The quenching process is one of the most important steps in steel heat treatment and has a significant impact on the quality of the product. In systems with water cooling rings, there are several operational parameters that affect the performance of the heat treatment, such as the longitudinal and angular speeds of the pipe, the water flow rate and pressure. Mathematical models have been developed to predict the temperature profile in the steel pipe, to investigate the causes of defect formation and to allow a better control of the cooling conditions. In the present work, two-dimensional (2D) and tridimensional (3D) models were developed to simulate the heat transfer during the cooling of the pipe. The models were verified, and the tridimensional model was validated by industrial tests. Several simulations have been carried out to compare the 2D and 3D approaches and the effect of the angular speed of the pipe was investigated. The results indicated that the temperatures calculated using the 2D and 3D models are different. It was demonstrated that an increase in the rotation speed of the pipe leads to a more uniform temperature distribution. However, for rotation speeds superior to 6rad/s, no significative effect in the temperature profile is obtained by increasing the speed of rotation.

Steel pipes are used in the automobile and construction industries, in petroleum extraction and in fuel transport. The market for steel pipes has become more and more competitive and the demand for high performance materials has been growing. One of the most crucial steps in the production of pipes with the required mechanical properties such as hardness and mechanical resistance is the heat treatment, which involves controlled cooling of the heated material. One of these processes is the steel quenching that is used to change the steel microstructure from austenite to martensite and to provide the required properties. In the quenching process, the pipe is normally heated to 800–1000°C to guarantee that the entire pipe has an austenitic structure. Subsequently, the pipe is cooled by a fluid that can be air, oil or water, depending on the steel grade. Cooling can be carried out in different types of installations, like air cooling beds, water or oil tanks and water sprays. When water sprays are used, the pipe passes through a system with one or more rings with water jets. The water flow rate and pressure are controlled, and, in some installations, the pipe is rotated. These variables affect the rate of heat transfer and the final properties of the tube.

Inadequate conditions during cooling can cause ovalization and warping of the tube and generate cracks. To avoid these quality problems, it is essential to specify the adequate cooling conditions and to have a precise control of the operational parameters. This specification is usually performed based on previous experience or on experimental tests, which are expensive and affect the productivity of the plant. In this context, mathematical modeling of the heat transfer during cooling can be a powerful tool to reduce the cost of quality.

The first models for steel quenching using water impinging jets were developed for steel plates. The models developed by Chan [1] and Wang et al. [2] are examples of approaches to calculate the heat transfer in steel plates. More recently, Volle et al. [3] developed a model to predict the temperature distribution in a rotating cylinder cooled by water sprays. In this work, the predictions of 1D and 2D models were compared and the effect of the rotation speed was studied. Devynck [4] developed a 2D model for the heat transfer in a steel pipe quenched in a system with six circular modules with water sprays. The rotation of the pipe was neglected, and this simplification probably contributed to increase the errors in the predictions of the model. The heat transfer coefficient in the process was calculated using an inverse heat transfer technique based on experimental tests carried out in laboratory. Chen et al. [5] developed a mathematical model for heat transfer during cooling of a steel tube quenched in an installation with nine cooling rings of water sprays. The predictions of the model matched the experimental measurements and the heat transfer coefficient was obtained as a function of the water flux density.

In the present work, two and tridimensional models were developed. Both models were verified, and the 3D version was validated using industrial tests. To reduce the computational effort, some authors [1–4,10] proposed two-dimensional models, neglecting the heat transfer in the axial direction. In the present work the predictions of two and tridimensional models were compared. The effect of the speed of rotation of the pipe on the temperature distribution was also analyzed.

2MethodologyIn the present work, two and tridimensional models for heat transfer during quenching of steel pipes were developed. The 2D model considers the heat transfer in the radial and angular directions while the 3D model also includes the heat transfer in the axial direction. Both models considered the rotation of the pipe.

2.1Mathematical modelingThe domain for the 2D mathematical model is a slice of the cross section of the pipe and considers the heat transfer in the radial and angular direction. The general energy balance is expressed by Eq. (1):

ρ is the steel density (kg/m3), cp is the specific heat (J/kgK), T is the temperature (K), t is the time (s), vθ is the rotation speed (m/s), r is the radial direction (m), k is the thermal conductivity (W/mK) and θ is the radial direction (rad).The domain for the 3D model is the entire tube, as shown in Fig. 1.

In this case, the partial differential equation included the heat transfer in the axial direction, as seen in Eq. (2):

z is the axial direction (m).To solve these partial differential equations, the domain was divided in control volumes and the equation was discretized using the finite volume method. The explicit technique, described by Patankar [6], was adopted.

The steel properties adopted in the solution of the equations are presented in Table 1.

2.2Initial and boundary conditionsThe initial temperature, Ti, in all control volumes, was considered uniform and equal to the temperature at the exit of the reheating furnace:

For boundary conditions, heat fluxes were specified in all the surfaces of the pipe. These heat fluxes considered the contributions of convection and radiation, using a global heat transfer coefficient, as seen in Eq. (4):

q is the heat flux (W/m2), hglobal is the global heat transfer coefficient including convection and radiation (W/m2K), Ts is the surface temperature (K), T∞ is the temperature of the cooling fluid and of the surroundings (K).The hglobal was specified considering the cooling fluid. For air (internal surface and edges of the tube) it was assumed 150W/m2K [5]. For water (external surface of the tube), values in the range of 4000–8500W/m2K [5] were adopted, depending on the water flux density.

where Re and Ri are the external and internal radius of the tube, respectively, and L is its length.In both, 2D and 3D models, the dimensions of the pipe (diameter, wall thickness and length) and the cooling conditions are specified. Therefore, the model can be used to calculate the heat transfer for different dimensions of the pipe and quenching machines. It also permits to calculate transient temperatures profile along the wall thickness.

2.3Numerical procedureBefore running the simulations, grid and time step independence tests were carried out to determine the adequate number of divisions in each direction and the time step to be used in the simulations. Based on these tests, a mesh with 15, 36 and 50 divisions in the radial, angular and axial direction, respectively, were selected. The time interval adopted in all the simulations was 0.01s.

The computer program developed to solve the differential equations was verified comparing its predictions with situations for which analytical solutions are available. The predictions of the models were also compared to results obtained using the commercial package Ansys-CFX. In both cases, very good agreement was attained.

The tridimensional model was validated using results of industrial tests provided by engineers of the Thermal Department of the Vallourec Research Center in France. They also provided values for the hglobal estimated using an inverse heat transfer technique [3,10].

After verification and validation of the models, several simulations were carried out. To do that, the configuration of the system studied by Chen et al. [5] was considered. In these simulations, the predictions of the 2D and 3D models were compared. The effect of the speed of rotation of the pipe was analyzed.

2.4Configuration of the cooling systemThe system considered in the present work consisted of 9 water cooling boxes. In each water box, there is a cooling ring with water sprays as shown in Fig. 2. The external radius, thickness of the wall and length were, respectively, 139.7mm, 7.72mm and 10m.

Cooling ring [5].

The operational parameters are presented in Table 2.

3Results and discussionIn all the simulations, the initial temperature was fixed at 800°C. This temperature is sufficiently high to achieve the austenite field and to allow the martensitic transformation for steels with carbon content close to the eutectoid composition. It is noteworthy that the eutectoid temperature in the iron - carbon equilibrium diagram is 727°C [11]. During cooling in the water boxes, the global heat transfer coefficient was assumed as 8000W/m2K for the surfaces in contact with the water sprays. It was also considered that the pipe moves 10m after exiting the quenching line.

3.1Variation of the temperatures along the length of the pipeThe purpose of developing two and tridimensional models was to investigate if the introduction of heat transfer in the axial direction would significantly affect the predictions of the temperature profiles. Fig. 3 shows the pipe positions in the quenching line during 20s of simulation, the vector vZ indicates the direction of the pipe movement along the longitudinal direction in the cooling line. Fig. 4 depicts the external surface temperature variation (r=Re) along the length of the pipe at different positions in the cooling system calculated by the 3D model.

After 5s, almost half of the tube had entered in the cooling zone. In the region that is outside the cooling zone, the temperatures are practically uniform. In the region inside the cooling system, the temperatures vary significantly depending on the location of the pipe, in or out of the water boxes. Reheating occurs in the space between the cooling boxes. After 10s, almost all the pipe is inside the quenching machine except for the positions close to 10m, which have higher temperatures. For 15s, part of the tube is already out of the quenching line and presents more uniform temperatures along the length. Finally, after 20s, only the final part of the tube is in the quenching zone. These significant variations of the temperatures can lead to thermal stresses that could cause deformation and cracks in the pipe. These effects are currently being analyzed.

3.2Comparison between the predictions of the two and tridimensional modelsMost of the models found in the literature used a 2D approach to simulate the heat transfer in the pipe. In this case, a two dimensional slice of the cross section of the pipe is analyzed as it moves along the quenching line [3–5]. The differences between the results of the 2D and 3D approaches were analyzed. Fig. 5 shows the predictions for the temperature on the surface using the 2D and 3D models, both for a position close to the head of the pipe, which is the first part to be cooled, and for a position in the middle of the pipe.

It can be noticed that the temperatures calculated by each approach are different. The maximum difference between the temperatures in this simulation reached 140°C. This difference would certainly increase if the pipe were thicker. In this situation, conduction in the axial direction would be more significant.

3.3Effect of the speed of rotationDuring quenching, the pipe is usually rotating. Volle et al. [3] showed the relevance of the rotation of the pipe to equalize the temperatures in the angular direction. The variation of the cooling rate around the pipe affects the microstructure, mechanical properties and the presence of the residual stress. High values of the speed of rotation can promote a uniform cooling, but might increase the cost of the cooling line. Thus, it is interesting to determine a minimum speed of rotation that can provide similar cooling around the circumference of the pipe to guarantee that the entire product achieves the specifications. To investigate this speed rotation effect, the 2D approach was used to simulate a system in which the global heat transfer coefficient was a function of the angular position, as indicated in Eq. (9):

Fig. 6 shows the surface temperatures as a function of the angular position for different speeds of rotation after 5s of cooling.

The surface temperatures when the pipe is not rotating present a significant variation. As the speed of rotation increases, these variations tend to decrease. Above 6rad/s, the effect of rotation on the predicted temperature decreases. The values for 6 and 12rad/s are practically the same and the temperatures are almost uniform along the angular direction. The temperatures in the internal surface and in the middle of the thickness were the same for 6 and 12rad/s. Volle et al. [3] calculated the temperatures during the one water jet cooling process of a rotational cylinder. The results were similar to those of the present work, 6rad/s could promote an uniform cooling along the angular direction. Simulations considering speeds of rotation higher than 6rad/s presented results very similar to those for 6rad/s. Therefore 6rad/s is an adequate value for the speed of rotation.

3.4Validation of the mathematical modelExperimental data from industrial tests carried out at Vallourec Research Center in France were used to validate the predictions of the models. The 3D model developed in the present work was adapted to represent the Vallourec quenching system. This system has 6 modules with 600 nozzles, each row has 20 nozzles and the total length is 9m. The modules are assembled in sequence without separation between them, but the water flux can be controlled on each row separately.

In this system, the pipe is heated to 900–1000°C and transported by foot rolls that have a fixed inclination. During the cooling process, temperatures of the pipe were measured using three type K thermocouples with 1.5mm of diameter that were inserted 1mm below the surface of the pipe and 8m away from the head of the pipe. The external radius, thickness of the wall and length of the pipe were, respectively, 139.7mm, 7.72mm and 14m.

The Vallourec team provided the results of temperature and heat transfer coefficients calculated using an inverse heat transfer model described in [4]. Figs. 7 and 8 compare the predicted and measured temperatures as a function of time inside the cooling line for two tests. In industrial test 1 the initial temperature of the pipe was 905°C and in test 2 it was 907°C. The water temperature was 16°C.

The results demonstrated that the model developed reproduced very well the experimental data in both industrial experiments.

The cooling rate predicted by the model was also compared to the cooling rate obtained in the experimental measurements and a similar behavior was verified.

These results validated the model that can now be used as a tool to adjust cooling conditions and to analyze defect formation.

4Summary and conclusionsIn the present work, two and tridimensional models for conduction heat transfer during quenching of steel pipes by water cooling rings were developed. The predictions of the 3D model were validated using experimental temperature data obtained in plant trials. It was demonstrated that the results of the 2D and 3D models are different, which confirms the relevance of adopting the 3D approach. It was determined that an increase in the rotation speed of the pipe leads to a more uniform temperature distribution, when the water distribution is uneven along the angular direction. Nevertheless, for rotation speeds superior to 6rad/s, no significative improvement in the temperature profile is achieved increasing the rotating speed.

Conflicts of interestThe authors declare no conflicts of interest.

The authors are grateful to the (PPGEM) at UFMG, to the CAPES-PROEX, CAPES and FAPEMIG research programs for their support to the project. They also thank the Vallourec group for supplying the data used in the validation of the model developed.