JP4804841B2 - Method for producing strip material by continuous casting using melt spinning method - Google Patents

Description

  The present invention relates to a method for producing a strip-shaped material such as a metal strip by a melt spinning type continuous casting machine. This belt-like substance is widely applied industrially as various machine parts and as various energy conversion equipment parts.

  The melt spinning method is widely applied in the manufacturing industry as a continuous casting method capable of producing a relatively thin strip material at a high speed. The formation mechanism of the plate thickness in the melt spinning method will be described with reference to FIG.

  The melt 2 flowing in from the inflow boundary 6 and sprayed from the nozzle 4 solidifies from the cooling roll 5 side by coming into contact with the rotating cooling roll 5. A liquid phase of the melt 2 formed as a puddle between the nozzle 4 and the cooling roll 5, a part of the solidified phase 3 in the vicinity of the liquid phase, and a solidification interface 7 that is a boundary between the solidified phase 3 and the liquid phase Are collectively referred to as paddle 1 by those skilled in the art.

  The liquid phase of the paddle 1 is surrounded by the upstream free surface 8, the downstream free surface 9 and the solidification interface 7. In the most downstream part of the paddle 1, the solidified phase 3 grown from the cooling roll 5 side reaches the downstream free surface and becomes a complete belt-like substance. Hereinafter, the solidified phase 3 grown from the cooling roll 5 side will be referred to as a band-shaped substance 3 and described. The solidified phase thickness L0 at this position has a strong relationship with the finally obtained product plate thickness L1. The factors that determine the plate thickness are conventionally the gap G between the outlet of the nozzle 4 and the cooling roll 5, the inflow pressure Pin that is the melt pressure at the nozzle 4, and the inflow that is the melt temperature at the nozzle 4. It has been said that there are plate thickness affecting factors such as the temperature Tin, the speed uR of the cooling roll 5, and the cooling characteristic value of the cooling roll 5 (for example, the cooling water temperature Tw of the cooling roll 5). In many cases, the strip-like material obtained by this production method is required to have high thickness accuracy. For this reason, various methods for plate thickness stabilization and plate thickness control have been proposed.

  In “Patent Document 1” below, the thickness immediately after being formed on the cooling roll 5 is measured through measurement of the gap G, and the deviation from the target thickness value is corrected for the gap G or the inflow pressure Pin. A feedback control method for correcting by this is shown. In addition, a method for independently correcting the speed uR of the cooling roll 5 and the cooling water temperature Tw of the cooling roll 5 is also known.

  Conventionally, the PID control method shown in the following “Patent Document 2” or a simple method based on the PID control method has been employed as a method for calculating the correction amount of the plate thickness affecting factor. In the case of these methods, the thickness deviation amount obtained from the thickness gauge (for example, in Patent Document 2, the melt weight change rate, ie, the melt flow rate actual measurement value under the assumption that the plate width is constant, (There is a one-to-one correspondence with the measured plate thickness), and there is no physically valid method to calculate how much the plate thickness influencing factor is corrected. The gain was only set.

  However, there are innumerable combinations between the thickness influencing factors of the thickness influencing factor group that satisfy the same thickness, and changes in each factor are inevitable during melt spinning operation. It is not appropriate to apply the feedback control gain value adjusted in step 1 in common to other operating conditions with different conditions of the plate thickness influencing factors. For this reason, in the prior art, it is necessary to set the feedback gain in PID control to be extremely small in order to avoid excessive thickness variation, and it takes a long time from occurrence of thickness deviation to completion of correction due to control response delay. was there.

  In this method, in principle, it is indispensable to use a highly accurate thickness measurement value, and a generally expensive thickness gauge must be installed. Furthermore, even if the plate thickness can be measured, for example, plate thickness measurement generally required for products, for example, that always satisfies the accuracy of 1 μm or less is generally difficult in the vicinity of the cooling roll 5 where vibration is severe, and low It is necessary to manufacture high-precision products, or to install high-precision plate thickness far from the cooling roll, which has a better measurement environment. When a thickness gauge is installed far from the cooling roll 5, the problem of control response delay becomes more serious.

  In addition, a technique is disclosed in which the plate thickness is stabilized by controlling a specific factor among the plate thickness influencing factors to be constant instead of using the plate thickness measurement value. For example, in the following “Patent Document 3”, the cooling water temperature Tw of the cooling roll 5 is controlled so as to make the surface temperature of the cooling roll 5 constant in order to make the gap G which is one of the plate thickness influencing factors constant. The technique is shown. However, this method is indirect from the viewpoint of plate thickness control, and even if a plate thickness deviation occurs due to the influence of uncontrolled plate thickness influencing factors, it cannot be corrected. There was a problem that it was difficult to control the plate thickness.

  Furthermore, as a method of changing the plate thickness without the plate thickness measurement value, the gap G (described in the prior art) and the inflow pressure Pin shown in the following “Patent Document 4” are corrected at specific times, Attempts have also been made to control the thickness in order to obtain a desired thickness transition. However, in this method, since the relationship between the plate thickness influencing factor and the plate thickness is not modeled, there is a problem that the obtained plate thickness largely fluctuates due to variation in the plate thickness influencing factor for each operation.

  In the following “Non-Patent Document 1”, an offline experimental regression equation for calculating the plate thickness from the operation factor is proposed. Further, in the following “Non-Patent Document 2”, the paddle 1 is regarded as an area surrounded by two free surfaces on the upstream side and the downstream side, the surface of the cooling roll, the nozzle outlet, and the outflow boundary of the solidified substance. Attempts are being made to calculate the plate thickness tendency in the melt spinning method by a numerical analysis method that expresses the solidification phenomenon by continuously increasing the viscosity coefficient of the melt as the temperature drops. However, these methods are not directly applied to the plate thickness control, and the method applied to the control is not disclosed.

JP-A-10-128504 Japanese Unexamined Patent Publication No. 1-448652 JP 58-35047 A JP 2002-283008 A Fiedler, H .; J. et al. of Material Science, Vol9. , P3229-3235 (1984). Kawahara, H .; Other, Modeling of Casting Welding and Advanced Solidification Processes VII, Minerals Materials Soc. , P833-840 (1995) J. et al. M.M. McCormick et al., Numerical calculation program by FORTRAN IV / 77, Science, 1970 Computational Fluid Dynamics Editorial Committee, Moving Boundary Flow Analysis, The University of Tokyo Press, 1995 Toshiaki Nakae, Rheological Engineering and its Applied Technology, Fuji Techno System, 2001

  Therefore, in the present invention, a melt spinning method for manufacturing a strip material having a small thickness deviation by correcting the control value of the thickness influence factor at high speed and with high accuracy without necessarily requiring a thickness gauge. It aims at providing the manufacturing method of the strip | belt-shaped substance by the used continuous casting.

  As a result of research on melt spinning by the present inventors, the following solution has been invented.

According to a first aspect of the present invention, a melt continuously jetted onto a cooling roll rotating from a nozzle of a casting machine forms a paddle on the surface of the cooling roll, and the melt in the paddle is connected to the cooling roll. In the method for producing a strip material by continuous casting using a melt spinning method that solidifies by contact cooling and moves along with the cooling roll and continuously forms a solid strip material, it is a plate thickness influencing factor of the strip material The gap between the outlet of the nozzle and the cooling roll, the melt pressure at the nozzle, the melt temperature at the nozzle, the peripheral speed of the cooling roll, and the cooling characteristic value of the cooling roll are continuously obtained. A calculation step of continuously measuring during casting and calculating the thickness of the strip material at the downstream end of the paddle according to a predetermined function using the measured value as an input value, and a target of the product thickness of the strip material The value Thermal expansion coefficient of the strip material, or to calculate the thickness reduction amount by the thermal expansion coefficient and a predetermined function using the tension applied to the strip material, the plate thickness reduction amount of the product thickness of the strip material target A conversion step of subtracting the value into a target value of the plate thickness of the strip material at the downstream end of the paddle, and the strip material at the downstream end of the paddle calculated in the calculation step Feed forward control is performed by adjusting one or more selected from the plate thickness influencing factors so that the plate thickness becomes the target value of the plate thickness after conversion in the conversion step. This is a method for producing a strip material by continuous casting using a melt spinning method.

The difference between the first invention and the prior art will be described.
First, the first difference will be described.
In the prior art, a plate thickness meter is generally required to control the plate thickness. However, there is a problem that the apparatus is expensive and inevitably complicated. Among the conventional technologies, the method of indirectly controlling the plate thickness by controlling a specific plate thickness affecting factor or the method of changing the plate thickness affecting factor unconditionally at a specific timing increases the plate thickness in principle. It cannot be controlled to accuracy.

  On the other hand, in the first invention, since the plate thickness is calculated by covering all the main plate thickness influencing factors, complete plate thickness feedforward control is possible, and feedback from the plate thickness meter is not necessarily required. Therefore, in the first invention, a thickness gauge is not necessarily required. In the first invention, it is necessary to measure or estimate each plate thickness affecting factor during operation. However, the apparatus for this purpose is essentially essential for operation management (for example, a cooling roll perimeter), or is generally less expensive than a plate thickness meter, and therefore the entire system according to the first invention. Is less expensive than the conventional method using a thickness gauge.

Next, the second difference will be described.
In the prior art, there is no idea of correcting a plurality of plate thickness influencing factors at the same time, and only a single plate thickness influencing factor is always corrected. This is disadvantageous in the following points.
For example, when the inflow pressure Pin decreases and the plate thickness becomes equal to or less than the target value, in the related art, the flow rate is generally increased by increasing the gap G with high responsiveness instead of the inflow pressure Pin with low responsiveness. Is increased to increase the plate thickness. By the way, there are restrictions on the combinations between the plate thickness influencing factors that are allowed to form the paddle stably. The fact that the paddle is stable means that the distance between the paddle upstream free surface and the downstream free surface can be constantly maintained at a constant distance.

  The correction of the gap G alone is an effective operation in the case of a combination of thickness influencing factor groups far from the paddle stability limit, but in the case of a combination of thickness influencing factor groups close to the limit of paddle stability, The operation exceeds the constraints of the plate thickness influencing factor group and may cause the paddle to collapse. In this way, in the prior art, only a single plate thickness influencing factor can be manipulated. Therefore, in the vicinity of the paddle stability limit, is it possible to correct the gap G and take the risk of paddle collapse, or leave it uncontrolled? There was only one of the two options.

  On the other hand, in the first invention, a plurality of plate thickness influencing factors can be corrected simultaneously. For this reason, in the first aspect of the invention, in operation near the limit of the paddle stability, for example, the inflow pressure Pin is gradually increased and the cooling roll rotation speed, which is relatively quick, is decreased. The plate thickness is quickly corrected to the vicinity of the target value, and the cooling roll rotational speed is gradually returned to the original speed as the inflow pressure Pin thereafter increases. By doing so, it is possible to ensure the thickness accuracy while maintaining the stability of the paddle. That is, in the vicinity of the limit of paddle formation, it is important to simultaneously correct a plurality of plate thickness influencing factors in order to control the plate thickness while ensuring the stability of the paddle. This fact has been found by the present inventors.

  As the “predetermined function”, a known method, for example, a plate thickness interpolation factor may be obtained by multiple regression equations of the plate thickness influencing factors and used. In addition, as a method of correcting the plate thickness change after calculating the plate thickness of the strip material at the downstream end of the paddle, for example, continuously measuring the tension on the plate by a winder, the actual tension value and the actual measurement An instantaneous plate thickness change correction value corresponding to the instantaneous tension during operation may be calculated using an empirical model formula obtained from the relationship between the plate thickness change rates.

According to a second aspect of the present invention, the melt continuously jetted onto the cooling roll rotating from the nozzle of the casting machine forms a paddle on the surface of the cooling roll, and the melt in the paddle is connected to the cooling roll. In the method for producing a strip material by continuous casting using a melt spinning method that solidifies by contact cooling and moves along with the cooling roll and continuously forms a solid strip material, it is a plate thickness influencing factor of the strip material The gap between the outlet of the nozzle and the cooling roll, the melt pressure at the nozzle, the melt temperature at the nozzle, the peripheral speed of the cooling roll, and the cooling characteristic value of the cooling roll are continuously obtained. A calculation step of continuously measuring during casting and calculating the thickness of the strip material at the downstream end of the paddle according to a predetermined function using the measured value as an input value, and a target of the product thickness of the strip material The value Thermal expansion coefficient of the strip material, or to calculate the thickness reduction amount by the thermal expansion coefficient and a predetermined function using the tension applied to the strip material, the plate thickness reduction amount of the product thickness of the strip material target A conversion step of subtracting the value into a target value of the plate thickness of the strip material at the downstream end of the paddle, and the strip material at the downstream end of the paddle calculated in the calculation step And adjusting one or more selected from the plate thickness influencing factors so that the difference between the plate thickness and the plate thickness of the band-shaped material measured by the meter is within a predetermined target range, and feedback This is a method for producing a strip-like material by continuous casting using a melt spinning method.

The difference between the second invention and the prior art will be described.
First, the first difference will be described.
In the prior art, the plate thickness is mainly controlled by feedback control. In this feedback plate thickness control method, the quantitative relationship between the plate thickness influencing factors and the plate thickness was not clearly conscious. For this reason, an excessively small feedback gain is always used in order to prevent the control from being overcompensated even in the condition of the highest sensitivity of the thickness variation with respect to the feedback correction amount among the combinations of the thickness affecting factor groups. There was a problem that control response was slow. Even in the prior art, there is an example of using a relational expression between a specific thickness influencing factor and the thickness and using it for control, but the influence of other thickness influencing factors is ignored, so the prediction accuracy of these relational expressions is It was low and could not realize a large feedback gain.

  On the other hand, in the second invention, all the main plate thickness influencing factors are covered, and the plate thickness is calculated using all the measured values or estimated values of these plate thickness influencing factors during operation. For this reason, the accuracy of predicting the plate thickness is high, and the feedback gain when performing feedback control can be set large.

Next, the second difference will be described.
Similar to the first invention, in the second invention, a plurality of plate thickness control factors can be corrected simultaneously, so that the plate thickness accuracy can be ensured while maintaining the paddle stability.

According to a third aspect of the present invention, as the predetermined function used when calculating the plate thickness of the strip material at the downstream end of the paddle in the calculation step, the melt pressure at the nozzle that is the plate thickness affecting factor, And, using the plate thickness affecting factor of one type or two or more types selected from other plate thickness affecting factors other than this, the following formula:
Melt flow _{casting machine} = function _{casting machine} (melt pressure at nozzle, other factors affecting plate thickness)
Melt flow rate _paddle = Function _paddle (melt pressure at nozzle, other factors affecting plate thickness)
(Here, the “function _{casting machine} ” uses “melt pressure at the nozzle” and “other thickness influencing factors” (may be plural) as independent variables, and the flow rate of the melt at the nozzle “melt The “function _paddle ” is a function that represents “ flow _{casting machine} ”, and “melt pressure at the nozzle” and “other plate thickness influencing factors” (which may be plural) are independent variables. (This is a function representing the melt flow rate “melt flow rate _paddle. ” The above relational expressions are referred to as a flow rate characteristic function on the casting machine side and a flow rate characteristic function of the paddle, respectively.)
In the casting machine side flow rate function and the paddle flow rate characteristic function expressed independently of each other, and for the specific combination in the input value of the plate thickness influencing factor, the casting machine side flow rate characteristic The melt flow rate when the function value coincides with the flow rate characteristic function value of the paddle is the melt flow rate in the combination of the plate thickness affecting factor groups, and the melt flow rate in the combination of the plate thickness affecting factor groups A method for producing a strip-like material by continuous casting using the melt spinning method according to the first or second invention, wherein a function for calculating the plate thickness at the downstream end of the paddle is used.

    According to a fourth aspect of the present invention, as a calculation method of the flow characteristic function of the paddle, in the combination of the selected thickness affecting factor groups, numerical calculation is performed by modeling behavior in the paddle for a plurality of different values. Calculate the flow rate value and the strip thickness of each of the different values inside the paddle, from each combination of the strip thickness influencing factor groups different from each other and the strip thickness calculation value of the strip corresponding to each combination, According to a third aspect of the present invention, a regression equation of a plate thickness influencing factor group related to a plate thickness of the strip material in a flow rate characteristic function inside the paddle is obtained, and the regression equation is used as a flow rate characteristic function of the paddle. It is the manufacturing method of the strip | belt-shaped substance by continuous casting using the melt spinning method of this.

  Note that “the thickness of the strip material in the flow characteristic function inside the paddle from the calculated thickness values of the strip materials corresponding to the combinations of the thickness affecting factor groups different from each other” As a method of “determining the regression equation of the influential factor group”, a known method such as a polygonal line approximation method or an approximation method using a spline function can be applied.

  According to a fifth aspect of the present invention, in the numerical calculation modeling the behavior in the paddle, the region in the paddle is divided into two regions, a liquid phase and a solidification phase, and the behavior in the region is modeled independently in each region. The velocity of the liquid phase and the solidification phase coincides at the solidification interface that is the boundary between the liquid phase and the solidification phase, and the solidification interface stress of the liquid phase and the solidification phase at the solidification interface. The boundary condition of the solidification interface having the same size and opposite directions is obtained by repeated calculation, and the thickness of the strip material when the boundary condition at the solidification interface is satisfied in both the liquid phase and the solidification phase is determined by the paddle. It is used as a plate thickness of the strip material in an internal flow rate characteristic function, and is a method for producing a strip material by continuous casting using the melt spinning method according to the fourth invention

  Here, the cooling roll cooling characteristic value is an index for expressing the influence of cooling roll cooling on the strip material thickness, for example, the roll cooling water temperature, the heat transfer coefficient value between the cooling roll and the melt, These are the heat resistance value between the chill roll and the melt, the heat flux value on the chill roll, and the winding temperature of the belt-like substance.

  In addition, simply expressing “plate thickness” means the thickness of the strip material at the most downstream part of the paddle, and expressing “product thickness” means the thickness of the strip material plate at the time of winding. That is. In general, the plate thickness and the product plate thickness do not match due to effects such as a difference in thermal expansion and a reduction in the plate thickness due to the winding machine tension.

  The flow rate characteristic function is a function that relates the pressure at a predetermined reference point to the flow rate. Assuming that the solidification phase velocity in the paddle is approximately equal to the roll peripheral speed value, when the plate width is constant, [plate thickness] = [flow rate] / [roll peripheral speed], the flow rate characteristic function is the pressure It can also be expressed as a relational expression between thickness and thickness.

  According to the present invention, by applying to a melt spinning method using a melt spinning type continuous casting machine, a thickness gauge is not necessarily required, and a thickness affecting factor can be corrected at high speed and with high accuracy. A strip-shaped substance with a small thickness deviation can be produced.

  Next, an embodiment of a method for producing a strip material by continuous casting using the melt spinning method of the present invention will be described.

  First, a specific embodiment of the first invention will be described.

(Device configuration)
FIG. 2 is a schematic configuration diagram of the casting machine. The first invention will be described with reference to FIG.
The melt 2 stored in the melt reservoir 10 is given a positive pressure at the outlet of the nozzle 4 by the application of an atmospheric pressure by a pressurizing device 15 such as a head of the liquid itself or a compressor, and is ejected onto the cooling roll 5. , Paddle 1 is formed. The band-shaped material 3 solidified in the paddle further decreases in temperature due to contact with the cooling roll 5 while moving downstream along with the cooling roll 5. The belt-like substance 3 is peeled off from the cooling roll 5 by a tension applied by the winder 11 at a certain point downstream and taken up by the winder 11.

  The melt reservoir 10 is provided with an inflow thermometer 20 such as a thermocouple and an inflow pressure gauge 21 such as a diaphragm, and measures an inflow temperature Tin and an inflow pressure Pin, respectively. The melt reservoir 10 can be kept warm and heated by a heater 34. The cooling roll 5 is driven by a variable speed cooling roll motor 13, and a cooling roll peripheral speed meter 19 such as a rotary encoder is installed. The entire melt reservoir 10 is movable in the direction perpendicular to the surface of the cooling roll 5 by the lifting device 16, and the gap G can be changed. The value of the gap G is measured by a gap meter 18 such as a transmitted light detector.

  The cooling roll 5 is cooled on the roll surface or inside the roll by a cooling roll cooling device 17 such as a water cooling nozzle, and the amount of cooling water is variable by a flow rate adjusting valve (not shown). The band-shaped material 3 peeled off from the cooling roll 5 is temperature-measured by a winding thermometer 22 such as a radiation thermometer before reaching the winder 11.

  With the above configuration, plate thickness feed forward control is possible. When the thickness feedback control is performed, a thickness gauge 23 is installed. Further, from the viewpoint of equipment costs, the pressurizing device 15, the heater 34, and the cooling roll cooling device 17 may be omitted. In the case where these devices are omitted, the following description of the control method corresponds to setting the correction amount of the control end to 0. Further, a gap G is calculated by substituting the gap meter 18 by using a device (such as a cooling roll thermometer (not shown)) for measuring the amount of expansion of the cooling roll and the position (setting value or actual value) of the lifting device 16. Also good.

(Thickness control method)
First, feedforward control will be described.
The product plate thickness target value of the band-shaped material 3 is determined by the band material temperature difference between the paddle 1 and the winder 11 (defined by the temperature difference between the winding thermometer 22 and the inflow thermometer 20). A value obtained by adding a value obtained by multiplying the linear expansion coefficient and further subtracting the thickness reduction amount due to the tension of the winder 11 obtained by an experiment is set as a thickness target value at the downstream end of the paddle 1. Hereinafter, when simply described as “plate thickness target value”, it is defined as that at the downstream end of the paddle 1.

  During casting operation, inflow thermometer 20, inflow pressure gauge 21, cooling roll peripheral speed meter 19, gap meter 18, and inflow temperature Tin, inflow pressure Pin, cooling roll input from winding thermometer 22 to arithmetic unit 12 Using the measured values of the speed uR, the gap G, and the winding temperature Tc of 5, the roll cooling characteristic value, the strip material product plate thickness calculation value hcal, and the plate thickness influence coefficient εi of each plate thickness influence factor are calculated. 12 is calculated.

  As the roll cooling characteristic value, for example, when the thermal resistance Hr between the cooling roll 5 and the solidified surface is used, it can be calculated from each input measurement value to the arithmetic unit 12 by developing the following equation for the thermal resistance Hr. Cp uses equivalent specific heat in consideration of latent heat.

dT _sol / dt = q _out / (ρ · Cp · h) (Formula 1)
q _out = 1 / Hr · (T _{sol, surf} −T _{rol, surf} ) [Equation 2]
Here, the boundary conditions are shown below.
T _sol = Tin (t = 0)
T _sol = Tc (t: winding temperature measurement time)
T _sol : Average temperature in the radial direction of the cooling roll of the strip material T _rol : Cooling roll temperature qout: Heat flux ρ: Band material density Cp: Strip material specific heat h: Strip material thickness Subscript surf: Interface between cooling roll and solidified phase

  As the strip material thickness h in Formula 1, the strip product product thickness calculation value hcal at that time expressed using the thermal resistance Hr is used. In this case, the strip thickness h becomes an implicit expression that becomes an unknown number shared by a plurality of equations, but this is solved by solving a general equation such as the Gauss-Jordan method shown in Non-Patent Document 3 above. The strip material thickness h can be obtained.

  Further, the thermal resistance Hr has a slow change over time compared to other plate thickness influence coefficients (because the thermal resistance Hr is the influential factor of the true contact area ratio between the plate and the roll, and the change in the roll surface properties. Roll surface properties change due to roll surface wear and wear during operation.In general, roll materials that do not wear or wear abruptly are selected for stable operation. Therefore, the time change rate of the true roll contact area due to roll wear and wear during operation is relatively small, and as a result, the time change rate of the thermal resistance Hr is also relatively small). 2 using an additional measuring instrument not shown in FIG. 2, for example, a winder weigh scale, on the premise of a constant plate width, from the history of past weight and cooling roll peripheral speed, Thickness h may be calculated

  In this case, it is not necessary to solve the strip material thickness h implicitly. Even in the conventional method, there was a method for obtaining the strip material thickness h from the change in the weight of the strip material, but in these methods, the obtained strip material thickness h is immediately regarded as the actually measured strip material thickness, Used for feedback control. On the other hand, the strip material thickness h obtained from the weight change in the present invention is merely used as an auxiliary to calculate the thermal resistance Hr, and the strip material product thickness used as a control object. The calculated value hcal differs from the conventional method in that the calculated value calculated from the total plate thickness influence coefficient including the thermal resistance Hr is used.

The thickness influence coefficient εi of the thickness influence factor is defined by the following equation 3.
εi = ∂h / ∂xi (Equation 3)
x: Plate thickness influencing factor Subscript i: Number of each plate thickness influencing factor

  Next, since the difference Δhcal between the strip material product plate thickness calculated value hcal and the target plate thickness value haim is generally not 0, each plate thickness affecting factor correction amount Δxi for correcting this is calculated.

The difference Δhcal can be expressed by the following equation 4.
Δhcal = Σ (βi · εi · Δxi) [Formula 4]
βi: Arbitrary value

  Here, since the plate thickness influencing factors can be changed independently of each other, there are innumerable combinations of arbitrary values βi satisfying Expression 4. Therefore, among these combinations, the most appropriate combination is adopted.

The specific method is as follows.
As a first step, prioritize the plate thickness influencing factors in the order in which they are easy to operate (determined as appropriate in consideration of engineering constraints and operational limitations of specific equipment / specific workers). Set.

As a second stage, it is considered that the difference Δhcal is satisfied only by the plate thickness influencing factor having the highest priority (for example, the gap G). That is, it is the following formula 5.
Δhcal = εi1 · Δxi1 (Equation 5)
i1: i with the highest priority

  As a third stage, xi1 to which Δxi1 in Formula 5 is corrected is a paddle formation limit range obtained by mapping paddle formation conditions that have been empirically or analytically obtained in advance for each combination of plate thickness influencing factor conditions. Check if it exists inside the. This limit range also takes into account the influence of the values of other plate thickness influencing factors at that time. If it is within the limit range, Δxi1 is output as a correction amount.

  As the fourth stage, if the correction result is out of the limit range, the second and third priority orders are repeated until the same check is satisfied.

  As a fifth step, when the final priority order is reached and the paddle formation limit range is not reached, a plurality of plate thickness affecting factors are simultaneously corrected according to the priority order to search for a correction amount so as to be within the paddle limit range. .

  When setting the above priority order, it is not always necessary to fix the same priority order. For example, standard conditions for each plate thickness influencing factor may be determined in advance, a priority order may be set in descending order from the standard value at a certain time, and used for control at that time. In addition, the plate thickness affecting factors that should not be corrected from the viewpoint of operation stability may be excluded from the priority order for correction and not corrected.

  Finally, the correction amount Δxi of each plate thickness influencing factor is corrected to calculate the actual correction amount, and the pressurizing device 15, the heater 34, the cooling roll motor 13, the winder motor 14, the lifting device 16, and The output to the cooling roll cooling device 17 is corrected. The reason for correcting the correction amount Δxi is that if the instantaneous correction amount is output as it is, an unstable phenomenon such as plate thickness self-excited vibration may occur, and the control is performed by correcting the raw correction amount. Stabilize. As a technique, general PID control or the like for Δh may be applied. When instability such as self-excited vibration is not recognized, proportional control with a gain of 1 may be used.

Next, feedback control will be described.
The plate thickness actual value haq is added as an input of the arithmetic unit 12 to the feedforward control. The plate thickness actual value haq is corrected to a value corresponding to the downstream end of the paddle according to the measurement position. Hereinafter, the plate thickness actual value haq is regarded as a value at the downstream end of the paddle 1. For the strip material thickness h required for calculating the thermal resistance Hr, the actual measured thickness haq is used. The other calculation methods of the strip material product plate thickness calculation value hcal and the plate thickness influence coefficient εi are the same as the feedforward control. As a method for calculating the correction amount of each plate thickness affecting factor, for example, Δh = haq−haim may be used, and the same processing as the feedforward control may be performed. Moreover, you may use the general learning control which correct | amends sequentially the model coefficient of a plate | board thickness prediction formula so that it may become haq-hcal = 0.

Next, the third invention will be described.
In the first invention, if it is attempted to obtain the relationship between the combination of the main plate thickness influencing factors and the plate thickness for every casting machine condition and every paddle condition, it takes a lot of labor and is uneconomical. Therefore, in the third invention, the flow rate characteristic on the casting machine side and the flow rate characteristic on the paddle side are separated with the nozzle 4 of the casting machine as a boundary, and the plate thickness according to the conditions of the plate thickness influencing factor group is independently evaluated. Thus, the labor required for the study can be greatly reduced.

  Here, the key point is the flow characteristics of the casting machine under the casting conditions of the actual machine that must be considered as a unit from the casting machine (which is defined as the upstream part including the nozzle 4 outlet) to the paddle 1. This is a method that combines the flow characteristics of the paddle. In the third aspect of the invention, the casting machine flow rate characteristic and the paddle flow rate characteristic are independently expressed as a function of the flow rate at the nozzle and the melt pressure as shown in the following equations 6 and 7.

Melt flow _{casting machine} = function _{casting machine} (melt pressure at nozzle, other plate thickness influencing factors) ... [Formula 6]
Melt flow rate _paddle = function _paddle (melt pressure at nozzle, other plate thickness influencing factors) ... [Formula 7]

The expression of Equation 6 indicates that the “function _{casting machine} ” uses “melt pressure at the nozzle” and “other thickness influencing factors” (may be plural) as independent variables, and the flow rate of the melt at the nozzle 4. This is a function representing the “melt flow rate _{casting machine} ”, and the same applies to Equation 7. Formula 6 is called a casting machine flow rate characteristic function, and Formula 7 is called a paddle flow rate characteristic function.

  For example, the paddle flow rate characteristic curve of FIG. 3 becomes a specific curve when “other plate thickness affecting factor” in Equation 7 is fixed to a specific value. At this time, when the pressure P1 is set in FIG. 3, the melt flow rate Q1 at the nozzle position is determined from Equation 7.

  “Other plate thickness influencing factors” in Equations 6 and 7 are one or more arbitrary combinations selected from the above plate thickness influencing factors excluding the melt pressure at the nozzle 4. 6 and Equation 7 are generally different combinations of plate thickness influencing factors. In the same combination of plate thickness influencing factors (some influencing factors are used in equations 6 and 7), the flow rate matching conditions, that is, the casting machine flow characteristic function and the paddle flow characteristic function in FIG. Let the flow rate and melt pressure at the intersection of the points be the point where the operation is established (operation point). The flow rate value and the melt pressure value at this point are regarded as the calculated values of the flow rate and the melt pressure at the nozzle 4 in the combination of the plate thickness affecting factors. In general, since the opening size of the nozzle 4 is fixed during operation, the flow rate at the nozzle 4 can be easily converted into a plate thickness if the solidification phase velocity at the downstream end of the paddle material 3 is known. The final strip-shaped product thickness is obtained by adding the thickness change due to deformation in the solidification phase to the thickness at the operating point.

This will be specifically described below.
First, the casting machine flow characteristic function will be described.
From the head of the melt 2 that can be expressed in terms of the static pressure of the melt 2 in the melt reservoir 10 (the pressure can be increased if a pressurizer is present) and the height of the liquid level, from the nozzle 4 What reduced pressure loss becomes the pressure (total pressure) at the nozzle 4. When the total pressure at the inlet of the casting machine is constant, the pressure loss in the flow path increases as the flow rate of the nozzle 4 increases due to the characteristics of the viscous fluid, so the total pressure at the nozzle 4 decreases.

  Therefore, in FIG. 3, the casting machine flow rate function is a curve that rises to the left starting from the coordinate point (inflow pressure = total pressure at the inlet of the casting machine, flow rate at the nozzle = 0). A specific curve shape can be determined by a small number of experiments by conducting a discharge experiment of the melt 2 and obtaining a pressure loss coefficient. If the melt physical property value changes due to a change in the melt temperature or the like, the pressure loss characteristic changes, so the flow rate characteristic function also changes. In addition, not all of the plate thickness influencing factors affect the casting machine flow characteristics. For example, the peripheral speed of the cooling roll does not directly affect the flow rate characteristic function of the casting machine. Therefore, this thickness influence factor may be ignored when the flow rate characteristic function is constructed.

Next, the paddle flow characteristic function will be described.
From many experimental results, it is found that the melt thickness properties at the nozzle 4 (for example, density, viscosity coefficient, specific heat, thermal conductivity, etc.), nozzle pressure, cooling roll peripheral speed, and cooling roll cooling characteristics are constant. Has been confirmed to be constant. Therefore, it can be expected that the relationship between the nozzle flow rate (= plate thickness / plate width / cooling roll peripheral speed) and the nozzle pressure, that is, the paddle flow rate characteristic function will be constant (in the case of the same physical properties and the same cooling conditions).

  Here, when the pressure distribution of the paddle 1 is sufficiently small, the paddle 1 cannot be formed, so that the flow rate = 0. On the other hand, when the pressure of the paddle 1 is large, a large paddle 1 can be formed, in which the solidified phase can grow thick, so that the plate thickness is large, that is, the nozzle flow rate is large. Therefore, the paddle flow rate characteristic function becomes a curve that rises to the right in FIG. The paddle flow rate characteristic function also changes depending on the difference in physical properties and cooling roll cooling conditions.

A method for obtaining the operating point from the casting machine flow rate characteristic function and the paddle flow rate characteristic function will be described.
In FIG. 3, since the nozzle flow rate and the nozzle pressure coincide at the intersection of both function curves, operation is possible only at this point, and the intersection of both function curves is the operation point. These flow rate characteristic functions are previously created off-line in advance, and in online setting / control, values can be easily derived simply by referring to formulas or tables.

Next, the fourth invention will be described.
In calculating the paddle flow rate characteristic function, it is necessary to know the plate thickness corresponding to a specific condition of the plate thickness affecting factor group. For this purpose, it is a reliable method to conduct experiments under a number of conditions. As the plate thickness experimental regression equation based on the plate thickness influencing factor obtained by such a policy, the non-patent document 1 is famous, and it is considered that there is no problem with the qualitative tendency.

  However, since the regression equation in Non-Patent Document 1 quantitatively has an error of several tens% or more of the plate thickness, the plate thickness prediction with high accuracy required by the plate thickness feedforward control of the first invention is performed. It cannot be applied as an expression. The reason for the large prediction error in the experimental regression equation is that it is difficult in the experimental technique to maintain the combination of the plate thickness affecting factor groups at a predetermined value in the experiment. Such errors can be reduced by conducting a number of experiments and performing statistical processing, but casting experiments are generally expensive and time-consuming, so it is economically unreasonable to conduct a huge number of experiments. is there. Further, since the behavior in the paddle is complicated as a transport phenomenon, it is impossible to solve the flow in the paddle by an analytical expression.

  Therefore, in the fourth invention, the plate thickness corresponding to the individual conditions of the plate thickness affecting factor group is calculated using a numerical calculation method that can easily fix the plate thickness affecting factor. In principle, numerical calculation methods tend to have less variation in results than in experiments, so it is economical to use a small amount of results in comparison with experiments and obtain regression equations with less variation. . As a numerical calculation method, the inside of the paddle is described by the transport equation of mass, momentum, and energy of the continuum, and solidification is modeled together. By converting these equations into discrete equations in the paddle and solving them by applying a finite element method or the like, the flow field, temperature field, and solidification distribution in the paddle can be calculated. The plate thickness is given from the calculation result as the solidified phase thickness at the most downstream portion.

  For a free interface consisting of a free surface (upstream and downstream) existing in the paddle and a solidification interface, for example, the general ALE method shown in Non-Patent Document 4 described above (through deformation of the calculation mesh, the free surface shape) By applying one of the methods for expressing the free interface position, the free interface position can be calculated. As the transport equation of the continuum, the fluid phase is assumed to be a viscous fluid, and the solid phase is applied to general transport equations such as high-viscosity viscous fluids, viscoplastic bodies, rigid plastic bodies, and elastoplastic bodies. can do. For the solidification model, general methods such as a method of increasing specific heat at the solidification temperature, an enthalpy method, and a method of absorbing latent heat at the solidification interface can be applied.

  The regression equation using the inflow pressure Pin as a parameter is obtained by fixing the conditions other than the inflow pressure Pin by using a plurality of combinations of the individual conditions and the thickness of the plate thickness influencing factor group. As a conversion formula for converting the plate thickness into the flow rate, one paddle flow rate characteristic function is used.

  Next, a condition other than the inflow pressure Pin is fixed to a condition different from the previous one, and another paddle flow rate characteristic function is similarly obtained. The flow rate characteristic function in an arbitrary combination of the plate thickness influencing factor groups is obtained by interpolation from the plate thickness influencing factor group obtained by the calculation under the condition closest to the combination of the plate thickness influencing factor groups. As this interpolation method, a known technique, for example, a linear interpolation method can be used.

The reason why the object to be returned must be a paddle flow rate characteristic function will be described below.
For example, in principle there is a method for obtaining a paddle characteristic equation by simply performing multiple regression from the calculation result without going through the paddle flow rate characteristic function, and obtaining a point where this characteristic equation and the casting machine flow rate characteristic function coincide with each other. . However, in such a method, depending on the calculation condition setting method, there may be a case where a plate thickness influencing factor other than the inflow pressure Pin cannot obtain a positive correlation between the inflow pressure Pin and the nozzle flow rate in a specific case. This is because when the operating point is to be obtained under such conditions, the intersection of the casting machine flow rate characteristic function and the paddle characteristic equation is likely to occur at an unrealistic point and is a problem.

  Further, in the present invention, in the case of a combination of thickness influencing factor groups in which paddle 1 cannot exist stably, the solution by numerical calculation diverges, and thus such a combination of thickness influencing factor groups cannot be paddle-formed. A paddle stabilization condition range in which a physically reasonable convergence solution exists is obtained as a condition. This paddle stabilization condition range is used when calculating the plate thickness affecting factor correction amount in the first invention.

Next, a fifth invention will be described.
Although the numerical calculation method for the flow in the paddle shown in the fourth invention is an effective plate thickness calculation method, in order to satisfy the highly accurate plate thickness prediction required for the present invention, the numerical calculation method is also used. Special consideration is required. For example, in the case of a method in which the entire paddle including the solidified phase is regarded as a uniform viscous flow (however, a continuous change in physical property value due to temperature is considered), the present inventor cannot obtain sufficient plate thickness accuracy. Found. The following is a list of problems with accuracy.

  First, in the conventional method, there is a problem in which no consideration has been given to discontinuities in pressure and velocity at the solidification interface. This will be specifically described.

  For the sake of simplicity, the solidification phase is regarded as a viscous flow with high viscosity. The mass conservation equation and the transport equation in the incompressible steady state are expressed by the following Equation 8 and Equation 9 throughout the paddle.

<Mass conservation type>
∂ (ρu _i ) / ∂x _i = 0 (Equation 8)
<Momentum transport type>
ρ · u _j ∂u _i / ∂x _j = −∂P / ∂x _i + ∂ (μ · (∂u _i / ∂x _j + ∂u _j / ∂x _i )) / ∂x _j . Equation 9]
ρ: Density u: Speed x: Length P: Pressure μ: Viscosity coefficient Subscript i, j: Coordinate axis direction number

Here, the conditions to be satisfied at the solidification interface are as shown in the following formulas 10 to 12.
ρ _L u ′ _L = ρ _s u ′ _s [Equation 10]
-P _L + 2 · μ _L ∂u ' _L / ∂x'
= −P _s + 2 · μ _s ∂u ′ _s / ∂x ′ (Equation 11)
μL (∂u _L '/ ∂x "+ ∂u _L " / ∂x')
= -Μs (∂u _s '/ ∂x "+ ∂u _s " / ∂x') (12)
u ': Solidification interface vertical velocity x': Solidification interface vertical length u ": Solidification interface parallel velocity x": Solidification interface parallel length Subscript L: Liquid phase Subscript s: Solidification phase

In the case of a general substance whose viscosity coefficient changes discontinuously due to solidification, μ _L << μ _s , and ∂u / ∂x is determined by the surrounding flow field, so ∂u ′ _L / ∂x ′ >> It is not necessarily ∂u ' _s / ∂x'. Therefore, from Equation 11, on the solidification interface, the pressure is generally different between the liquid phase side and the solidification phase side. That is, P _L ≠ P _s on the solidification interface.

For example, in the rapid solidification continuous casting machine shown in FIG. 4 where the solidified phase grows on the cooling roll 5 while being constrained by the rotating cooling roll 5 and the free slip boundary 24, the band-shaped substance 3 is subjected to plastic deformation processing. Since it solidifies, ∂u ′ _s / ∂x ′> 0, and on the solidification phase side of the solidification interface, the pressure value is several orders of magnitude larger than the liquid phase pressure, so the pressure discontinuity at the solidification interface cannot be ignored. It can be.

Further, since the material density generally changes due to solidification, ρ _L ≠ ρ _s , and from Equation 10, the speed must be discontinuous between the liquid phase and the solidified phase at the solidification interface. That is, u _L ≠ u _s on the solidification interface. However, in the solidification analysis of the conventional continuous casting, there is no one that clearly recognizes the discontinuity at the solidification interface at the pressure P and the velocity u and has taken measures to avoid it, and generates some error at the solidification interface. It was inevitable.

  For example, a method for solving a flow field, a temperature field, and a solidification state field (hereinafter collectively referred to as a state field) using a finite volume method with a liquid phase and a solidification phase coexisting in the same calculation region is as follows. Although it has often been used for numerical calculation of continuous casting, there are the following model error generation factors.

  First, in the case of the discretization method (coloke method) in which both the pressure P and the velocity u are most frequently used in the finite volume method, the solidification interface is defined at the cell boundary, and the equation of motion of the cell When solving, it is necessary to interpolate the pressure at the solidification interface. In the finite volume method, this pressure interpolation value is common to both side cells (liquid phase and solidification phase, respectively) at the cell boundary, so the pressure discontinuity at the solidification interface is ignored, and a large model error occurs. appear.

  Second, the method of defining the velocity u and the pressure P at different positions of the cell (staggered method) is a method of setting the velocity u on the solidification interface without interpolating the pressure on the solidification interface. Since it is not necessary to define the pressure above, the influence of the pressure discontinuity can be reduced as compared with the first method. However, in this method, since the stress acting on the basis of the speed difference of the speed u distributed on the solidification interface is almost ignored, generation of a model error due to this influence is inevitable. Furthermore, in this method, since the velocity u on the solidification interface must be shared by the liquid phase and the solidification phase, the velocity discontinuity due to density change due to solidification cannot be expressed, and a model error occurs.

  Third, as a solidification model, a method for expressing a solidification phase by applying a volume force of Cu · (u−uc) to a region corresponding to the solidification phase using a target velocity uc in the solidification phase and an arbitrary pressure loss coefficient Cu. Has often been adopted. However, in this method, since there is no general method for giving the target velocity uc in the solidification phase, a study system other than a study system that guarantees that the target velocity uc in the solidification phase is constant (for example, uc = uR), For example, in the system of FIG. 4, it is not even possible to evaluate the model error. Therefore, the application of this method is not appropriate.

  Similarly, a method of solving the state field using the finite element method in which the liquid phase and the solidified phase coexist in the same calculation region is also frequently used. This method also has the following model error problem at the following solidification interface.

  First, in the method (pressure continuous primitive function method) that defines the velocity u and the pressure P on the node of the finite element and solves both the velocity u and the pressure P, which is generally recommended as high accuracy in the finite element method. , There is a node that defines the pressure P on the solidification interface (in the finite element method, the solidification interface is generally located at the finite element boundary, so there is a node on the solidification interface), and there is a liquid on the solidification interface. The same pressure value is shared through nodes shared by the phase finite element and the solidification phase finite element. As a result, the discontinuity of the pressure P at the solidification interface is ignored and a model error occurs. Specifically, in the case of FIG. 4, the pressure on the solidified sea surface of the solidified phase is supposed to be >> 0, but in this method, the pressure of the solidified interface is approximately zero.

  Second, a penalty method that removes the pressure P from the equation of motion may also be applied. In the case of this method, since the pressure P does not exist, the problem regarding the discontinuity of the pressure does not occur. However, in the penalty method, the physical meaning of the penalty coefficient introduced into the equation of motion instead of pressure is poor, and the penalty coefficient value is highly arbitrary, so the validity of the analysis result is often questioned. As a numerical calculation method for quantitative prediction, the problem of basic model error is large in the first place.

  Thirdly, a separate solution method is often used in which the pressure P is not directly solved with the velocity u, but the Poisson equation of the pressure P is separated and solved in conjunction with the equation of motion. In this method, the pressure value of the adjacent finite element is required at the stage of solving the Poisson equation of pressure, and in the discrete equation between adjacent cells across the solidification interface, a huge pressure difference must be used. This kind of processing is not intrinsically physical (since the Poisson equation of pressure P is implicitly assumed to be uniform in the physical property value), a model error occurs. There is a problem to do.

  Furthermore, since all of the first to third methods share the velocity u among the nodes on the solidification interface, a model error due to the discontinuity of the velocity u always occurs.

  Therefore, in the fifth invention, as shown in FIG. 5, at the time of continuous casting, the calculation area is separated into a liquid phase calculation area 29 and a solid phase calculation area 28, and after performing numerical calculation in each area, Connect to explore the final state field. Thereby, the problem of the discontinuity of the velocity u and the pressure P at the solidification interface can be avoided, and the state field can be predicted with high accuracy.

The specific procedure is as follows.
As a first step, as shown in FIG. 5, the calculation region is separated into a liquid phase calculation region 29 and a solidification phase calculation region 28 with the solidification interface as a boundary, and the solidification interface is set as a boundary between the calculation regions.

  As the second stage, the state field in the solidification phase calculation region 28 shown in FIG. 6 is numerically calculated. Here, the inflow boundary 26 is fixed, and the boundary conditions are as shown in the following Expressions 13 to 15.

σ = Pm _L [Equation 13]
τ = μ _L (∂u _L "/ ∂x '+ ∂u _L ' / ∂x") | _{solidification interface} ... [Formula 14]
T _s = Tw (um ′ _s ) (Equation 15)
σ: Solidification interface vertical stress Pm _L : Liquidus pressure at solidification interface τ: Solidification interface horizontal stress Tw: Solidification temperature um ′: Speed in solidification interface vertical direction (solidification rate)

Here, the solidification temperature Tw is a function of the solidification rate um ′, which reflects that the influence of freezing point depression cannot be ignored during rapid solidification. The melting point may be set. As the boundary condition at the contact surface between the outflow boundary 25 and the cooling roll 5, the discretization formula, and the like may be used. For example, the finite element method using the pressure continuous primitive function method, which is a highly accurate calculation method, may be used, with the outflow boundary condition being a constant pressure and the surface boundary condition of the cooling roll 5 being a constant temperature. As a result of the numerical calculation, the velocity at the solidification interface (combined vertical direction and horizontal direction of the solidification interface) um _s and heat flux qm _s are obtained.

  As a third stage, numerical calculation of the state field in the liquid phase calculation region shown in FIG. 7 is performed. Using the state field calculation result in the solidification phase, the boundary condition at the outflow boundary 27 is given by the following equations 16 and 17.

um _L = ρ _s / ρ _L · um _s [Equation 16]
qm _L = qm _s −γ · ρ _L · Cp _L · um ′ _L (Equation 17)
γ: Solidification latent heat

For the inflow boundary condition, the boundary condition on the contact surface with the cooling roll 5, and the discretization formula, the conventional method may be applied as it is. For example, the inflow boundary condition may be a constant pressure, the roll heat flux distribution may be specified as the boundary condition on the surface of the cooling roll 5, and the finite element method using the pressure continuous primitive function method may be used. The solidification interface temperature obtained as a result of the numerical calculation in this way is generally different from the solidification temperature Tw (um ′ _s ) and does not satisfy the conditions to be satisfied by the solidification interface.

Therefore, the position of the solidification interface is corrected so that Tm _L = Tw (um ′ _s ). As a method for correcting the solidification interface, for example, the ALE method is used to limit the moving direction of the node on the solidification interface to the main flow direction. Assuming the system of FIG. 4, when Tm _L > Tw (um ′ _s ), the direction of increasing heat input to the liquid phase, that is, the direction of extending the solidification interface length to make the solidification interface slope more gentle. In the case of the opposite temperature condition, the interface position satisfying Tm _L = Tw (um ′ _s ) can be found by correcting the solidification interface so that the slope of the solidification interface becomes steeper. it can.

  As the fourth stage, based on the corrected solidification interface, the numerical calculation of the solidified phase is performed again in the same way as in the second stage, and using the result, the numerical calculation of the liquid phase is continued as in the third stage. To implement. The above-described numerical calculation of the solidification phase and the liquid phase is repeated until the correction amount of the solidification interface position is equal to or less than a predetermined allowable value. The state fields of the liquid phase and the solidified phase thus obtained are physically valid because they satisfy all the given transport equations and boundary conditions.

Instead of the above method (first method), the following second method may be used as a cooperation method for numerical calculation of the solidified phase and the liquid phase. In the third stage, qm _L obtained from the numerical calculation result of the solidified phase is not directly used as the boundary condition of the liquid phase numerical calculation, but the solidification interface boundary condition in the liquid phase numerical calculation is constant temperature (Tw). And the solidification interface position is corrected so that the heat flux calculated at the solidification interface becomes the target heat flux qm _L.

  The solutions obtained by the first method and the second method are the same, but in a system with a large freezing point depression, that is, a system with a large Tw variation, the numerical stability of the first method is better. In small systems, the stability of the second method tends to be good.

  The first point of the fifth invention is that the velocity u and pressure P at the solidification interface are different values in the liquid phase numerical calculation and the solidification phase numerical calculation, respectively. The point is that continuity can be expressed without problems.

  In the calculation of the solidification interface position in the conventional continuous casting numerical calculation, regarding the heat transfer, set the liquid phase and the solidification phase in the same calculation region (use different constitutive equations for the liquid phase and the solidification phase, respectively) It was common knowledge to search for the solidification interface position in the process of solving all the temperature fields around the liquid phase and the solidification phase (even when solving the equation of motion separately).

  On the other hand, in the second point of the fifth invention, the liquid phase numerical calculation and the solidification phase numerical calculation are performed independently of each other, and the coupling between them is simply the velocity on the solidification interface, the heat flux, In addition, only the solidification interface shape needs to be exchanged with each other, and calculation efficiency is improved in that the amount of information required in the coupled partner calculation area is significantly reduced. That is, when searching for a solidification interface by liquid phase numerical calculation, there is no need for temperature field information inside the solidification phase, and it is only necessary to know only the heat at the solidification phase side at the solidification interface and the velocity at the solidification interface.

  Further, in the present embodiment, the solidification temperature is set to a single temperature assuming a pure substance or a quenching material. A general model to be used may be used. Even in the case of assuming a semi-solid phase, in the case of a general substance, the viscosity rises to the same level as the solid phase, which is limited to a solidification rate close to 100%. If the point is replaced, the above method can be applied as it is.

  Also in melt spinning, a tension is applied to the band-shaped material 3 and a stress such as a residual stress due to shrinkage during cooling of the solidified phase is applied. For this reason, the flow in the solidification phase cannot be regarded as a uniform flow by melt spinning, and a technique for avoiding the discontinuity of the pressure P and the velocity u at the solidification interface as in FIG. 4 is necessary. The basic concept of the technique for this can be dealt with by the method described above, in which the liquid phase and the solidified phase are separated and numerical calculation is performed, and the two are coupled. However, in melt spinning, since a free interface exists, special measures in numerical calculation are required in addition to the system of FIG.

FIG. 8 shows an example in which the concept of the fifth invention applied to the system of FIG. 4 is simply applied to melt spinning.
In this case, the solidification interface is in contact with the two free surfaces on the upstream side and the downstream side. In principle, if the solidification interface is searched while the position of the contact point between the interfaces is moved two-dimensionally, a converged solution should be obtained as in the case of FIG.

  However, as shown in FIG. 8, it is known that the numerical calculation of a region surrounded by an interface where a plurality of moving interfaces that are longer than the plate thickness are in contact with each other becomes extremely unstable. For this reason, even if a solution as shown in FIG. 8 is obtained, it is difficult to apply it from the viewpoint of economy because it requires an enormous calculation time. Therefore, in the fifth invention improved for melt spinning, as shown in FIG. 9, the fixed boundary 30 is sandwiched between the solidification interface and the two free surfaces, and the solidification interface and the free surface are not in direct contact with each other. The numerical instability can be reduced.

  However, when a fixed boundary is installed between the solidification interface and the upstream free surface, the solidification phase should originally develop on this fixed boundary, but the fixed boundary itself cannot be deformed. Adversely affects the prediction of sheet thickness. Therefore, this problem is avoided by using the following solidification model in a region where solidification can occur where the fixed boundary and the solidification interface are combined. In this solidification model, solidification is not expressed only by the shape of the solidification interface, but the true solidification surface shape is substituted by combining the solidification speed and the solidification interface shape.

  The solidification speed is defined by a speed um ′ perpendicular to the solidification interface. In melt spinning, since the inclination of the solidification interface is generally gentle, the speed um ′ perpendicular to the solidification interface is regarded as the cooling roll radial speed vA. However, the error is small. Therefore, first, the speed um ′ perpendicular to the solidification interface is set to the cooling roll radial speed vA to eliminate the influence of the solidification interface gradient so that the solidification speed can be calculated even at a fixed boundary of the gradient 0.

Next, consider searching for the cooling roll radial velocity vA so as to satisfy the temperature condition and the heat transfer condition at the solidification interface. The conditions to be satisfied are the following expressions 18 and 19.
Tm _L = Tw (vA) (Equation 18)
qm _L = qm _s -γ · ρ _L · Cp _L · vA [Equation 19]

Here, the solidification phase thickness Δh increased by the cooling roll radial speed vA in a certain solidification interface section Δx is expressed by the following Expression 20.
Δh = vA / umr _L · Δx (Equation 20)
umr _L : Cooling roll parallel component of solidification interface velocity

On the other hand, since the solidification interface speed can be regarded as umr _L , this speed is also given as a boundary condition to the solidification interface. As described above, when the solidification interface is flat, Δh is expressed only by the cooling roll radial velocity vA, and vA = 0 should be true for the true solidification interface. This is because the solidification interface vertical direction component of umr _L coincides with the solidification speed, and there is no room for newly giving the cooling roll radial speed vA on this interface.

Therefore, the plate thickness in the specific section i downstream along the solidification interface can be generalized by the following equation (21).
h _i = h _i-1 + dhm / dx · Δx _i
+ VA / umr _L · Δx _i [Equation 21]
hm: Solidification interface height Δxi: Section length of section i

By the way, it is possible in principle to express the plate thickness only by the cooling roll radial velocity vA with hm = 0 in the entire section of the solidification interface. However, this method has the following drawbacks. That is, the change in the downstream free interface height hf is in the order of the following expression 22 at least in the most downstream portion.
dhf / dx≈vA / umr (Equation 22)

  Therefore, when the cooling roll radial velocity vA suddenly changes in the x direction, the downstream free interface height hf also changes abruptly. However, the expression of the solidification speed based on the cooling roll radial speed vA originally simulates the change in the solidification interface height hm, and even if the solidification interface height hm changes suddenly, the downstream free interface height hf changes suddenly. There is no need to do. Therefore, the sudden change in the downstream free interface height hf due to the sudden change in the cooling roll radial velocity vA is physically unreasonable, and at the sudden change portion of the downstream free interface height hf, the internal pressure Also drastically hinders the computational stability of the flow field at this site.

  Therefore, in the downstream region of the paddle 1 adjacent to the solidification interface and the downstream free interface, the expression by the change in the solidification interface height hm should be used instead of the expression of the solidification speed by the cooling roll radial speed vA. As described above, on the fixed boundary in the vicinity of the upstream free interface, the expression of the solidification speed by the cooling roll radial speed vA is indispensable. It is necessary to use in combination with the change in interface height.

  Further, the method for expressing the solidification rate only by (dhm / dx) without using the cooling roll radial direction velocity vA is the following procedure.

  As a first step, the two free interface positions and the velocity on the free interface are fixed. As a second stage, on the premise of the solidification interface height hm distribution at this time, the cooling roll radial velocity vA distribution is obtained so as to satisfy Expression 18 and Expression 17. As a third step, the solidification interface height hm distribution is corrected using Equation 21 so that the cooling roll radial velocity vA = 0 in this section. As a fourth stage, after fixing the cooling roll radial velocity vA and the solidification interface height hm distribution, the restriction of the two free interfaces is released, and the flow field in the paddle including the free interface position is calculated. By repeating the above four steps, a convergent solution can be obtained stably even in a state field having a plurality of free interfaces.

Next, the handling of the fixed boundary between the solidification interface and the downstream free interface will be described.
An outflow velocity umr _L corresponding to this position is given to the outflow boundary as a boundary condition. Since this fixed boundary substitutes the contact portion between the solidification interface and the free interface, the fixed boundary should be set at a position where the solidification interface and the free interface are sufficiently close. However, it is difficult to know in advance where the solidification interface and the free interface are close enough on the downstream side. Therefore, this fixed boundary is set sufficiently downstream, and when the solidification interface and the free interface approach within a predetermined value upstream of this boundary, the height of the solidification interface and free interface downstream is fixed. . A region from a position where the solidification interface and the free interface approach within a predetermined value to a fixed boundary is referred to as a paddle outflow region 31.

Since the flow is substantially uniform in the paddle outflow region, an average velocity of the fixed boundary outflow velocity umr _L is given to the upstream end of this region. In addition, since the free interface height is fixed, the pressure field inside this region is independent of the free interface curvature, so that negative pressure corresponding to the outflow velocity can also be expressed. However, in this region, since the solidification interface height is fixed, the temperature field here is not physically meaningful. In other words, the paddle outflow area is set for the convenience of numerical calculation work and is used to appropriately represent the state field at the upstream end of the area where the physical meaning exists.

If the heat flux qms and the outflow velocity umr _L applied to the solidification interface are given by the above modeling, the state field in the melt spinning liquid phase can be solved by numerical calculation. For other boundary conditions, discretization methods, and numerical solutions, traditional methods may be used. For example, the inflow boundary condition is a pressure and temperature constant condition, the nozzle inner wall boundary condition is speed = 0 and the adiabatic condition, and the boundary condition at the free surface is the adiabatic condition for the temperature field and the following for the flow field: Conditional expression 23 is used.

P = 2 · σ / R (Equation 23)
σ: Surface tension R: Radius of curvature of free surface Further, the finite element method may be used for the numerical solution, and the ALE method or the like may be used for searching the position of the free interface.

Next, the state field of the solidification phase is solved.
A system as shown in FIG. 10 is set. The shape of the inflow boundary is given by a solidification interface height hm distribution when all solidification rates are expressed by solidification interface height hm using Equation 21. However, since there is no physical validity in the solidification interface position of the paddle outflow region, it is regarded as a fixed boundary having a physical meaning and connecting the upstream end of the paddle outflow region to the solidification interface. In the calculation of the solidification phase, the position of the solidification interface is always fixed, and the pressure distribution on the solidification interface obtained by the liquid phase numerical calculation is used as the inflow condition at the solidification phase solidification interface at the corresponding position.

  The main stress applied to the solidification phase by melt spinning is the tension acting between the winder 11 and the boundary normal stress corresponding to the tension is given to the outflow boundary. The lower boundary of FIG. 10 is divided into a solidified phase roll boundary 32 corresponding to the adhesion portion of the cooling roll 5 and a solidified phase free boundary 33 corresponding to the free surface. The boundary condition at the free boundary of the solidification phase is given by the adiabatic condition and Equation 23.

The boundary at the upper end of FIG. 10 is a solidified phase free boundary corresponding to the free surface, and is given by the adiabatic condition and Equation 23. Based on the above boundary conditions, if the solid state phase field is solved by a traditional numerical method such as the finite element method using the ALE method, the plate thickness (height of the upper boundary at the outflow boundary), the solidification interface The heat flux qm _s and the solidification interface velocity um _s can be obtained.

The heat flux qm _s and the solidification interface velocity um _s obtained here are the heat flux and solidification interface velocity at the liquid phase solidification interface used before the numerical calculation in the solidification phase (actually, umr = generally used as ρ _s / ρ _L · um _s ). Accordingly, again using the and updated heat flux qm _s solidification interface velocity um _s, implementing the liquid phase numerical calculation. By repeating the above process until the heat flux qm _s and the solidification interface velocity um _s do not change, the final state quantity field in the entire paddle including the liquid phase and the solidification phase can be obtained.

  By the way, in the case of melt spinning, since the two free interface positions fluctuate during the calculation, the number of repetitions of the coagulation calculation between the solidification phase and the liquid phase necessary for obtaining the convergent solution is compared with the case of the system of FIG. And there are much more. Therefore, a numerical calculation method obtained by simplifying the above model, which can be applied only when the deformation of the solidified phase is relatively small, will be described.

  In melt spinning, generally, the plate thickness is sufficiently smaller than the paddle length, and the state quantity change rate in the plate thickness direction is much larger than the state quantity change rate in the cooling roll circumferential speed direction. If the deformation in the solidified phase is small, the error is small even if the speed of the solidified phase is regarded as the cooling roll peripheral speed uR.

  Here, a time change of the solid phase state quantity in a specific cross section of the paddle on the Lagrangian coordinate system moving with the cooling roll is considered. At this time, since the solidification phase is stationary on the Lagrangian coordinate system, the continuous equation and the equation of motion are unconditionally satisfied, and the state quantity change in the circumferential direction of the cooling roll is the state quantity in the plate thickness direction. It can be ignored compared to changes. As a result, the state quantity field of the solidified phase can be simplified to a one-dimensional unsteady heat conduction problem (stefan problem) in the thickness direction in consideration of solidification.

  Even in the prior art, there was a calculation method that considered the paddle cross section as a Stefan problem, but since a calculation region including both the solidified phase and the liquid phase was set, in order to obtain the temperature field of the solidified phase, Information on temperature distribution was essential. However, since the liquid phase of melt spinning is not a uniform flow rate, it is impossible to consider it as a Stefan problem, and the model error in the prior art has to be large.

  In the present invention, the Stefan problem is applied only to the solidification phase that is guaranteed to have a uniform flow rate, and the influence of the liquid phase is expressed simply by the solidification phase thickness and the solidification rate at the solidification interface. Therefore, it is possible to avoid a model error caused by assuming a temperature distribution in the liquid phase that is difficult to predict.

The solidification phase temperature field is obtained by solving the following equation 24.
dT _s / dt = κ · (d ² T _s / dy ² ) (Equation 24)
t: Time κ: Temperature conductivity y: Plate thickness direction position on Lagrange coordinate system

Further, the boundary conditions are as shown in Expression 25 and Expression 26 below.
T _s = T _R cooling roll surface (y = 0) ··· [Equation 25]
T _s = Tw (vA) Solidification interface (y = hm) (Equation 26)
T _R : Cooling roll surface temperature

The boundary condition may include the influence of the surface thermal resistance of the cooling roll 5. Cooling roll surface temperature T _R is the solidification phase may be obtained by solving forms communication with the cooling roll 5, when the thermal conductivity of the cooling roll 5 compared to the coagulation phase is sufficiently large, the cooling roll surface temperature T _R a cooling roll initial temperature (constant value) not a large model errors as. In the above equation, the solidification rate vA represents the total solidification rate including the influence of dhm / dx, and the solidification interface height hm is given from the liquid phase numerical analysis result.

On the other hand, by using Ts obtained by solving the equation 24, the solidification phase heat flow rate target value q _L required for the liquid phase numerical analysis can be given from the following equation 27.
q _L = k _s (dT _s / dy) | _{y = hm} −ρCpvA [Equation 27]
k _s : Solid phase thermal conductivity

  When the above method is used, the state field in the paddle 1 can be obtained without performing (multi-dimensional) numerical calculation of the solidified phase, and the calculation load can be greatly saved. Alternatively, the temperature field calculation may be solved as a Stefan problem, and only the solidification phase flow field having a large influence on the plate thickness may be subjected to multidimensional numerical analysis of the solidification phase to give umr used in the fluid numerical calculation.

  Numerical calculation can be used for solving the equation 24. However, in order to obtain the temperature gradient of the equation 27 with high accuracy, it is necessary to set the mesh division extremely finely, in terms of calculation load and calculation stability. It is not necessarily advantageous. Therefore, the following method that gives an analytical solution of the temperature gradient at the solidification phase solidification interface is used.

When the physical properties are uniform in the solidification phase and the solidification interface height hm is always constant, the following steady equation 28 is obtained.
T _s = (Tm−T _R ) · y / hm (Equation 28)

_Let T _s distribution in Equation 28 be T _s0 . The solidification phase solidification interface temperature gradient at this time is as shown in the following equation 29.
(DT _s / dy) | _{y = hm} = (Tm−TR) / hm (Equation 29)

Further, since Expression 24 is linear with respect to T, when Ts = Ts0 + Ts ′ is substituted into Expression 24, a heat conduction equation of only the deviation temperature Ts ′ shown in the following Expression 20 is obtained.
dT _s ′ / dt = κ · (d ² T _s ′ / dy ² ) (Equation 30)

The boundary condition is expressed by the following Expression 31 and Expression 32.
T _s ′ = 0 y = 0 (Equation 31)
T _s ′ = 0 y = hm (Equation 32)

When an arbitrary temperature deviation T _s 'i0 is given at an arbitrary y point yi (0 ≦ yi ≦ hm) and a linear temperature distribution as shown in FIG. 11 is given (this temperature distribution is denoted as T _s ' i). Since Expression 30 is linear with respect to T and T _s ' i satisfies the boundary conditions of Expression 31 and Expression 30, it can be expressed as the following Expression 33 and Expression 34.
_{Ts '= T s' i +} Ts "··· [ formula 33]
Initial _{conditions: Ts '| t = 0 =} T s' i | t = 0 + Ts "| t = 0 ··· [ formula 34]

Solving the heat conduction equation separately with T _s 'i, Ts "and adding the temperatures of those solutions is the same as the temperature solution obtained by solving Equation 30. Therefore, Ts' _{| t = 0} T _s in terms of a plurality of different y 'i _| by adding the _{t = 0} approximates the individual T _s' after solving the heat conduction equation of i, if the addition the temperature of these solutions, Ts An approximate solution of 'is obtained.

There is an analytical solution of the following equation 35 in the heat conduction equation of T _s ' i.
Ts '(y, t) = Σ {2T s' i0 / ((nπ) 2 α (1-α)) · sin (αnπ)
Sin (nπy / h) exp [− (nπ / h) ² κt]} [Expression 35]

Here, Σ takes values from n = 1 to ∞. Α is a dimensionless position in the y direction defined by α = y / hm. Next, the differentiation of the expression 35 with respect to y becomes the following expression 36.
dTs '(y, t) / dy = Σ {2T s' i0 / (nπhα (1-α))
Sin (αnπ) cos (nπy / h)
Exp [− (nπ / h) ² κt]} [Equation 36]

Therefore, the average deviation heat flux q _s ′ _a between time 0 and te at y = hm is expressed by the following Expression 37.
q _s ' _a = -Σ {2 khT _s ' i0 / ((nπ) ³ κα (1-α) te)
・ Sin (αnπ) ・ cos (nπ)
(1-exp [-(nπ / h) ² κte])} [Expression 37]

Further, the initial temperature T _s ′ is expressed by the following equation 38 by combining a plurality of T _s ′ i 0.
T _s '= Σ {αi / αj · T _s ' i0}
+ Σ {(1-αi) / (1-αj) · T _s ' i0} [Equation 38]

Here, Σ in the first term on the right side of Equation 38 takes the sum from j = i to N, and Σ in the second term on the right side takes the sum from j = 1 to i−1. N is the number of T _s ' i used for approximation. When T _s ' initial values at N points are given and the simultaneous linear equations of Equation 38 are solved, T _s ' i0 initial values can be expressed as a function of T _s ' initial values.

  Furthermore, when actually solving the Stefan problem, the temperature distribution at the next time step is calculated from the approximate temperature distribution at a certain step, but the solidification interface height hm changes between these steps. The y value for a particular α changes.

Therefore, the temperature approximation point is changed for each step so as to keep the value of α constant, and Ts ′ in the previous step is interpolated to the value at the new approximation point. Also, it is necessary to predetermine a pattern of hm change within the step. In this method, the change of the solidification interface height hm is calculated as T _s ′ i in the following equation (39).

In addition, the maximum value of n in Formula 35 and the like is usually about 50. This is because the case where n larger than this is necessary is limited to the case where the solidification rate is extremely low. The calculation of the temperature deviation is important when the solidification rate is fast and the temperature distribution in the solidification phase deviates significantly from the steady solution, and when the solidification rate is slow, the temperature distribution is close to the steady solution in the first place. This is because no calculation is necessary.

The deviation heat flux q _s ′ _a obtained from the above deviation temperature calculation and the average heat flux obtained by multiplying Equation 29 by k are added together to obtain the heat flux q _s at the solidification interface. The above calculation processes are all configured by a combination of analytical solutions, and problems of calculation stability and calculation load that tend to occur when a numerical analysis method is adopted can be avoided.

(Example 1)
The strip-shaped material 3 was manufactured with the casting machine shown in FIG.
The melt reservoir 10 was opened at the top, and a melt spray pressure was applied by a head based on the weight of the melt. The height of the melt 2 was measured using a laser-type hot water meter installed at the upper end of the melt reservoir 10. A general sheath thermocouple was installed in the melt to measure the inflow temperature Tin. The speed uR was measured with a rotary encoder type tachometer. Moreover, coiling temperature Tc was measured using the radiation thermometer. Further, the gap G was measured with a transmitted light type gap meter.

  These measurement results are input to the arithmetic unit 12 in real time, and the set value of the instantaneous gap G obtained from the arithmetic result is output to the linear guide type nozzle moving device, and the value of the gap G becomes a predetermined value. The position was corrected every moment. Further, a Ti-48 at% Ni alloy was used as the strip material 3. The melt 2 maintained at 1350 ° C. by a heater installed outside the melt reservoir 10 is sprayed for 12 minutes onto a pure copper rotary cooling roll whose average temperature is maintained at room temperature by internal water cooling, and has a target plate thickness of 50 μm. The strip material 3 having a target plate width of 300 mm and a mass of 200 kg was continuously cast.

  During casting, it is obtained in advance by using the inflow temperature Tin, the gap G, the speed uR, the winding temperature Tc, and the inflow pressure Pin calculated from the measured surface height in the melt reservoir 10 during the casting. The instantaneous plate thickness is calculated using the casting machine flow rate characteristic function and the paddle flow rate characteristic function, and the difference in thermal expansion is corrected to this (the plate thickness deviation due to the tension of the winder 11 is very small. As a result, correction was not carried out in particular), and the calculated product thickness was determined. The inflow pressure Pin may be measured by installing a pressure gauge.

  The value of the gap G was set from time to time using general feedforward control so that the calculated product thickness matches the target product thickness. As a method for obtaining the flow characteristic function of the casting machine, an experiment for discharging the melt without using the cooling roll 5 in an actual casting machine was performed under a plurality of melt back pressure conditions, and a relational expression between the flow rate and the pressure was obtained. In the method of obtaining the paddle flow rate characteristic function, a strip material production experiment by melt spinning is performed on the cooling roll 5 using a casting machine having a simple structure whose pressure loss characteristic is known, and a combination of a plurality of thickness affecting factor groups is performed. The measured product thickness of the strip material obtained in each experiment is converted to the thickness at the paddle outlet corrected for the amount of change due to thermal expansion and winder tension, and the flow rate is expressed by pressure from these thickness data groups. A regression equation was obtained and used as a paddle flow rate characteristic function. The deviation of the obtained product plate thickness from the target plate thickness was 1% of the target product plate thickness as an average value and 0.6% as the standard deviation.

(Example 2)
Numerical calculation was used as a method for obtaining the paddle flow characteristic function. As a numerical calculation method, a constitutive equation is given by using the paddle 1 as a viscous fluid, the solidification phase is regarded as a highly viscous liquid phase, the transport equation of the continuum (for example, Non-Patent Document 5 described above), and the finite element method are used. The flow field, the temperature field, and the solidification interface position were obtained by solving them. Measured values were used as physical property values of the constitutive equation. Boundary conditions were given as speed or stress conditions at the outlet of the nozzle 4, the solidified phase cross section at the downstream end of the paddle 1, the upstream free surface, the downstream free surface, and the surface of the cooling roll 5. The ALE method was used for calculation of the free surface. The calculation domain was set to a single domain including the liquid phase and the solidified phase, and the transport equation was solved simultaneously for the liquid phase and the solidified phase.

  Regarding the calculation method of the solidification interface position, a finite element corresponding to the solidification temperature was searched in the temperature field calculation, and this series of finite elements was regarded as the solidification interface. As a result of carrying out a casting experiment in the same manner as in Example 1 except for the other conditions, the deviation of the obtained product plate thickness from the target plate thickness was 0.9% of the target product plate thickness as an average value, and the standard deviation It was 0.5%.

(Example 3)
Numerical calculation was used as a method for obtaining the paddle flow characteristic function.
As a numerical calculation method, the paddle 1 was divided into a liquid phase calculation region shown in FIG. 9 and a solidification phase calculation region shown in FIG. In the liquid phase calculation domain, the continuum transport equation (for example, Non-Patent Document 5) given a constitutive equation as a viscous fluid in the solid phase calculation domain and the viscoplastic body in the solid phase calculation domain is solved by using the finite element method, The temperature field and the solidification interface position were determined. The boundary conditions in the liquid phase calculation region are the exit of the nozzle 4, the solidification interface, the upstream free surface, the downstream free surface, and the surface of the cooling roll 5, and are given as speed or stress conditions. Further, the boundary conditions in the liquid phase calculation region were a solidification interface, a solid phase cross section at the downstream end of the paddle, a free surface on the downstream side, and a cooling roll surface, and were given as speed or stress conditions.

  The ALE method was used for calculation of the free surface. The numerical calculation was performed according to the following procedure. As a first step, the state field in the solidification phase calculation region was calculated under the boundary conditions of Equations 13 and 12. As the second step, the state field in the liquid phase calculation region is calculated based on the boundary conditions of Equations 16 and 15, and the solidification interface is used by using the above method so that the temperature at the solidification interface becomes the solidification temperature. Fixed the position. As the third stage, the calculation of the solidification phase and the calculation of the liquid phase were alternately performed again to obtain a convergent solution of the state field. As a result of performing a casting experiment under the same conditions as in Example 1 except for the above conditions, the deviation of the obtained product plate thickness from the target plate thickness was 0.8% of the target product plate thickness as an average value, and the standard deviation And 0.45%.

(Comparative Example 1)
In the casting machine of Example 1, a γ-ray thickness gauge is installed between the cooling roll 5 and the winder 11, and the value of the gap G is corrected by PID feedback control based on the measured thickness. The band-shaped material 3 was cast in the same manner as in Example 1 except that the production conditions were as follows. As a result of performing casting several times and adjusting each gain of PID control optimally, the deviation from the target plate thickness of the product plate thickness in the best casting is 1.2% of the target product plate thickness on average, the standard The deviation was 2%.

  Therefore, the present invention was able to demonstrate that the plate thickness can be controlled with higher accuracy than the prior art.

It is a schematic diagram for demonstrating melt spinning type continuous casting. It is a schematic block diagram of a casting machine. It is a characteristic figure of a paddle flow rate. It is the schematic diagram which showed the example in which a pressure and speed become discontinuous in a solidification interface. It is a schematic diagram which shows the concept which isolate | separated the calculation area | region into the liquid phase calculation area | region and the solidification phase calculation area | region. FIG. 6 is a schematic diagram of a solidified phase model in a solidified phase calculation region shown in FIG. 5. It is a schematic diagram of the liquid phase model in the liquid phase calculation area | region shown in FIG. It is a schematic diagram of the liquid phase model using the melt spinning method. It is a schematic diagram of the liquid phase model using the melt spinning method. It is a schematic diagram of the solidification phase model using the melt spinning method. It is the figure which showed the distribution function of deviation temperature.

Explanation of symbols

1 Paddle 2 Melt (or liquid phase)
3 Band-shaped substance (or solidification phase)
4 Nozzle 5 Cooling roll 6 Inflow boundary 7 Solidification interface 8 Upstream free surface 9 Downstream free surface 10 Melt reservoir 11 Winder 12 Arithmetic unit 13 Cooling roll motor 14 Winder motor 15 Pressure device 16 Lifting device 17 Cooling roll cooling device 18 Gap meter 19 Cooling roll peripheral speed meter 20 Inflow thermometer 21 Inflow pressure gauge 22 Winding thermometer 23 Plate thickness gauge 24 Free slip boundary 25 Outflow boundary 26 Solidification phase side solidification interface (or inflow boundary)
27 Liquid phase side solidification interface (or outflow boundary)
28 Solidification phase calculation area 29 Liquid phase calculation area 30 Fixed boundary (or outflow boundary)
31 Paddle outflow area 32 Solidified phase roll boundary 33 Solidified phase free boundary 34 Heater

Claims (5)

The melt continuously ejected from the nozzle of the casting machine onto the rotating cooling roll forms a paddle on the surface of the cooling roll, and the melt in the paddle is solidified by contact cooling with the cooling roll. In the method for producing a strip material by continuous casting using a melt spinning method that continuously forms a solid strip material while moving with the cooling roll,
The strip thickness affecting factor of the belt-shaped substance, the gap between the outlet of the nozzle and the cooling roll, the melt pressure at the nozzle, the melt temperature at the nozzle, the peripheral speed of the cooling roll, and A calculation step of continuously measuring the cooling characteristic value of the cooling roll during continuous casting, and calculating the thickness of the strip material at the downstream end of the paddle according to a predetermined function using the measured value as an input value; ,
The target value of the product thickness of the strip material is calculated by calculating the plate thickness reduction amount by a predetermined function using the thermal expansion coefficient of the strip material or the thermal expansion coefficient and the tension applied to the strip material, A conversion step of subtracting the amount of reduction from the target value of the product thickness of the strip material to the target value of the thickness of the strip material at the downstream end of the paddle,
One or more kinds selected from the plate thickness influencing factors so that the plate thickness of the strip material at the downstream end of the paddle calculated in the calculation step becomes a target value of the plate thickness after conversion in the conversion step A method for producing a strip-like material by continuous casting using a melt spinning method, wherein two or more kinds are adjusted and feedforward control is performed. The melt continuously ejected from the nozzle of the casting machine onto the rotating cooling roll forms a paddle on the surface of the cooling roll, and the melt in the paddle is solidified by contact cooling with the cooling roll. In the method for producing a strip material by continuous casting using a melt spinning method that continuously forms a solid strip material while moving with the cooling roll,
The strip thickness affecting factor of the belt-shaped substance, the gap between the outlet of the nozzle and the cooling roll, the melt pressure at the nozzle, the melt temperature at the nozzle, the peripheral speed of the cooling roll, and A calculation step of continuously measuring the cooling characteristic value of the cooling roll during continuous casting, and calculating the thickness of the strip material at the downstream end of the paddle according to a predetermined function using the measured value as an input value; ,
The target value of the product thickness of the strip material is calculated by calculating the plate thickness reduction amount by a predetermined function using the thermal expansion coefficient of the strip material or the thermal expansion coefficient and the tension applied to the strip material, A conversion step of subtracting the amount of reduction from the target value of the product thickness of the strip material to the target value of the thickness of the strip material at the downstream end of the paddle,
The plate thickness so that the difference between the plate thickness of the strip material at the downstream end of the paddle calculated in the calculation step and the plate thickness of the strip material measured by a meter is within a predetermined target range. A method for producing a strip-like material by continuous casting using a melt spinning method, wherein one or more selected from influential factors are adjusted and feedback controlled. As the predetermined function used when calculating the plate thickness of the strip material at the downstream end of the paddle in the calculation step,
Using the thickness influence factor of the 1 type or 2 or more types of combination selected from the melt pressure in the nozzle which is the thickness influence factor, and the other thickness influence factors other than this, the following formula,
Melt flow _{casting machine} = function _{casting machine} (melt pressure at nozzle, other factors affecting plate thickness)
Melt flow rate _paddle = Function _paddle (melt pressure at nozzle, other factors affecting plate thickness)
(Here, the “function _{casting machine} ” uses “melt pressure at the nozzle” and “other thickness influencing factors” (may be plural) as independent variables, and the flow rate of the melt at the nozzle “melt The “function _paddle ” is a function that represents “ flow _{casting machine} ”, and “melt pressure at the nozzle” and “other plate thickness influencing factors” (which may be plural) are independent variables. (This is a function representing the melt flow rate “melt flow rate _paddle. ” The above relational expressions are referred to as a flow rate characteristic function on the casting machine side and a flow rate characteristic function of the paddle, respectively.)
In the casting machine side flow rate function and the paddle flow rate characteristic function expressed independently of each other, and for the specific combination in the input value of the plate thickness influencing factor, the casting machine side flow rate characteristic The melt flow rate when the function value coincides with the flow rate characteristic function value of the paddle is the melt flow rate in the combination of the plate thickness affecting factor groups, and the melt flow rate in the combination of the plate thickness affecting factor groups The method for producing a strip-like material by continuous casting using the melt spinning method according to claim 1 or 2, wherein a function for calculating a plate thickness at the downstream end of the paddle is used.   As a calculation method of the flow rate characteristic function of the paddle, in the combination of the selected thickness influence factor groups, the different values inside the paddle are calculated by performing numerical calculation modeling behavior in the paddle for a plurality of different values. The flow rate value and the plate thickness of the strip material are calculated, and the flow rate characteristics inside the paddle are calculated from the plate thickness calculation values of the strip material corresponding to each combination of the plate thickness influencing factor groups different from each other. The melt spinning method according to claim 3, wherein a regression equation of a plate thickness influencing factor group related to a plate thickness of the strip material in a function is obtained, and the regression equation is used as a flow rate characteristic function of the paddle. A method for producing a strip-like material by continuous casting.   In the numerical calculation modeling the behavior in the paddle, the region in the paddle is divided into two regions, the liquid phase and the solidification phase, and in each region, the numerical calculation in which the behavior in the region is modeled independently is performed. In addition, the speeds of the liquid phase and the solidified phase coincide at the solidification interface that is the boundary between the liquid phase and the solidified phase, and the solidification interface stresses of the liquid phase and the solidified phase are the same in the solidification interface. The boundary condition of the solidification interface in the opposite direction is obtained by repeated calculation, and the thickness of the strip material when the boundary condition at the solidification interface is satisfied in both the liquid phase and the solidification phase is expressed in the flow rate characteristic function inside the paddle. The method for producing a strip-like material by continuous casting using the melt spinning method according to claim 4, wherein the strip-like material is used as a plate thickness of the strip-like material.

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