing the computer’s memory requirements.
- The simulation begins at the meniscus level inside the mold. Then, the simulation time is (t = 0).
- The step time (Δt) is calculated as a function of the billet dimensions (lx) and (ly) using (Nx) and (Ny) nodes, and the steel thermal diffusivity (α) is given in Equation (1) where k is the thermal conductivity, ρ is density, and C
_{p}is heat capacity.$$\alpha =\frac{k}{\rho {C}_{p}}\text{}$$

_{Liq}, temperature of solidus, T

_{Sol}, upper transformation temperature, T

_{AR1}, and lower transformation temperature, T

_{AR3}) is through Equations (3)–(6) as a function of the steel chemical composition [22]. Therefore, Equation (7) calculates the corresponding energy required to melt the steel (H

_{i,j}) [3,15,22]. Here, (w) is the weight of each discretized steel element obtained using Equation (8). A graphical representation of the energy calculated for a steel volume is in Figure 2.

## 3. Heat Transfer and Conduction inside the Billet Core

_{i,j}). Then, the new values are stored in a new array for the next iteration and the previous array is deleted and updated for efficient use of the computational resources. The calculation of the applied heat removal for every node in the billet surfaces (q

_{i,j}) is a function of the mechanism involved according to the billet position and the CCM [5,6,7,24,25,26].

_{i,j}) corresponds to a value of temperature (T

_{i,j}) [13,15,26,27,28,29]. The main routine updates the heat that remained after each step time (Δt) during the simulation. If the cast speed is known, it is possible to calculate exactly the time at which each node changes from liquid to a mushy structure and from a mushy structure to a solid-state, as is indicated in Equation (10), after which data is stored in a pair of 2D computational arrays, namely (t

_{sol i,j}and t

_{liq i,j}). A comparison routine works for this purpose. This routine is applied to all external and internal nodes and is included in the main calculation routine to update the information. Here, the superscripts (t) and (t − 1) represent the corresponding values of the latest and previous iterations during the simulation time. Computationally, these values correspond to the actual time (t) and the previous simulated time (t − Δt).

_{w}) and (T

_{i,j}). Equation (13) is the Stefan–Boltzmann law and calculates the heat flux value as a function of the steel emissivity (ε). These two heat removal conditions work during the simulation when the geometrical conditions of the CCM are verified and validated. The result is a temperature curve that goes down when steel is under a sprayed area and goes up when steel is under a non-sprayed area. The entire process to calculate the coefficient (h) solves Equations (14)–(17). The sub-indexes “ns” and “side” indicate that these values can differ for every segment of the SCS and every billet side. The sub-index “w” identifies the liquid used as water. (ε) is the emissivity, (µ) is the dynamic water viscosity, and the sub-indexes (i) and (j) identify the nodal positions of the billet surface.

## 4. Process Simulation

#### 4.1. Case 1

#### 4.1.1. Operating Conditions and Assumptions

- The steel composition is homogeneous.
- The cast speed is constant during the simulation.
- The heat removal inside the mold is constant and equal on each side of the billet.
- The operating and quenching conditions are constant during the casting operation.

_{0}) is the first angle of the SCS and is measured from the end of the mold. (Ω) is the shooting angle of every spray.

#### 4.1.2. Simulations and Results

_{Liq}) and (T

_{Sol}), depending on the steel chemistry.

#### 4.2. Case 2

#### 4.3. Case 3

## 5. Conclusions

- The direct approach of calculating the heat transfer coefficients through the appropriate dimensionless numbers, rather than through other reported empirical correlations, is suitable to predict the temperature fields in slab and billet machines.
- The surface temperatures along the casting length of slabs and billets using this algorithm match acceptably well the temperature measurements.
- The matching between the measured temperatures and those simulated indicate that the mesh size of 200 × 200 nodes is large enough to obtain reliable thermal predictions.
- The algorithm is versatile as it permits the friendly changes of different casting machines, including the use of different types of water spray nozzles.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

AR1 | upper austenite-ferrite transformation temperature |

AR3 | lower austenite-peralite transformation temperature |

C_{p} | heat capacity |

D | diameter |

h | heat transfer coefficient |

H | enthalpy |

k | thermal conductivity |

l | billet side length |

Nu | Nusselt number |

q | heat flux |

Pr | Prandtl number |

Re | Reynolds number |

T | temperature |

t | time |

W | mass of steel |

Greek symbols: | |

α | thermal diffusivity |

Δx, Δy, and Δz | increments of distance and time |

μ | viscosity |

Ω | shooting angle of every spray |

ρ | density |

${\theta}_{0}$ | the first angle of the secondary colling system |

Subindexes: | |

bs | length of billet side cooled by the spray |

i,j | nodes in the computational mesh |

Liq | liquidus temperature |

m | mold |

mushy | two-phase region of solidification depending on the steel chemistry |

n and side | indexes to indicate that these values can differ for every segment of the SCS and every billet side |

s | water spray |

sol | solidus temperature |

w | water |

Acronyms: | |

CCP | continuous casting process |

PCS | primary cooling system, i.e., the mold |

SCS | secondary cooling system, i.e., the water spray segments of a machine |

CCM | continuous casting machine |

TCO | casting temperature at the meniscus level |

RC | machine radius |

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**Figure 1.**A typical temperature profile in the cross section of a steel billet [23].

**Figure 2.**Relationship between enthalpy and temperature for a steel billet [23].

**Figure 3.**Billet faces used in the mold. Internal and external faces refer to the internal and external radius of the machine radius of the curvature. The het flow is symmetric [23].

**Figure 8.**(

**a**,

**b**) Temperature on the billet surfaces due to the symmetrical heat removal applied (consequence of a non-symmetrical spray distribution and differential water flow rates).

**Figure 9.**(

**a**–

**c**) Temperature profiles inside the mold (assumed symmetrical removal). The positions of these profiles are a function of the distance, taken as a reference at the meniscus level; time (min) and distance (m).

**Figure 10.**(

**a**–

**i**) Temperature profiles in the SCS. Here, the temperatures’ profiles became non-symmetrical due to different water flows rates applied to quench every side; time (min) and distance (m).

**Figure 13.**Temperature profiles and surface temperature of a squared billet with no homogeneous heat removal.

**Figure 16.**Temperature profiles calculated computationally for case 3 considering a slab cast running at four different cast speeds (perpendicular views to the cast direction for the slabs).

**Figure 17.**Surface temperatures on the narrow and wide slab faces (

**a**) for a casting speed = 1.0 m/min and (

**b**) 1.30 m/min.

Cast Speed (m/min) | Casting Temperature (°C) | RC (m) | ${\mathit{\theta}}_{0}$ | ${\mathit{l}}_{\mathit{x}}$ | ${\mathit{l}}_{\mathit{y}}$ |
---|---|---|---|---|---|

2.40 | 1535 | 7.45 | 5.8 | 130 | 130 |

T_{ariq}(°C) | T_{sol}(°C) | T_{AR3}(°C) | T_{AR1}(°C) |
---|---|---|---|

1524.38 | 1507.85 | 844.07 | 721.04 |

C | Al | Cr | Cu | Mn | Nb | Mo |

0.380 | 0.003 | 0.05 | 0.040 | 1.050 | 0.002 | 0.002 |

Ni | P | Ti | S | Si | Sn | V |

0.006 | 0.014 | 0.002 | 0.018 | 0.200 | 0.001 | 0.002 |

**Table 4.**Operating conditions of the SCS (segments 1, 2, and 3). Note: Int. = internal and Ext. = external.

Segment | 1 | 2 | 3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Surface | Internal | External | Left | Right | Internal | External | Left | Right | Internal | External | Left | Right |

Water flow rate (L/min) | 7 | 10 | 7 | 10 | 7 | 10 | 7 | 10 | 7 | 10 | 7 | 10 |

Sprays on cast direction | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 13 | 12 | 9 | 9 |

Sprays on the lateral direction | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Nozzle diameter (m) | 0.003 | 0.003 | 0.003 | |||||||||

${\mathsf{\Omega}}_{castdir}$ | 50 | 60 | 60 | |||||||||

${\mathsf{\Omega}}_{lateraldir}$ | 60 | 50 | 50 | |||||||||

${D}_{bs}$ (m) | 0.083 | 0.100 | 0.100 | |||||||||

$\theta $ | 3 | 17 | 25 |

Segment | 1 | 2 | 3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Surface | Internal | External | Left | Right | Internal | External | Left | Right | Internal | External | Left | Right |

Water flow rate (L/min) | 15 | 15 | 25 | 22 | 12 | 12 | 17 | 15 | 9 | 9 | 12 | 10 |

Sprays on cast direction | 8 | 11 | 8 | |||||||||

Sprays on the lateral direction | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Nozzle diameter (m) | 0.003 | |||||||||||

$\theta $ | 7.50 | 22.5 | 30 | |||||||||

$\mathsf{\Omega}$ | 80 | 60 | 60 | |||||||||

${D}_{bs}$ (m) | 0.060 | 0.110 | 0.075 |

Cast Temperature (°C) | R_{C}(m) | θ_{0} | Narrow Side D_{x}(mm) | Broad Side D_{y}(mm) | Mold Length (m) | Meniscus Level (%) |
---|---|---|---|---|---|---|

1545 | 10.5 | 23.5 | 200 | 1100 | 1.10 | 82 |

Zone | θ | Σθ | Rd (m) | Ω | Sprays on Casting Direction | ds′_{ZN}(mm) | dw′_{ZN}(mm) | dnw′_{ZN}(mm) |
---|---|---|---|---|---|---|---|---|

Curved Zone | ||||||||

1 | 12 | 18 | 3.29 | 60 | 11 | 500 | 329 | 171 |

2 | 8 | 26 | 4.76 | 55 | 5 | 500 | 467 | 33 |

3 | 8 | 34 | 6.23 | 50 | 5 | 500 | 467 | 33 |

4 | 11 | 45 | 8.24 | 50 | 5 | 750 | 660.5 | 89.5 |

5 | 11 | 56 | 10.26 | 50 | 5 | 750 | 660.5 | 89.5 |

6 | 11 | 67 | 12.28 | 50 | 5 | 750 | 660.5 | 89.5 |

7 | 11 | 78 | 14.29 | 50 | 5 | 750 | 660.5 | 89.5 |

8 | 10 | 88 | 16.30 | 50 | 5 | 750 | 660.5 | 89.5 |

Straight Zone | ||||||||

Zone | Rd (m) | Ω | Sprays | ds′_{ZN}(mm) | dw′_{ZN}(mm) | dnw′_{ZN}(mm) | ||

9 | 18.61 | 55 | 5 | 750 | 630.4 | 119.6 | ||

10 | 20.92 | 55 | 5 | 750 | 630.4 | 119.6 | ||

11 | 23.22 | 55 | 5 | 750 | 630.4 | 119.6 | ||

12 | 25.52 | 55 | 5 | 750 | 630.4 | 119.6 |

Zone | Sprays on the Lateral Direction | Nozzle Diameter (mm) | Ω | D_{bs}(mm) |
---|---|---|---|---|

1 | 5 | 2.5 | 40 | 180 |

2 | 5 | 2.5 | 40 | 180 |

3 | 4 | 2.5 | 45 | 150 |

4 | 4 | 2.5 | 45 | 150 |

5 | 3 | 3 | 30 | 120 |

6 | 3 | 3 | 30 | 120 |

7 | 3 | 3 | 30 | 120 |

8 | 3 | 3 | 30 | 120 |

9 | 3 | 3 | 30 | 120 |

10 | 3 | 3 | 30 | 120 |

11 | 3 | 3 | 30 | 120 |

12 | 3 | 3 | 30 | 120 |

Distance below Meniscus (m) | Temperature Measured (°C) | Temperature Simulated (Lowest) 200 × 200 Mesh (°C) | ΔT (°C) | Temperature Simulated (Highest) 200 × 200 Mesh (°C) | ΔT (°C) |
---|---|---|---|---|---|

2.25 | 1072 | 1065 | −7 | 1080 | 8 |

3.75 | 1111 | 1070 | −41 | 1125 | 14 |

4.8 | 1080 | 1050 | −30 | 1098 | 18 |

5.0 | 1054 | 1017 | −37 | 1089 | 35 |

5.4 | 1050 | 1020 | −30 | 1075 | 25 |

7.0 | 1035 | 995 | −40 | 1052 | 17 |

7.5 | 1033 | 1002 | −31 | 1045 | 12 |

8.0 | 1024 | 1010 | −14 | 1037 | 13 |

8.5 | 1024 | 1012 | −12 | 1031 | 7 |

9.0 | 1022 | 1012 | −10 | 1025 | 3 |

9.5 | 1025 | 1010 | −15 | 1022 | −3 |

10 | 1015 | 1006 | −9 | 1017 | 2 |

10.5 | 1010 | 1004 | −6 | 1014 | 4 |

Distance Below Meniscus (m) | Temperature Measured (°C) | Temperature Simulated (Lowest) 200 × 200 Mesh (°C) | ΔT (°C) | Temperature Simulated (Highest) 200 × 200 Mesh (°C) | ΔT (°C) |
---|---|---|---|---|---|

1 | 975 | 920 | −55 | 1040 | 65 |

2.5 | 1095 | 1040 | −55 | 1140 | 45 |

3.0 | 1095 | 1030 | −65 | 1135 | 40 |

4.1 | 1090 | 1025 | −65 | 1120 | 30 |

4.9 | 1056 | 1001 | −55 | 1108 | 52 |

5.5 | 1045 | 980 | −65 | 1099 | 54 |

6.0 | 1035 | 976 | −59 | 1084 | 49 |

6.5 | 1031 | 970 | −61 | 1068 | 37 |

7.0 | 1035 | 980 | −55 | 1095 | 60 |

7.5 | 1030 | 962 | −68 | 1072 | 42 |

8.0 | 1048 | 995 | −53 | 1051 | 3 |

8.5 | 1025 | 1001 | −24 | 1047 | 22 |

9.0 | 1025 | 1003 | −22 | 1041 | 16 |

9.5 | 1020 | 1002 | −18 | 1034 | 14 |

10 | 1021 | 999 | −22 | 1028 | 7 |

Distance below Meniscus (m) | Temperature Measured (°C) | Temperature Simulated 550 × 100 Mesh (°C) | ΔT (°C) |
---|---|---|---|

24 | 800 | 810 | 10 |

26 | 795 | 807 | 12 |

28 | 850 | 842 | −8 |

30 | 860 | 851 | −9 |

32 | 882 | 874 | −8 |

34 | 890 | 883 | −7 |

36 | 886 | 881 | −5 |

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**MDPI and ACS Style**

Ramírez-López, A.; Dávila-Maldonado, O.; Nájera-Bastida, A.; Morales, R.D.; Rodríguez-Ávila, J.; Muñiz-Valdés, C.R.
Analysis of Non-Symmetrical Heat Transfers during the Casting of Steel Billets and Slabs. *Metals* **2021**, *11*, 1380.
https://doi.org/10.3390/met11091380

**AMA Style**

Ramírez-López A, Dávila-Maldonado O, Nájera-Bastida A, Morales RD, Rodríguez-Ávila J, Muñiz-Valdés CR.
Analysis of Non-Symmetrical Heat Transfers during the Casting of Steel Billets and Slabs. *Metals*. 2021; 11(9):1380.
https://doi.org/10.3390/met11091380

**Chicago/Turabian Style**

Ramírez-López, Adán, Omar Dávila-Maldonado, Alfonso Nájera-Bastida, Rodolfo D. Morales, Jafeth Rodríguez-Ávila, and Carlos Rodrigo Muñiz-Valdés.
2021. "Analysis of Non-Symmetrical Heat Transfers during the Casting of Steel Billets and Slabs" *Metals* 11, no. 9: 1380.
https://doi.org/10.3390/met11091380