2015 Volume 55 Issue 5 Pages 976-983
In continuous casting process, solidification should evenly proceed to have as good steel quality as possible. Molten steel starts to solidify in a water-cooled mold to create solidified shell followed by shell growth and termination in a secondary water cooling zone. Visualization of the flow pattern of spray water greatly helps to analyze how to have even shell. Computational fluid dynamics is useful represented by the grid based methods of FVM, FDM, and FEM. However, they are not appropriate for simulation of spray water flow because of complex free surfaces. So, the particle based method of MPS has been applied. A typical roll arrangement was modeled where spray water flow was particularly focused on. As a result, standing water on rolls overflows according to the water flow rate of spray accounted for in this study. Accuracy of the numerical model has been verified by water model experiments equipped by acrylic plates, rolls and spray nozzles. The computational results with a practical condition agreed well with the experimental results. Heat transfer coefficients between water and slab surface were estimated by the calculated results to simulate how solidification proceeded in practice. It was found that uneven water flow significantly affected unevenness in temperature distribution of a slab.
To ensure evenness of solidification in continuous casting process is one of the important technologies for the production of high-quality slabs. In a secondary cooling zone, multiple sprays spaced in width between rolls cool down the surfaces of slabs to solidify molten steels. However, research of cooling by multiple spray units has never been done so far. It is only by using simple spray units1) that have been studied. Therefore, problems of unevenness as for cooling in width by sprays are so difficult to be solved completely because no quantitative analyses have been conducted on the behavior of spray water. In particular, they include dripping water passing through roller bearing portions of split rolls with intricately arranged pattern, and standing water on rolls. In order to support a slab with thinner shell during casting at higher speeds, roll pitches have to be narrower because of weaker shell than at conventional speeds. This may induce an issue to use rollers in smaller diameter resulting in lower rigidity leading to larger deformation for rolls.
Therefore, rolls divided into plural pieces in width have been recently used to have rolls less deflective. However, it is necessary for a couple of roller bearings to be arranged as a result of division. Because of these roller bearings, spray water for slab cooling may unevenly flow downstream through the space caused by roller bearings and a slab.
In order to analyze how this flow pattern affects uneven solidification, it is important to visualize the flow pattern of spray water quantitatively. Computational fluid analysis is effective, however, since the conventional method of employing grids (meshes) is judged to be inadequate for treating complex free surfaces, a particle-based, meshless method2) has been tried in this study.
In fact, this is the first trial to simulate the spray water behavior in a secondary cooling zone in a continuous casting machine by means of the particle-based method. Therefore, the followings were unknown; how large the diameter of particles should be, how to treat the contact angle of water with the slab/rolls and how spray jet shapes influence the fluid flow. That was why these factors were specifically examined in the process of modeling.
Heat transfer coefficients between cooling water and a slab under various conditions were measured using the flow of the spray water obtained from the analyses. Solidification progress of a slab was analyzed with the measured heat transfer coefficients employed as the boundary conditions.
Solidification analyses3,4) have been carried out by one dimensional calculation at the center of a slab and two dimensional analysis by a cross section perpendicular to casting direction. But, quantitative analyses were usually considered to be difficult to clarify unevenness in slab width.
Therefore, a model has been developed considering separate sprays and rolls, position of roller bearings and effect of standing water and dripping water. Finally, the origin of unevenness caused by cooling in slab width could be determined.
Figure 1 shows the surface temperatures of a slab inside the strand of a continuous casting machine of a common vertical bending type.
Measured surface temperature at 18 m below the meniscus.
This was measured by a radiation thermometer with scanning along the width direction, which was installed at 18 m below the meniscus. The slab size was 300 mm × 2200 mm, and the casting speed was 1.0 m/min. Secondary cooling zones of a continuous casting machine typically consist of plural segments with multiple rolls arranged. Thereby, surface of a slab cannot be generally observed except for the spaces between the segments.
Measurement was conducted on the upstream side of an unbent segment with a comparatively large gap due to the extraction of a segment at the time of maintenance. A scanning radiation thermometer of mono-color (measurement wave length: 1.0 μm) was used to measure the surface temperatures. As realized in Fig. 1, the temperatures at the width center of the slab are 100°C lower than those at the vicinity of both edges. Similar tendency can be also seen under the different conditions of casting speeds, steel grade and secondary cooling patterns.
The origins to bring uneven solidification has been so far considered by “effects of fluid flow of molten steel” and “operations saving water at the edge portions to prevent over-cooling in secondary cooling”.5) However, the extreme temperature drop in the width center as seen in Fig. 1 can’t be clarified by the conventional concept.
Accordingly, a study has been performed to understand how uneven cooling takes place. Especially, we focused on the influence of the arrangement of the split roller bearings shown in Fig. 2 along with the behavior of spray water flowing in secondary cooling zones.
Schematic view of the flow pattern of spray water.
The particle-based method (Moving Particle Simulation; MPS)6,7) was applied to calculate the spray water flow pattern. Further, necessary functions were added to the commercial computational fluid dynamics software of Particleworks.8) The key point of the particle-based method is meshless resulting in easily calculating free surfaces. The particle-based method can be classified into two types; one is DEM method used to solve the target particles in analysis as particles, and the other is MPS method applied in this study, that is good at solving a continuum as the calculation points of particles as shown in Fig. 3.
MPS Method and formulation of gradient ϕ.
As for the two formulas of continuation formula and Navier-Stokes formula (Eq. (1)) which are also solved with the finite volume method, the gradient term is discretized using an inter-particle interaction model as shown in Fig. 3.
| (1) |
Where, u, p, ρ, ν and f denote the flow velocity, the pressure, the density, the dynamic viscosity coefficient and the external force (gravity), respectively. Figure 3 displays the discretization of gradient vectors in the physical quantity ϕ at the i-th position of particles. r, d, n0 and w denotes the position of particles, the dimensional number of space, the density of particles and the weight function defined for the influences to be smaller with increasing the distances between particles, respectively. <> denotes the symbol expressing an inter-particle interactional model.
The following three matters were examined in detail before modeling the spray water flow with the particle-based method.
(1) Study of effects of the particle diameter on the spray water flow
Diameter of particles must be defined to calculate the fluid flow by the particle-based method. A smaller diameter of particles induces an enormous number of particles and longer time is needed for calculation. A larger diameter of particles causes the problem that they can not flow into small gaps.
Accordingly, the effects of the various diameters on the flow pattern were analyzed using the model shown in Fig. 4.
Simulation model to set particle diameters (unit: mm).
The flow rate was measured at the positions indicated in Fig. 4 with the respective diameters of particles; 2 mm, 3 mm and 4 mm.
The time average of flow rates (that for 3 seconds between 2 and 5 seconds) and the standard deviation of flow rates at which the cooling water flows in Region A (the edges of the slab) and Region B (the intermediate portion of the roller bearings) are shown in Figs. 5(a) and 5(b), respectively.
Relationship between particle diameters and time-averaged water flow rates or standard deviation of water flow rate at (a) region A and (b) region B defined by Fig. 4.
The results in Region C presented the same behavior as in Region A due to the symmetry of the model.
The minimum gap that allows water to flow is 10 mm between the top plate on the roller bearings and the slab. The average flow rate is not so greatly affected by the diameter of particles, however, the larger the particles becomes, the greater the variation of the flow rate does resulting in unstable flows into gaps. This is also attributed to higher standard deviation of flow rates ranges. The number of particles and calculation time employed for calculation are shown in Fig. 6. The particle diameter was determined as 3 mm after considering the practicability of the calculation time and stability to flow into the narrow gap. The gap of 10 mm between the roller bearing and the slab can accommodate three particles with this condition.
Relationships between particle diameters and number of particles or computational time.
(2) The effects of the contact angle between water and the roll/slab on fluid flow
A study was made to see how the contact angle between cooling water and the roll/slab affects fluid flow focusing on whether the spray water through roller bearings portion flow along the rolls or not.
Analysis was done on the water flow that freely falls down in a space of 10 mm between the roller bearings and the slab using the model shown in Fig. 7(a). There are no remarkable differences between the contact angles of 30° and 60°. (Figs. 7(b) and 7(c)) It is assumed that the flow is dominated by inertia due to the higher flow speed of the water with free fall. Therefore, 30° was applied for every calculation.
Effect of contact angles on water flow. (a) Simulation model. (b) Calculated flow pattern (Contact angle is 30 degree). (c) Calculated flow pattern (Contact angle is 60 degree).
(3) Shape expressions of spray jets
In a secondary cooling, it is preferable to minimize the number of sprays in order to reduce maintenance costs. Therefore, sprays should be only in sufficient number to achieve even cooling within the reachable positions.
For this reason, the nozzles are set to be the spray nozzles expanded in width relative to the casting direction jetting ovally with a large angle. The commercial software used in this analysis did not assume oval jets but common circular jets. Therefore, the program was properly modified to permit independent setting of different angular spreads in two orthogonal directions.
Usually, the density of water droplets in a unit volume has a distribution to some extent within the reachable positions of an oval jet. It is therefore assumed that particles are allowed to be jetted at random angles by providing angles against the casting and width directions. The difference between circular (Fig. 8(a)) and oval jets (Fig. 8(b)) is shown in Fig. 8.
Spray patterns. (a) Circle-shaped pattern. (b) Oval-shaped pattern.
It should be noted that particles do not express actual water droplets but calculation points. The purpose of the present analysis is to simulate the macroscopic behavior of fluid flow after the spray water has impinged with a slab. Therefore the density distribution of water droplets in a unit volume within an oval was not taken into consideration. The above three considerations can suggest the conditions of a particle diameter of ϕ 3 mm, a contact angle of 30° between water and rolls/slab and oval spray jets. The properties of water and the boundary conditions are shown in Table 1.
| Density (kg/m3) | 1000 |
| Dynamic coefficient of viscosity (m2/s) | 1.0×10–6 |
| Coefficient of surface tension (N/m) | 0.072 |
| Boundary condition of wall | Non-slip |
| Turbulence model | None |
Analysis was performed through modeling with a characteristic pattern extracted from roller bearings arranged in the strand; three steps of rolls and two steps of sprays as shown in Fig. 9.
Simulation model for spray water flow between rolls (unit: mm).
The sprays with upper steps of eight pieces and the lower ones of seven pieces were reciprocally arranged along the casting direction. The rolls are divided into three pieces while the intermediates roller bearings are provided in the two places at the central part of the width. The distance between the sprays and the slab, jetting angles of sprays were set as 155 mm, 100° in width and 30° in the casting direction, respectively.
An analysis result arranged as in Fig. 9 is shown in Fig. 10. This is the view from the side of the slab at 5 seconds after the injection of the spray.
Calculated spray water flow (View from slab side).
This output with the flow rate of 20 L/min a spray has revealed that the cooling water jetted onto the slab unevenly flows down as the dripping water. Obviously, it passes through the roller bearings portions with a lot of water standing still on the upper part of the rolls of the downstream side. In addition, some droplets spill over toward the back side. Accuracy of the model was verified by using a water model, in which water was jetted from the sprays placed between acryl pipes attached to an acryl sheet that simulates a slab. Figure 11 shows a typical example of the experimental result. The flow rate for each spray unit was 20 L/min the same as the calculation of Fig. 10. The behavior of the spray water was successfully simulated; water flowed downward passing through the roller bearings along with some water standing still on the rolls. For more quantitative evaluation, comparison was made on the flow rate of water flowing out from the roller bearings portion and the edges of the slab obtained by the calculation with the experiments as shown in Fig. 12.
Measured spray water flow from the water model (View from slab side).
Comparison between the calculated and the measured water flow rates. (a) Water flow rate of each nozzle is 5 L/min. (b) Water flow rate of each nozzle is 10 L/min. (c) Water flow rate of each nozzle is 20 L/min.
At Regions 1-8 defined in Fig. 9, the flow rates obtained by the calculation agreed well with the measurements. Three figures of (a), (b) and (c) display the results with the flow rates per spray piece of 5, 10 and 20 L/min, respectively.
The difference can be seen in the flow rates between the actual measurements and the calculations at Regions 6 and 7 when the water flow rates are higher. This might be attributed to the variation of water amount that overflowed from the system, since the amount of standing water on the rolls varied owing to the slight difference in diameter of the acryl pipes and the rolls of the calculation.
The subsequent analyses were performed only by calculation, because it was proved that fluid flow of spray water could be more accurately analyzed by the particle-based method.
Further analysis was carried out to see the effects of the dripping water stood on the downstream side with increasing the step number of the rolls and spays. The behavior of spray water was analyzed at a flow rate of 20 L/min per spray piece in the model which consists of the five-step rolls and the four-step sprays between the rolls. The results from the slab and roll sides are shown in Figs. 13 and 14 at 5 seconds after the injection of the spray jets, respectively.
Calculated spray water flow (View from slab side).
Calculated spray water flow (View from roll side).
The standing water on the intermediate rolls overflows to the rear side. The water amount becomes larger as it approaches to the downstream side. On the other hand, the outflow becomes larger at the edges of the slab with no significant effects on the flow rate at the roller bearing positions in the vicinity of the central part in width. This means that modeling the characteristic sprays of around two-step allows us to analyze the flow of spray water sufficiently.
This analysis allows us to understand the generation of a large quantity of water kept standing on the intermediate rolls of the split rolls that flows downward from the positions of the roller bearings. A question arises if this behavior of the cooling water affects over-cooling in the vicinity of the central part of slab width. Therefore, solidification analysis was performed taking the spray water flow into consideration.
Solidification calculation was made to understand the solidification conditions of the strand considering above-mentioned dripping water etc.
Before calculation, heat transfer coefficients were measured by the following method. Temperature variation of a heated steel plate cooled by a spray was measured with multiple thermocouples simulating the dripping water and the standing water.9) The procedure was that the plate was heated to 900°C in a furnace under a controlled atmosphere followed by cooled with sprays immediately after being extracted from the furnace. An image during the measurement with a single spray is shown in Fig. 15. Heat transfer coefficients were determined by reverse analysis10) of the measured temperatures of the steel plate. Figure 16 shows the temperature variation of the steel plate as time at the positions of the center and 210 mm away from the center. Figure 17 shows the heat transfer coefficient distribution with a water flow rate of 10.45 L/min when the surface temperature reached 800°C. Reverse analysis of a result makes it possible to calculate heat transfer coefficients as functions of time implying that coefficients can be determined at various surface temperatures. However, at the lower temperatures, in general, obtained heat transfer coefficients are not accurate enough due to the effect of the three-dimensional heat conduction in the steel plate. Accordingly, heat transfer coefficients were measured at a temperature 100°C below the initial temperature of the plate. For instance, when the heat transfer coefficients at 700°C are required, the plate was initially heated to 800°C.
Experimental image (Water is sprayed on the heated steel plate). (Online version in color.)
Measured temperatures at the center and 210 mm from the center of the steel plate.
Heat transfer coefficients (W/m2K) calculated from the measured temperatures at the cooling test. (Online version in color.)
Heat transfer coefficients were thus obtained as functions of the density of sprayed water in a local unit volume, surface temperature of the steel plate and the impingement pressure of the sprays by the experiments under various water flow rates from the sprays in conjunction with various temperatures of the steel plate.
The effect of the dripping water on heat transfer coefficients was determined by providing a water flow from above the sprays simulating the dripping water on the roller bearings.
The amount of dripping water was set referring to the flow rates provided in Fig. 12. In consideration of the effect of water standing on the rolls, cooling experiments were performed with a steel sheet assembled with the steel plate simulating an actual situation.
It was found that the water calmly standing on the rolls did not show significant cooling effects. On the other hand, an important finding was that cooling was facilitated when the standing water was interfered with the spray water, that leads to vigorous agitation. These experiments could successfully determine the heat transfer coefficients in the portions of sprays with and without interference of water standing on the rolls, dripping water, and so forth. Therefore, these values were substituted as the boundary conditions of the solidification calculation in the strand.
The boundary conditions of heat transfer were given to the four separated regions between the rolls. (Fig. 18) I, II, III, and IV denote the roll cooling region, the air cooling region or roller bearing dripping water region, the spray cooling region, and the region of water standing on the rolls or spray dripping water region, respectively. As for the variation of the heat transfer coefficient in width, in the portion (II) of the dripping water interfered with the spray water, heat transfer coefficient was given as 1.1 times higher than that in the portion only with the dripping water. Similarly, heat transfer coefficient in the portion (IV) of the standing water interfered with the spray water was given as 1.5 times higher than that in the portion only with the standing water. These coefficients of 1.1 and 1.5 were determined by the experiments described above. Various phenomena during solidification were calculated by inputting the heat transfer coefficients as the boundary conditions for every region shown in Fig. 18. Here, the two-dimensional sections vertical to the casting direction were considered under a fixed casting speed. The enthalpy method was used for the solidification calculation. Table 2 shows the values of material properties and the boundary conditions. Calculation was succeeded accounting for the effects of the split rolls, the dripping water and standing water created due to the roller bearings and the rolls. As explained above, these effects were identified by the particle-based method providing heat transfer coefficient distributions.
Boundary conditions for the simulation model of solidification.
Simulation model of solidification.
| Density (kg/m3) | 7800 |
| Reference specific thermal conductivity (W/mK) | 59 |
| Reference specific heat (kJ/kgK) | 0.47 |
| Latent heat (kJ/kg) | 260 |
| Reference temperature (°C) | 30 |
| Heat transfer coefficient (W/m2K) | |
| Roll-Slab | 1700 |
| Spray water-Slab | Measured data |
As a result, it was found that there was an over-cooled part at the central portion in the width center of the slab. It could clearly reveal the uneven solidification as shown in Figs. 20(a), 20(b) and 20(c), indicating surface temperature, heat transfer coefficient and solid fraction at the center of the slab in thickness, respectively.
Calculated results of solidification. (a) Surface temperatures. (b) Heat transfer coefficients. (c) Solid fraction at the center of the slab in thickness.
Heat transfer coefficients in Fig. 20(b) are larger in the central portion of the width in the vicinity of a point at 5 m below the meniscus due to the previously mentioned interference between the standing water on the rolls and the spray water. Surface temperatures at 5 m below the meniscus are relatively lower due to this effect. Further, distribution of the center solid fractions in Fig. 20(c) shows that solidification at the center portions complete prior to the vicinity of the slab edges.
Figure 21 shows comparison of the surface temperature between the measurement results obtained by the thermometer and the calculation results. This indicates that the temperatures of the central portions tend to be 100°C lower or less than those of the vicinity of the edge portions. The over-cooled phenomena at the center are considered to be caused by the higher heat transfer coefficients due to the vigorously agitated water originated from the dripping water passing through the roller bearings in the central portion of the split rolls.
Comparison between the measured and the calculated surface temperatures at 18 m below the meniscus.
A study was undertaken to understand how uneven solidification phenomena occur along the width direction of a slab during secondary cooling in continuous casting process. Numerical analyses was performed by the particle-based method to see the behavior of spray water inside the strand. In addition, solidification analysis was conducted taking heat transfer coefficients into consideration, which is affected by the fluid flow of spray water.
The following words conclude this study.
(1) The optimal diameter of particles used for calculation was found to be ϕ3 mm. It was further found that the effects of the contact angle between the water and the roll/slab on the behavior of spray water were negligibly small.
(2) The present simulation by the particle-based method was verified by the experiments using a water model because of good agreement with each other.
(3) The present model successfully simulated the behaviors of dripping water flowing to the downstream side in the roller bearing portion and water standing still on the rolls.
(4) Determinations of heat transfer coefficients were performed accounting for the interference of dripping water with water standing on the rolls leading to vigorous agitation.
(5) Solidification calculation using the measured heat transfer coefficients revealed that the dripping water between the roller bearings and standing water on the rolls contributed to the unevenness along the width direction.
(6) Temperature distribution along the width direction obtained by the analyses well agreed with the measured values by a radiation thermometer.
(7) It was understood that interference of water standing still on the intermediate roll portion among the split rolls with the spray water caused temperature drop in the central portion of the slab width.
The present study was assisted by Professor Seiichi Koshizuka, who majored in the Department of Systems Innovation, Graduate School of Engineering, the University of Tokyo and the staff of Prometech Software Company Ltd. for the particle-based method analysis; Mr. Toshihiro Kawano of Meitec Company Ltd. for construction/processing and visualization of the calculation models; and the staff of Kyoritsu Gokin Company Ltd., for the measurement of heat transfer coefficients. I am sincerely grateful for their contributions.
