2025 Volume 65 Issue 5 Pages 666-675
Flotation dryer systems are widely used to dry liquid layers on substrates such as films, paper and steel strips, and many reports discussing design optimization for better heat transfer characteristics and strip stability are available.
As an advantage of this type of system, surface defects caused by contact between a support roll and the strip are prevented by floating the strip with a jet flow. However, since the friction force between the jet flow and the strip is smaller than that between a support roll and strip, flotation systems are prone to strip walking.
This tendency is noticeable in case of bad shape strip. Thus, it is important to improve the strip centering force. To our knowledge, no systematic in-depth study on prediction of the strip centering force with flotation dryers exists in the literature, and in particular, literature which compares experimental and analytical results is very rare.
In the present study, the centering force acting on a steel strip in a flotation dryer was investigated by experiments and simplified two-dimensional fluid analyses in order to evaluate the influence of the side plate geometry and the off-center value from the center of the floatation dryer on the centering force.
The centering force in the experiment and analysis showed a good correlation. Therefore, it is thought that the centering performance of actual floatation dryers can be estimated by simplified experiments and analyses.
In thin steel strip manufacturing processes, stable strip conveyance is required in order to achieve high quality and production efficiency. Although rolls are generally used to transport thin steel strips, rolls that rotate in contact with a steel strip may cause scratches or roll mark defects on the strip. Because a beautiful surface appearance is a particularly important requirement for thin steel strips used in the outer panels of products such as home appliances and automobiles, these materials must be thoroughly inspected and managed to prevent scratches. In addition, when a chemical solution is applied to a steel strip, contact with the roll is not allowed so as to avoid peeling or damage to the liquid film or soft film before drying. To solve these problems of contact conveyance by rolls, a non-contact conveyance technology, i.e., floater technology,1,2,3,4,5) has been developed, in which the steel strip is floated and conveyed by using the pressure of a gas jet.
Floaters also have the advantage of enabling efficient heat treatment such as heating, cooling and drying simultaneously with strip conveyance by utilizing the high heat transfer coefficient6,7) of the gas jet. On the other hand, floaters have the disadvantage of increased complexity, in that the device design and operational parameters affect transport stability, heat transfer, film quality during drying, energy costs, etc. Therefore, optimization research8) has been conducted using both experiments and analytical models.
Two important issues in strip conveyance by floaters are the stability of floating and strip walking in the width direction. If the floating height is not appropriate and large vertical fluctuations occur, the strip will be damaged by contact with the floater, and since there is no large frictional force due to roll contact during floating, strip walking can occur easily due to the influence of poor strip shape. Regarding flotation, Fujiwara et al.9) showed that the shape of the outer edges of the floater nozzle has a large effect on the flotation force. In another study, Takeda and Watanabe10) conducted an analysis and experiments that took into account the compressibility of air, and showed that self-excited vibration is caused not only by the air flow from the gap between the strip and the floater, but also by the compressibility of air. Where the centering force against strip walking is concerned, Shimokawa et al.11) showed that the centering force could be increased by providing a step on the rib plate installed on the floater surface.
Chang and Moretti12) demonstrated through a simple analytical model and experimental verification that the hydrodynamic characteristics of a floater are determined by four factors: the injection angle of the jet, the thickness of the jet, the width of the surface to which the static pressure for flotation is applied and the static pressure. Kim et al.13) conducted a three-dimensional numerical fluid analysis and experiments, and showed that these factors determine the hydrodynamic properties of the floater.
As described above, the basic characteristics and design of floaters for strip floating and conveying have already been clarified. In order to obtain a more accurate understanding of the floating behavior and centering force of a floater, which has various design elements, it is necessary to conduct full-scale experiments and numerical analyses. However, the overall structure of the floaters used to convey long strips is huge, making it difficult to conduct actual-scale experiments, and numerical analyses are costly and time-consuming.
Therefore, the purpose of this study was to predict the behavior of the strip on a floater and the floater centering force by a simple two-dimensional numerical fluid analysis.14) A two-dimensional numerical fluid analysis model was developed, and its accuracy was evaluated by experiments. As part of this study, an actual-scale numerical analysis using the developed analysis method was also conducted, and the differences between the actual-scale and reduced-scale phenomena were clarified.
As an example, Fig. 1 shows a schematic illustration of a floater drying furnace in which a thin steel strip coated with a chemical solution is continuously conveyed. To avoid damaging the coating film, the steel strip which has been coated with the chemical solution by the coater must not come into contact with the roll until the film is completely dry. Therefore, in the drying oven, the steel strip is floated and conveyed while maintaining a non-contact state by the pressure of the gas jet from the floater, and drying is performed simultaneously with conveying. The length of the floater drying furnace is determined based on the required running speed and the strip drying capacity, and the number of floaters and their installation interval required to support the weight of the strip are determined. For example, as shown at the lower left in Fig. 1, the static pressure between the strip and the floater top plate can be increased efficiently by jetting the flotation gas in countercurrent directions from two slit nozzles, enabling stable floating conveyance. The static pressure can also be increased more stably by arranging baffle plates called side plates and rib plates on the floater top plate to suppress the lateral flow of the gas in the strip width direction.
Next, the mechanism for controlling strip walking, i.e., strip centering, will be explained using Fig. 2, which shows the structure on the floater top plate and the strip in a cross section in the width direction. The side plates on the two sides of the strip are designed to be higher than the rib plates. When the strip position walks to the left from the width-direction center of the floater, the gap (A section in Fig. 2) between the left side plate and the strip edge indicated by the broken line becomes narrower. Then, as the static pressure below the left side of the strip increases, the strip tilts at an angle θ in the direction of the arrow (from the broken line to the position of the solid line). For a more detailed explanation, Fig. 3 shows a schematic diagram of the pressure distribution and flow before the strip tilts. On the left side, the flow path is narrow and it is difficult for the gas to escape, so the static pressure under the strip increases, while on the right side, the flow path is wide and the gas can escape easily, so the static pressure decreases. The strip is tilted by controlling the distribution of the static pressure applied to the lower surface of the strip in this way. In Fig. 2, a force perpendicular to the strip surface acts on the undersurface of the strip due to the static pressure after tilting. Depending on the inclination of the strip, the perpendicular force is divided into a flotation force, which supports the strip’s self-weight, and a horizontal force. This horizontal force acts as a centering force that corrects strip walking, and the strip returns to the center. The centering force varies depending on the design of the floater ribs, side plates, and nozzles.
In order to verify whether the strip floating and walking phenomena caused by the floater can be predicted by a simple numerical analysis, a two-dimensional numerical fluid analysis was carried out using the analysis software ANSYS Fluent 14.15) The analysis model and mesh are shown in Fig. 4. In this analysis, the change in the centering force depending on the height of the floater side plates was investigated.
For comparison with reduced-scale experiments, the two-dimensional analytical model was reduced to 1/10 of the actual size, which is a size at which experiments can be conducted to examine the width-direction cross-sectional structure on the floater top plate and the height of the side plates and rib plates. The strip thickness was also adjusted in this model. For example, if the strip thickness was 1/10 of the thickness of an actual thin steel strip, the strip would be difficult to handle in experiments due to bending and deformation. The analysis conditions were set so as to float the strip with inlet gas injected from below.
The gas inlet velocity from below was set to 0.153 m/s. A numerical analysis was performed under the condition that the steel strip was placed in the center of the floater, as shown in Fig. 4, and this inlet velocity was confirmed to result in stable floating at a floating height of approximately 5 mm.
The flotation height can be set to any value in the full-scale and reduced-scale experiments. However, if the flotation height is too small, there is a risk of contact with the floater, and if the flotation height is too large, flotation will be unstable. Therefore, the height was set to 5 mm, which is about 1/10 of the flotation height that can be expected in a full-scale experiment.
As shown in Fig. 1, the gas jets from the actual floater are injected from narrow slits at an angle close to horizontal. In the two-dimensional numerical flow analysis, the width-direction cross section of the floater is modeled, but because there are limitations to reproducing the floater structure in the strip longitudinal direction, a vertical inflow velocity was set.
The flow that stabilizes the strip at a floating height of 5 mm will stably increase the static pressure without significantly disturbing the flow below the steel strip, and is thought that this flow does not differ significantly from the flow field below the steel strip in the actual phenomena.
The analysis domain was set wide enough to reduce susceptibility to the effects of the domain boundaries, at a distance of 50 mm or more from the floater structure in both the width and height directions, and the surrounding boundaries were set as free inflow and outflow boundaries. The steel strip was treated as a rigid body that could move freely, and the constraints of strip tension were not considered. An overview of the analysis model is shown below.
Physical model: Two-dimensional unsteady incompressible flow
Turbulence model: Realizable k-ε model
Solver: 6-degree-of-freedom solver
Fluid: Air at 30°C
Since the floater pressure was small and the gas velocity was also slow, air was used as an incompressible fluid. The Realizable k-ε model16) was selected as the turbulence model, as it has a proven track record of application to a wide range of flows and can accurately predict the jet spread angle.
The analysis was conducted using a dynamic mesh model,17,18,19) in which the mesh moves, deforms, and is regenerated at each time step so that the steel strip can move freely without being fixed in analytical space. The total number of mesh cells was approximately 60800, although the number varied slightly due to the mesh regeneration caused by the movement of the steel strip.
The steel strip is treated as a rigid body, and deformation is not considered. However, since thin steel strips may warp in the width direction due to variations in the pressure distribution under the strip, it would be preferable to consider strip deformation in the analysis in order to improve analytical accuracy. For example, Huang et al.17) reported that, in a floater analysis that considered strip deformation in the longitudinal direction, deviations occurred in the flow field and pressure field compared to when the strip was assumed to be a rigid body.
The steel strip dimensions were 120 mm in width and 0.26 mm in thickness. Two side plate height conditions were examined, 5 mm and 10 mm.
Here, the strip walking behavior when the initial position of the steel strip was offset in the width direction 10 mm from the floater center and the floating height was 5 mm was analyzed.
3.2. Analysis Results and DiscussionFigure 5 shows the analysis results of the steel strip behavior when the side plate height was 5 mm.
The two-dimensional numerical fluid analysis was able to simulate the process in which the thin steel strip floats up due to the fluid force, tilts, and strip walking is controlled. The steel strip, which was initially shifted to the left, tilted clockwise due to an increase in static pressure on the underside of the left side, where the gap with the side plate was narrow. The analysis successfully simulated the behavior of the steel strip moving to the right due to the horizontal component of the static pressure acting on the undersurface of the strip, in other words, the centering force.
Next, the pressure and velocity distribution will be explained in detail. Figure 6 shows the pressure and velocity distribution at 0.34 s, when the steel strip has moved approximately 3 mm in the width direction from the initial position in Fig. 5. In the central part, it can be seen that the pressure on the undersurface of the steel strip is high, the velocity is slow, and the flow is stable. Because the flow is separated by rib plates arranged in the width direction, the pressure on the underside of the steel strip gradually decreases toward its edges. Focusing on the left and right pressure balance, the pressure is higher on the left side, where the steel strip edge is close to the side plate and the flow path is narrow. This pressure distribution causes the steel strip to tilt, generating a centering force that corrects strip walking. The gas velocity on the underside of the steel strip is slow in the center and increases toward the edge of the steel strip, where the gas flow escapes. It can also be seen that the flow on the left side flows out at higher direction than on the right side due to its closer proximity to the tall side plate.
The history of the steel strip position is shown in Fig. 7. The relationship between the centering force and the steel strip position is shown in Fig. 8. The centering force is the horizontal component of the force calculated from the relationship between the inclination of the steel strip and the pressure acting on its undersurface. In this figure, the center of the floater is denoted as 0 mm, and the initial position of the center of steel strip is denoted as −10 mm. The centering force Fc can be calculated from the geometric relationship shown in Fig. 2 using the following formula.
| (3-1) |
Here, P is the pressure acting on the undersurface of the steel strip, and θ is the inclination angle of the steel strip.
It is clear that the strip corrects to the center faster when a higher side plate is used, and the maximum centering force is approximately 2.15 times larger with the 10 mm side plate height than with the 5 mm side plate. This result shows that the higher side plate suppresses the lateral flow and improves the ability to control strip walking. Thus, this analysis demonstrated that strip walking behavior can be simulated qualitatively and correctly by a simple two-dimensional numerical fluid analysis.
It is thought that the strip walking behavior when the side plate height was changed could be evaluated correctly by the two-dimensional numerical fluid analysis because the rib plates on the floater top and the side plates have a uniform structure in the longitudinal direction. Similarly, in the actual phenomenon, there are thought to be areas where the static pressure under the steel strip maintains a uniform distribution in the longitudinal direction. That is, since there is little change in the longitudinal direction of the steel strip and the phenomenon is limited to the behavior in the width direction, i.e., strip walking, the phenomenon could be simulated successfully by a two-dimensional numerical fluid analysis.
Although dimensional constraints must also be taken into account, these results suggest that two-dimensional numerical fluid analysis methods using a dynamic mesh model may be able to predict not only the behavior of strips supported by floaters, but also the general motion phenomena of other types of objects in three-dimensional flow fields.
Next, in order to investigate the accuracy of the two-dimensional numerical fluid analysis, we devised and carried out an experiment using the analysis conditions and dimensions described in the previous chapter.
However, it would be beneficial if full-scale phenomena could be predicted directly from reduced-scale experiments. Experiments also have the advantage of being able to verify phenomena such as changes in strip tension, oscillation, and longitudinal direction that cannot be considered in a two-dimensional numerical fluid analysis. Therefore, as another objective, we carried out studies and investigations, including the law of similarity, to clarify whether it is possible to predict the centering force of strip walking on a full-scale basis from a reduced-scale experiment.
In this experiment, the geometric dimensions other than the steel strip thickness were adjusted to 1/10 of the actual size, and the conditions were the same as those of the numerical analysis in the previous chapter. The experimental set-up is shown in Fig. 9.
In actual production lines, flotation and conveyance of steel strips are often performed by using multiple floaters. Therefore, the floating support of the strip by the adjacent floaters in the longitudinal direction was simulated by hoisting the strip with a wire, so that only one floater could simulate three floaters arranged in the longitudinal direction of the strip. Although the wire cannot simulate the vertical movement caused by the floater floating the strip, it can ensure freedom of lateral movement in the width direction. In addition, both ends of the steel strip were connected to wires, and the equipment was designed so that any desired tension could be applied in the strip longitudinal direction by using an air cylinder and load cell. The steel strip between the wires forms a catenary curve due to the relationship between tension and gravity, and the strip will sag and come into contact with the floater unless the floater supports it. In other words, in this experiment, it is possible to arbitrarily simulate the weight conditions of a steel strip supported by one floater by adjusting the distance between the two lifting wires and the tension applied in the longitudinal direction.
The catenary curve20) is generally expressed by the following equation, where the strip longitudinal coordinate is x and the gravity coordinate is y.
| (4-1) |
| (4-2) |
Here, C1 and C2 are integral constants, T is strip tension, and M is the weight per unit length of the steel strip. In this experiment, the tension was set so that the steel strip sagging matched that of 1/10 of actual-scale conditions. The floating height of the steel strip was also set in the same way. The internal pressure of the floater at which the steel strip floated stably about 5 mm over the floater top plate was set in the experiment, similar to the numerical analysis conditions. The experimental conditions are shown in Table 1.
| Strip size | Thickness 0.26 × width 120 [mm] |
| Tension | 9.8 × 105 [Pa] |
| Pressure | 600 [Pa] |
| Off-center | 0, 10 [mm] |
| Side plate height | 5, 10 [mm] |
A schematic cross-sectional view of the experimental floater in the strip length and width directions is shown in Fig. 10. The width of the slit gap of the air injection nozzle is 2 mm, and the gas is injected in countercurrent directions from two sides. Rib plates and tall side plates are installed on the floater top. The rib plate height is 2.5 mm, which is the same as in the numerical analysis.
In order to accurately simulate the actual-scale phenomena in a reduced-scale experiment, it is necessary to consider the law of similarity. Geometric similarity was consistent except for the thickness of the steel strip. However, matching the geometric similarities results in deviations in mechanical similarities such as the flow and strip oscillation compared to the full-scale phenomena. Based on the mechanism of floating conveyance of steel strips by gas, the mechanical similarities to be considered in floating by the floater21) are the flow, oscillation and floating similarity. Their differences are discussed below.
The match of the Reynolds number Re, which should be considered in the similarity of the flow, must be excluded because the jet flow velocity is determined by the amount of floating of the steel strip (geometric similarity). It is important to note that there are differences in the flow around the object, the velocity distribution in the boundary layer, and the effects of drag when compared with a full-scale model.
Next, the similarity in the oscillation phenomenon of steel strips is considered. A steel strip on a floater, when fixed at both ends, constantly fluctuates in the direction of gravity due to fluid forces, and the amplitude of the fluctuation increases due to resonance at the natural frequency of the steel strip.22) Therefore, it is important to consider the fluctuation phenomenon as a parameter that should take precedence over the flow around the strip, and to make the Strouhal number St consistent between the full-scale and reduced-scale experiments. The definition of the St number is as follows:
| (4-3) |
Here, f indicates the frequency of the vibration phenomenon in the flow, but in the floating phenomenon in which the steel strip fluctuates due to its natural frequency, the flow field also changes in accordance with the fluctuation of the strip. Therefore, f was replaced with the fluctuation frequency of the strip, i.e., the St number, which is the converted fluctuation frequency.21) H is the characteristic length, which is defined here as the width of the flow path through which the gas flows in the strip width direction, that is, the floating height of the strip, and V is the gas velocity.
To match the St number at the 1/10 scale, if the gas velocity V is the same, the frequency must be 10 times larger.
The fluctuation frequency (i.e., natural frequency, resonant frequency) f of a strip with a length L fixed at both ends can be expressed by the following equation.17)
| (4-4) |
Here, n is an integer, and g is the acceleration due to gravity. On a reduced scale where L and the weight per unit length of the steel strip M are 1/10 of the actual values, f can be increased by a factor of 10 by reducing the tension T to 1/10. However, because T is determined from the geometric similarity and the scale ratio of the steel strip catenary curve, it is difficult to match the St number perfectly. As a supplementary note, since strip tension is not considered in the two-dimensional numerical fluid analysis proposed here, it is not necessary to consider the similarity of the St number between the reduced scale and the actual scale in the analysis.
Next, let us consider the similarity of the flotation phenomenon. It is preferable to set an equal dimensionless Froude number Fr, which expresses the ratio of the inertial force of the gas to the gravity acting on the steel strip.
| (4-5) |
Here, V is the gas velocity, g is gravitational acceleration, and W is the characteristic length of the strip width. As in the case of the Reynolds number, since the jet velocity is determined by the floating height of the steel strip, it is difficult to match the Froude numbers.
Geometric similarity is considered in this experiment, but the fact remains that the mechanical similarity parameters differ from the actual large-scale phenomena. Based on the conditions described above, we evaluated the effect of changing the floater structure on the strip walking centering force.
As in the numerical analysis, the floater side plate height was set to two conditions, 5 mm and 10 mm, and the change in the inclination of the steel strip was examined under two conditions, that is, when the steel strip was set at the center of the floater width and when it was set off-center by 10 mm.
The method for measuring the inclination angle of the steel strip will be explained with reference to Fig. 11, which shows the width-direction cross-sectional structure of the steel strip, the rib plates on the floater top plate, and the side plates. The inclination of the steel strip was calculated from the relationship of the measured distance by arranging two laser distance meters in the strip width direction, using ZX-LD100 laser distance meters manufactured by OMRON Corporation.
Since the steel strip is restrained at both ends in the longitudinal direction, vibrations occur in the thickness direction and width direction, and the inclination angle changes constantly. Therefore, the inclination angle was defined as the average value of a sufficiently long measurement time (30 s, measurement period 0.1 s) for the width-direction oscillation period (approximately 0.4 s), which is longer than the thickness-direction fluctuation period.
The method for evaluating the change in the inclination of the steel strip when the strip is set off-center will be explained with reference to Fig. 12. The inclination angle when the steel strip is placed at the center was a°, the inclination angle when it is placed 10 mm off-center is b°, and (b−a)° is defined as the “Increase of tilt angle” and evaluated. When the steel strip is at the center of the floater, under ideal conditions, the time-averaged angle a° should be nearly zero, although strip fluctuations will occur. However, the longitudinal elongation of actual thin steel strips is distributed unevenly within the strip width due to the rolling and annealing processes, and the strip is not completely flat. In addition, the horizontal angle adjustment of the experimental set-up was limited to about 0.1° to 0.2°. In the reduced-scale experiment, position adjustment was performed in units of mm, but a slight tilt of the strip nevertheless occurred when the strip was positioned at the floater center. Therefore, the influence of the strip inclination at the floater center due to the strip shape and equipment adjustment was disregarded, and the increase of the tilt angle when the strip was off-center, that is, the increase of the centering force (b−a)° was evaluated.
As a supplementary note, in this experiment, the position of the strip in the width direction was fixed by a wire, so the displacement in the width direction could not be evaluated. In order to evaluate the continuous strip walking behavior when a strip moves in the width direction, it will be necessary to improve the experimental equipment, for example, by using a running steel strip instead of a cut steel strip.
4.2. Experimental Results and DiscussionThe history of the steel strip tilt angle is shown in Fig. 13. The figure at the left shows the results when the side plate height was 5 mm, and that at the right shows the results when the side plate height was 10 mm. The tilt angle history is shown for two conditions, one in which the steel strip is placed at the center of the floater, and one in which it is offset 10 mm from the center.
As is clear from this figure, the steel strip is constantly vibrating and fluctuates at various angles at short intervals. The standard deviation of the strip tilt angle was approximately 0.18° to 0.28°. Moreover, even when the strip was positioned at the center of the width direction, an average tilt angle of about 0.7° occurred. As mentioned above, this inclination occurs due to the steel strip shape and initial adjustment. The tilt angle increases when the strip position is shifted 10 mm off-center because the flow path between the side plate and the strip becomes narrower when the strip is off-center, and the static pressure acting on the undersurface of the strip increases on the off-center side. Similarly, increasing the side plate height from 5 mm to 10 mm also significantly increases the tilt angle because the flow path between the side plate and the strip becomes narrower when the strip is off-center, and the static pressure acting on the undersurface of the strip on the off-center side becomes larger.
Next, the results of the tilt angles a° and b° and the Increase of tilt angle b−a° of the steel strip are shown in Fig. 14.
It was found that the tilt angle increased when the steel strip was off-center from the floater center, and the increase of the tilt angle, i.e., the centering force of strip walking, increased by 2.21 times when the side plate height was increased from 5 mm to 10 mm. Here, the tilt angle of the steel strip is proportional to the centering force. Strictly speaking, however, as shown in Fig. 2, the centering force is the weight of the steel strip multiplied by tanθ. In the small angle range, tanθ is substantially proportional to the tilt angle, so it was evaluated based on the tilt angle. In the reduced-scale experiment, the tilt angle of the steel strip was increased by increasing the side plate height, and the increase in the centering force could be quantitatively evaluated.
Furthermore, the centering force of 2.21 times obtained in the reduced-scale experiment is in relatively good quantitative agreement with the 2.15 times obtained in the same-scale numerical analysis. This result indicates the possibility of quantitatively predicting the change in the centering force of the steel strip due to the change in the floater structure with good accuracy by a simple two-dimensional numerical fluid analysis.
The fact that a qualitative evaluation of the centering force is possible has been described in a previous report.14) However, even when the steel strip is positioned in the center of the floater, there is a slight difference in the tilt angle when the side plate height is changed. Considering this point, further improvement in the accuracy of the experimental results may be possible by improving the dimensional accuracy of the experimental equipment. In addition, considering the discrepancy in mechanical similarity in the reduced-scale experiments compared to the full-scale experiments, it may be possible to improve the accuracy of full-scale predictions by adjusting the experimental parameters.
The comparison of the results of the reduced-scale experiments with the analysis results demonstrated the possibility of quantitatively predicting strip walking behavior by a two-dimensional numerical fluid analysis. Therefore, a numerical analysis was carried out to investigate the strip walking behavior at the actual scale. The analysis model and analysis mesh structure are shown in Fig. 15.
The steel strip dimensions were 1200 mm in width and 0.23 mm in thickness. The floater rib plate height was 25 mm. The gas was air at 450°C. The inflow velocity from below the strip was set to 0.254 m/s, at which stable flotation was confirmed at a flotation height of about 50 mm based on a numerical analysis performed under the condition that the steel strip was placed at the center of the floater. The analysis domain was set wide enough to avoid the influence of boundaries, and a distance of 500 mm or more was secured from the structure in both the width and height directions. The total number of meshes was approximately 57600 cells, although the number changed slightly due to mesh merging and regeneration. The other prerequisites and methods of the analysis were the same as those of the two-dimensional numerical fluid analysis described previously.
For the full-scale analysis, the initial position of the steel strip was offset 100 mm to the left from the floater center, and the strip walking behavior was analyzed under two conditions, i.e., with the side plate heights of 50 mm and 100 mm.
5.2. Analysis Results and DiscussionThe strip walking behavior under the 100 mm side plate condition is shown in Fig. 16. It can be seen that the steel strip, which was initially offset, tilts significantly clockwise after 0.4 s, the static pressure acting on the undersurface of the steel strip is distributed into a vertical lifting force and a horizontal centering force, and the centering force causes the steel strip to move to the right after 1.0 s. It was possible to analyze the behavior of steel strip tilting and centering without any problems, even under full-scale conditions.
Figure 17 shows the history of the strip position. Figure 18 shows the relationship between the horizontal component of the force (centering force) calculated from the relationship between the tilt angle of the steel strip and the pressure acting on the undersurface and the steel strip position. In the graph, the initial strip center position is indicated as −100 mm, and the floater center is indicated as 0 mm.
As shown in Fig. 18, strip walking was controlled earlier under the condition of a high side plate height of 100 mm, and the maximum centering force was about 2.48 times larger than that with the side plate height of 50 mm. As a supplementary note, in the previous report,14) the centering force was evaluated at the maximum angle. However, in the numerical analysis, the steel strip moves and the static pressure acting on the lower surface of the steel strip also changes. Therefore, it is more accurate to evaluate the horizontal component of the pressure acting on the undersurface of the steel strip, that is, the centering force shown in Eq. (3-1).
Comparing this result with the result of the 1/10 scale analysis in Fig. 7, the time required for the strip to return to the floater center is longer. This can be explained by the difference in the dimensionless Froude number Fr (Eq. (4-5)).
If the Froude numbers are the same, the time scale of the steel strip floating motion will be the same. In order to make the Froude number consistent with the reduced-scale analysis at full-scale, where the width of the steel strip is 10 times larger, the gas velocity needs to be multiplied by √10. However, the gas velocity in the full-scale analysis is less than twice that in the reduced-scale analysis, and the Froude number is smaller, and for this reason, the phenomenon occurs on a longer time scale than in the reduced-scale analysis.
Figure 19 shows the increase in the centering force when the side plate height is doubled in the reduced-scale experiment, reduced-scale analysis, and full-scale analysis. The increase of 2.48 times in the full-scale analysis is larger than the 2.15 times in the reduced-scale analysis.
The reason for this may be considered from the perspective of the change in the peak value of the centering force. In the reduced-scale analysis in Fig. 8, the maximum value of the centering force is clearly seen at the side plate height of 5 mm. However, at the side plate height of 50 mm in the full-scale analysis in Fig. 18, the maximum value is not clear, and the steel strip moves while maintaining a centering force close to the peak value.
As outlined above, the time scale in the reduced-scale analysis is short, and the change in the inclination of the steel strip also occurs in a short time. Therefore, it can be said that the steel strip tends to tilt more due to the fast acceleration and large inertial force. There is evidence that not only the static pressure acting on the undersurface of the steel strip, but also the inertial force of the change in inclination has an effect on the strip behavior. Figure 20 shows the pressure distribution acting on the undersurface of the steel strip at maximum inclination. The values of the steel strip width and pressure were normalized them by the maximum values under each condition. In this figure, the 0 position in the width direction is the steel strip edge on the offset side, and the 1.0 position is the steel strip edge on the floater center side.
In the reduced-scale analysis, the pressure distribution at 0.08 s, when the angle is at its maximum, shows a decrease in pressure on the offset side (left side). Although the steel strip tilted significantly due to inertial force, the static pressure distribution acting on its undersurface was already unable to maintain the tilt, so the tilt then decreased rapidly, and the maximum value of the centering force appeared clearly.
In contrast, in the full-scale analysis, the pressure distribution at 0.30 s, when the angle was at its maximum, maintained a high pressure on the offset side (left side). Therefore, it can be said that the steel strip moved while maintaining its inclination, without significant tilting by inertial force. Due to these characteristics when the side plate height was 50 mm, the maximum tilt angle, in fact, the maximum centering force, was limited to a small value. As a result, it is considered that the increase rate of the maximum centering force when the side plate height is increased is larger in the full-scale analysis than in the reduced-scale analysis.
When verifying the centering force of the floater, it is necessary to consider the time scale of the steel strip movement and tilt change, as well as the influence of inertial forces. Therefore, a full-scale evaluation is considered appropriate. In this respect, match of the Froude numbers is important for predicting the floater phenomena.
A two-dimensional numerical fluid analysis model that can easily predict the centering force of thin steel strips during floating conveyance was developed. A comparative verification was carried out by a 1/10 scale numerical analysis and experiments, and it was shown that the developed numerical analysis model can provide an accurate, quantitative evaluation of the floater centering force. The centering force of a full-scale floater was also predicted using the developed numerical analysis model. The conclusions obtained are as follows:
(1) A two-dimensional numerical fluid analysis model, which can analyze the behavior of steel strip floating, tilting, and centering by gas injection from a floater, was developed by using a dynamic mesh model that can handle mesh movement, deformation, and regeneration. The change in the centering force depending on the floater side plate height was evaluated by a numerical analysis using a 1/10 scale model of the actual-size floater. Increasing the side plate height from 5 mm to 10 mm improved the maximum centering force by 2.15 times.
(2) To investigate the accuracy of the numerical analysis, an experiment was conducted using a 1/10 scale floater to evaluate the change of the centering force depending on the floater side plate height. In this experiment, the centering force was improved by 2.21 times by increasing the side plate height from 5 mm to 10 mm.
(3) The two-dimensional numerical fluid analysis showed good quantitative agreement with the experimental results regarding the centering force when the side plate height was changed, demonstrating that a quantitative evaluation of the centering force is possible by using the developed two-dimensional numerical fluid analysis model.
(4) Using the developed two-dimensional numerical fluid analysis, the change of the centering force due to a change in the side plate height of a full-scale floater was evaluated. In this analysis, the maximum centering force was improved by 2.48 times by increasing the side plate height from 50 mm to 100 mm.
(5) As one problem of reduced-scale experiments, it is difficult to predict the centering force at full-scale due to in the mismatch of the similarity law. However, as advantages of the experimental method, it is possible to verify the effect of longitudinal strip tension, which is difficult to consider in a two-dimensional numerical fluid analysis, and the experimental approach is effective for elucidating the centering behavior of the floater. A comparison of the reduced-scale and full-scale analyses was also carried out. The results showed that the difference in the time scale and inertial force affects the strip walking behavior in an analysis of strip floating phenomena, indicating the importance of matching the Froude number as a similarity law.
