JP7319412B2 - GLASS SUBSTRATE MANUFACTURING DEVICE AND GLASS SUBSTRATE MANUFACTURING METHOD - Google Patents

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a glass substrate manufacturing apparatus having a circumferential wall portion and a tubular member for processing molten glass, and a glass substrate manufacturing method.

Glass substrates used in displays such as liquid crystal displays and organic EL displays are manufactured by melting glass raw materials and then transferring, clarifying, and homogenizing the molten glass, and then forming it into a plate. manufactured.

Tubing is used to process molten glass. The tubular member has a circumferential wall portion extending in the axial direction, and the molten glass is treated while flowing inside the wall portion. As such a tube member, for example, there is a clarification tube for fining molten glass. The fining tube is made of a highly heat-resistant metallic material containing a platinum group metal in order to heat the molten glass to a high temperature for fining. Refractory bricks are arranged around the fining tube so as to surround the circumferential wall of the fining tube to support the fining tube. A castable refractory such as alumina cement is placed in the gap between the fining pipe and the refractory bricks to suppress deformation of the fining pipe during operation and to suppress volatilization of platinum group metal components from the wall surface of the pipe. A protective layer is provided (Patent Document 1).

JP 2014-84253 A

From the viewpoint of firmly supporting the fining tube and suppressing deformation of the fining tube, it is preferable to reduce the gap between the fining tube and the support and to reduce the thickness of the protective layer. However, if the gap is small, it is difficult to fill the space between the fining tube and the refractory brick with the castable refractory during construction, and there is a problem that gaps such as air bubbles tend to remain in the protective layer. If the protective layer contains many voids, the effect of suppressing deformation of the clarifier and volatilization of platinum group metal components during operation cannot be sufficiently obtained. In addition, the amount of voids in the protective layer tends to vary depending on the procedure of the filling operation and the skill of the operator. When the gap between the fining tube and the refractory brick is small, such variations occur remarkably. Thus, it has been difficult to achieve both support of the clarification tube and formation of a protective layer with few voids.

Accordingly, the present invention provides a glass substrate manufacturing apparatus and a glass substrate manufacturing method capable of forming a protective layer with few gaps between the tubular member and the support while supporting the tubular member with a refractory support. intended to provide

One aspect of the present invention is a glass substrate manufacturing apparatus.
Glass substrate manufacturing equipment
a tubular member made of a material containing a platinum group metal, the tubular member having a circumferential wall for processing molten glass inside the wall;
a refractory support positioned to surround and support the wall;
a refractory protective layer formed in a gap between the wall and the refractory support so as to be in contact with the wall and the refractory support;
The wall portion has a first wall region located in a partial region on the circumference and a second wall region located below the first wall region,
The size of the second gap S2 between the second wall region and the support is 1 of the size of the first gap S1 between the first wall region and the support. .1 times or more .

The size of the second gap S2 is preferably 10 times or less the size of the first gap S1.

the outer peripheral surface of the wall portion has a circular shape in a cross section perpendicular to the axial direction of the tubular member;
Of the inner peripheral surface of the support that faces the outer peripheral surface, the inner peripheral region of the support that faces the wall region has an arc shape in the cross section,
A second inner peripheral region of the support facing the second wall region has a larger radius of curvature in the cross section than a first inner peripheral region of the support facing the first wall region. It is preferable to have

The first wall region and the second wall region are regions above and below the central axis of the pipe member in a cross section perpendicular to the axial direction of the pipe member. preferable.

Another aspect of the present invention is a method for manufacturing a glass substrate.
The manufacturing method of the glass substrate is
A treatment step in which molten glass is treated while flowing through a tubular member made of a material containing a platinum group metal and having a circumferential wall;
and a preparation step performed before the treatment step,
The preparation step includes
an arrangement step of arranging a refractory support so as to surround the wall; and a monolithic refractory in a gap between the wall and the refractory support so as to be in contact with the wall and the refractory support. a filling step of filling the
After the construction step, a temperature raising step of raising the temperature of the pipe member,
The wall portion has a first wall region located in a partial region on the circumference and a second wall region located below the first wall region,
The size of the second gap S2 between the second wall region and the support is 1 of the size of the first gap S1 between the first wall region and the support. .1 times or more .

According to the glass substrate manufacturing apparatus and the glass substrate manufacturing method of the above aspect, while supporting the tubular member by the refractory support, it is possible to form a protective layer with few gaps between the tubular member and the support. It is possible.

It is a figure which shows schematic structure of a glass substrate manufacturing apparatus. It is a figure which shows schematic structure of the manufacturing method of a glass substrate. FIG. 2 is a cross-sectional view showing an example of the layered structure of a clarifier tube, a refractory protective layer, and a refractory support. (a) and (b) are figures explaining a construction process.

Hereinafter, the method for manufacturing a glass substrate and the apparatus for manufacturing a glass substrate according to this embodiment will be described.

(Overview of the method for manufacturing a glass substrate)
FIG. 1 is a diagram showing an example of steps during operation in the method for manufacturing a glass substrate according to the present embodiment. The method for manufacturing a glass substrate includes a preparation step, a melting step (ST1), a clarification step (ST2), a homogenization step (ST3), a forming step (ST4), a slow cooling step (ST5), and a cutting step (ST6). Mainly have In addition, it may have a grinding process, a polishing process, a cleaning process, an inspection process, a packing process, and the like. The manufactured glass substrates are laminated in a packing process as necessary, and transported to a delivery destination.

A preparation process is a process performed before operation (operation) of a glass substrate manufacturing apparatus. During operation, a series of steps from the melting step (ST1) to the cutting step (ST6) are performed. The preparation process will be described later.
In the melting step (ST1), molten glass is made by heating glass raw materials. This molten glass usually contains bubbles containing CO ₂ , SO ₂ , O ₂ or N ₂ or the like.
In the clarification step (ST2), the temperature of the molten glass is raised to generate a gas such as oxygen through a reduction reaction of a clarifier (such as tin oxide) contained in the molten glass. The bubbles in the glass melt grow by absorbing the generated gas, rise to the liquid surface of the glass melt and are released. Thereafter, in the clarification step, the temperature of the molten glass is lowered to accelerate the oxidation reaction of the reducing substances produced by the reduction reaction of the clarifier. As a result, gases such as oxygen in the bubbles remaining in the glass melt are reabsorbed into the glass melt, and the bubbles disappear.

In the homogenization step (ST3), the glass components are homogenized by stirring the molten glass using a stirrer. As a result, uneven composition of the glass, which causes striae and the like, can be reduced. A homogenization process is performed in the stirring tank mentioned later. The homogenized glass melt is fed to a forming device.

The forming step (ST4) and the slow cooling step (ST5) are performed in a forming apparatus.
In the forming step (ST4), the molten glass is formed into a sheet glass, which is a strip of glass having a predetermined thickness, to create a sheet glass flow. For molding, a float method, a fusion method (overflow down-draw method), or the like is used.
In the slow cooling step (ST5), the formed and flowing sheet glass has a desired thickness and is cooled so as not to cause internal strain and warp.
In the cutting step (ST6), a sheet glass substrate is obtained by cutting the slowly cooled sheet glass into a predetermined length. Cutting sheet glass into blanks of a predetermined length is also called cutting. The glass substrate obtained by plate cutting is further cut into a predetermined size to produce a glass substrate of a target size.

(Overview of glass substrate manufacturing equipment)
FIG. 2 is a schematic diagram of a glass substrate manufacturing apparatus that performs the melting step (ST1) to the cutting step (ST6). The glass substrate manufacturing apparatus mainly includes a melting device 100, a forming device 200, and a cutting device 300, as shown in FIG. The melting apparatus 100 has a melting vessel 101 , a clarification tube 102 , a stirring vessel 103 , transfer tubes 104 and 105 and a glass supply tube 106 .
The melting tank 101 shown in FIG. 2 is provided with heating means such as a burner (not shown). A glass raw material to which a fining agent is added is put into the melting tank 101, and the melting step (ST1) is performed. The molten glass melted in the melting tank 101 is supplied to the clarification tube 102 through the transfer tube 104 .
In the clarification tube 102, the temperature of the glass melt MG is adjusted while flowing the glass melt MG, and the glass melt clarification step (ST2) is performed using the oxidation-reduction reaction of the clarifier. The glass melt MG is supplied to the clarification tube 102 so that a gas phase space is formed above the liquid surface of the glass melt MG. Specifically, when the temperature of the glass melt in the clarification tube 102 is raised, bubbles containing CO ₂ , SO ₂ , O ₂ , or N ₂ contained in the glass melt are reduced by the refining agent. It grows by absorbing gases such as oxygen generated by the process, floats on the liquid surface of the molten glass, and is released into the gas phase space. After that, by lowering the temperature of the molten glass, the reducing substances produced by the reductive reaction of the refining agent undergo an oxidation reaction. As a result, gases such as oxygen in the bubbles remaining in the glass melt are reabsorbed into the glass melt, and the bubbles disappear. The molten glass after refining is supplied to the stirring tank 103 through the transfer pipe 105 .
In the stirring tank 103, the molten glass is stirred by the stirrer 103a to perform the homogenization step (ST3). The molten glass homogenized in the stirring tank 103 is supplied to the forming apparatus 200 through the glass supply pipe 106 .
In the forming apparatus 200, the sheet glass SG is formed from molten glass by, for example, an overflow down-draw method (forming step ST4) and slowly cooled (slow cooling step ST5).
In the cutting device 300, a plate-like glass substrate cut out from the sheet glass SG is formed (cutting step ST6).

In the present embodiment, the clarification tube 102, the transfer tubes 104 and 105, and the glass supply tube 106 among the above components of the glass substrate manufacturing apparatus correspond to tube members. In the following description, the clarification tube 102 will be used as a representative example of the tube member.

(Clarifying tube, refractory protective layer, and refractory support)
FIG. 3 is a cross-sectional view perpendicular to the axial direction of the finer tube 102, showing the layer structure of the finer tube 102, the refractory protective layer, and the refractory support.
In addition to the clarification tube 102 , the glass substrate manufacturing apparatus has a refractory protective layer 113 and a refractory support 114 . As shown in FIG. 3, the clarification tube 102, the refractory protective layer 113, and the refractory support 114 have a laminated structure arranged sequentially from the inner peripheral side to the outer peripheral side.

The clarification tube 102 is a tubular member made of a material containing a platinum group metal. Platinum group metals refer to six elements: platinum (Pt), palladium (Pd), rhodium (Rh), ruthenium (Ru), osmium (Os), and iridium (Ir). Examples of materials containing platinum group metals include materials composed of a single platinum group metal or an alloy of two or more metals, such as platinum or a platinum alloy.

The clarification tube 102 has a circumferential (cylindrical) peripheral wall portion 112 extending in the axial direction of the clarification tube 102 . The molten glass is refined inside the circumferential wall 112 . The clarification tube 102 of the example shown in FIG. 3 has a circular outer circumference in a cross section orthogonal to the axial direction (hereinafter also simply referred to as a cross section). The cross-sectional shape of the outer periphery of the clarification tube 102 is not limited to a circular shape, and may be a flat shape such as an ellipse. The axial direction of the clarification tube 102 coincides with the longitudinal direction of the clarification tube 102 .

The clarification tube 102 is provided with a pair of flanges (not shown) connected to extend outward from the circumferential wall portion 112 and a vent tube (see FIG. 1). The flanges are connected to a power supply (not shown), and voltage is applied between the flanges to cause current to flow through the portion of the circumferential wall 112 between the flanges, heating the fining tube 102 . By this electrical heating, the clarification tube 102 is heated to, for example, about 1650.degree. C. to 1700.degree. The vent pipe is a pipe that communicates between the gas phase space and the space outside the clarification pipe 102 . The vent pipe has a function of discharging gases such as oxygen, CO ₂ and SO ₂ released from the molten glass MG into the gas phase space to the external space.

The refractory support 114 is arranged to surround the peripheral wall portion 112 with a gap between it and the outer peripheral surface of the peripheral wall portion 112 of the clarification tube 102, and supports the peripheral wall portion 112. is the body. The refractory support 114 supports the circumferential wall 112, insulates the finer tube 102, and protects the finer tube 102 from physical forces that may be applied from the outside. Refractory support 110 is constructed from refractory bricks. The refractory bricks are, for example, electroformed bricks made of a material containing alumina, AZS (Alumina-Zirconia-Silica), or zirconia. The refractory support 114 has an inner peripheral surface facing the outer peripheral surface of the peripheral wall portion 112 .

As shown in FIG. 3 , the surface of the refractory brick facing the outer peripheral surface of the circumferential wall portion 112 has an arc shape in cross section along the arc concentric with the circular shape of the clarification tube 102 . The arc shape is preferably a shape having an arc that is concentric with the circle forming the outer circumference of the clarification tube 102 . In this specification, a support made of refractory bricks is simply referred to as a "refractory brick" in accordance with common practice, but in many cases, a refractory brick is a plurality of refractory bricks (brick solids made of refractory material). ) are stacked in a predetermined shape, and in many cases, a support composed of a plurality of bricks which are fixed by applying a refractory filler such as refractory mortar between them. In the refractory bricks of the example shown in FIG. 3, among the plurality of stacked bricks, the bricks located in the portion facing the wall surface of the circumferential wall portion 112 are formed to have the arc shape.

The refractory protective layer 113 is a layer formed in a gap between the circumferential wall portion 112 and the refractory support 114 so as to be in contact with the circumferential wall portion 112 and the refractory support 114 . The refractory protective layer 113 plays a role of reliably supporting the circumferential wall portion 112 between the circumferential wall portion 112 and the refractory support 114 and suppressing deformation of the clarification tube 102 . In addition, the refractory protective layer 113 plays a role of suppressing the thinning of the peripheral wall portion 112 due to volatilization of the platinum group metal forming the peripheral wall portion 112 as a metal oxide such as _PtO2 . .

According to one embodiment, the average thermal expansion coefficient of the refractory protective layer 113 in the temperature range of 1300-1700° C. is preferably It is in the range of 0.1 to 10 times, more preferably in the range of 0.5 to 5 times. When the thermal expansion coefficients of the refractory protective layer 113 and the circumferential wall portion 112 satisfy the above relationship, the thermal expansion amount of the circumferential wall portion 112 and the shrinkage amount of the refractory protective layer 113 accompanying the temperature rise of the clarification tube 102 The effect of suppressing the occurrence of cracks in the refractory protective layer 113 due to the difference in the refractory protective layer 113 and reducing the voids in the refractory protective layer 113 is ensured.

A monolithic refractory is used for the refractory protective layer 113 . There are no particular restrictions on the type of monolithic refractory, but from the conditions of use, it is desirable to have a maximum use temperature of 1600° C. or higher, a compressive strength of 200 kgf/cm ² or higher, a dense structure, and low gas permeability. For example, a castable refractory compounded with alumina cement is suitable.

In the present embodiment, the peripheral wall portion 112 includes a first wall region 112a located in a partial region on the circumference of the peripheral wall portion 112 and a second wall region 112a located below the first wall region 112a. and two wall regions 112b. And the second gap S2 between the second wall region 112b and the refractory support 114 is wider than the first gap S1 between the first wall region 112a and the refractory support 114. . According to the study of the present inventor, it was found that the lower area of the area facing the circumferential wall tends to have more voids in the refractory protective layer 113 than the upper area. . It was also confirmed that by making the gap S2 larger than the gap S1, the gap in the refractory protective layer 113 in the region located below can be reduced. In this embodiment, the fining tube 102 can be reliably supported by the refractory support 114 through the thin refractory protective layer 113 formed in the gap S1, and the thick, less-void refractory protection layer formed in the gap S2 Layer 113 improves the effect of suppressing deformation of finer tube 102 during operation and volatilization of platinum group metal components.

When the temperature of the pipe member is raised for operation, the pipe member thermally expands and the outer diameter increases, while the monolithic refractory material that becomes the refractory protective layer 113 shrinks and becomes thinner. In this specification, the relationship between the sizes of the gaps S1 and S2 means the relationship at least before the temperature of the pipe member is raised for operation, for example, when the pipe member is at 25°C. , S2 does not change as the temperature rises.

The size of the second gap S2 is preferably 10 times or less the size of the first gap S1. When the size of the second gap S2 exceeds ten times the size of the first gap S1, the thickness of the refractory protective layer 113 in the second gap S2 becomes less than the thickness of the refractory protective layer 113 in the first gap S1. It is too large for the thickness, and when the fining tube 102 shrinks as the temperature rises, cracks occur between the two, protecting the peripheral wall 112 and suppressing the volatilization of the platinum group metal component. may be damaged. In addition, the clarification tube 102 cannot be suppressed from expanding in diameter due to temperature rise, and the clarification tube 102 may be deformed, resulting in a decrease in the strength of the clarification tube 102 or breakage of the clarification tube 102. be. On the other hand, the size of the second gap S2 is preferably 1.1 times or more the size of the first gap S1. If the size of the second gap S2 is less than 1.1 times the size of the first gap S1, the effect of reducing the gap in the refractory protective layer 113 is reduced. Preferably, the size of the second gap S2 is 1.5 to 5 times the size of the first gap S1, more preferably 1.8 to 2.4 times.

When the tube diameter of the clarification tube 102 (diameter in cross section of the circumferential wall portion 112) is, for example, 200 to 400 mm, the first gap S1 is, for example, 3 to 10 mm, and the second gap S2 is, for example, , 6 to 20 mm. It has been confirmed that the above effect of reducing the voids in the refractory protective layer 113 is enhanced when such a dimension example is satisfied.

A second inner peripheral region 114b of the refractory support 114 facing the second wall region 112b is, in cross-section, a refractory support facing the first wall region 112a, as in the example shown in FIG. Preferably, the body 114 has a radius of curvature R2 greater than the radius of curvature R1 of the first inner peripheral region 114a. By satisfying the above relationship with respect to the curvature radii R1 and R2, it becomes easier to obtain the above relationship of the gaps S1 and S2. Preferably, R2 is 101-130% as large as R1.

As in the example shown in FIG. 3, the first wall region 112a and the second wall region 112b are preferably regions above and below the central axis of the clarification tube 102 in cross section. . The central axis of the clarification tube 102 is a line passing through the center of the outer circumference of the circumferential wall portion 112 . By defining the wall regions 112a and 112b in this way, the refractory support 114 can adopt a form that is easy to install. In the example shown in FIG. 3, the refractory supports 114 comprise an upper support 114A located above the central axis of the finer tube 102 and a lower support 114B located below. Each of the upper support 114A and the lower support 114B consists of a plurality of stacked bricks and is made so that the surface facing the peripheral wall 112 has an arcuate shape.

(Preparation process)
Next, the preparation process of the manufacturing method of a glass substrate is demonstrated.
The preparation process has a construction process and a temperature raising process.

FIG. 4 is a diagram for explaining the construction process. FIG. 4(a) is a diagram for explaining the filling of the monolithic refractory into the gap S2, and FIG. 4(b) is a diagram showing the monolithic refractory to be filled into the gap S1.

The construction process includes an arrangement process and a filling process.
In the arranging step, refractory bricks to be the refractory support 114 are arranged so as to surround the circumferential wall portion 112 .
In the filling step, the monolithic refractory is filled in the gap between the circumferential wall portion 112 and the refractory bricks so as to be in contact with the circumferential wall portion 112 and the refractory support 114 .
The placement process and the filling process are performed in parallel.

In the arrangement step, specifically, as shown in FIG. 4A, the clarification tube 102 is arranged so that the circumferential wall portion 112 is positioned with a gap S2 with respect to the inner peripheral region 114b of the lower support 114B. . The gap S2 can be secured by holding the clarification tube 102 . The clarification tube 102 can be held, for example, by using jigs attached to a plurality of locations in the axial direction of the circumferential wall portion 112 . Also, other means such as cement may be used to hold the clarification tube 102 without being limited to such a method.

Next, in the filling step, as shown in FIG. 4A, the monolithic refractory 113' is filled so as to fill the gap S2 between the peripheral wall portion 112 and the inner peripheral wall portion 114b. The monolithic refractory 113' is, for example, poured into the gap S2 by a predetermined amount using a scoop or the like, and the air bubbles existing in the monolithic refractory 113' or between the monolithic refractories 113' are removed. In order to remove it, it is preferable to use a member such as a wire to stir in the gap S2. The monolithic refractory 113′ has a viscosity at 25° C. of 300 dPa·s or less, preferably 250 dPa·s or less, more preferably 30 A viscosity of up to 250 dPa·s, more preferably 50 to 250 dPa·s, and even more preferably 50 to 200 dPa·s is used.

The monolithic refractory 113' contains refractory oxide powder and water. Examples of refractory oxides include alumina cement. The mass ratio w/c (%) of the refractory oxide powder c and the water w of the monolithic refractory 113' is preferably 30% or less, more preferably 10 to 30%, still more preferably 16 to 24%, Especially preferred is 18 to 22%.

Next, in the filling step, as shown in FIG. 4(b), a monolithic refractory 113' is applied so as to cover the surface of the first wall region 112a of the peripheral wall 112, and then placed. In the process, the upper support 114A is arranged such that the inner peripheral wall surface 114a of the upper support 114A is arranged with a gap S1 from the first wall region 112a. The gap S1 can be secured, for example, by jigs attached to a plurality of locations in the axial direction of the circumferential wall portion 112 . The jig is removed from the clarification tube 102 after curing the monolithic refractory 113' and before the temperature rising step. The securing of the gap S1 is not limited to the above method using a jig, and may be performed using other means such as holding using a monolithic refractory with higher viscosity.

In the temperature raising step, after the construction step, an electric current is passed through the circumferential wall portion 112 to heat it, and the temperature of the clarification tube 102 is raised to the above temperature range for defoaming.

According to the method for manufacturing a glass substrate of the present embodiment, since the gap S2 is wider than the gap S1, the monolithic refractory 113' can be easily poured into the gap S2 in the construction process, and as a result, the refractory protective layer with few gaps can be obtained. Easy to get 3. Therefore, after heating the fining tube 102, the fining tube 102 can be reliably supported by the refractory support 114 during operation, as described above. On the other hand, a refractory protective layer 113 with few voids can be formed between the fining tube 102 and the refractory support 114 .

Although the glass substrate manufacturing apparatus and the glass substrate manufacturing method of the present invention have been described in detail above, the present invention is not limited to the above-described embodiments, and various improvements and modifications can be made without departing from the gist of the present invention. Of course you can.

100 melting device 101 melting tank 102 clarification tube 102a wall 103 stirring tank 103a stirrer 104, 105 transfer tube 106 glass supply tube 112 wall 112a first wall region 112b second wall region 113 refractory protective layer 113' Monolithic refractory 114 Refractory support 114a First inner peripheral wall surface 114b Second inner peripheral wall surface 114A Upper support 114B Lower support 200 Forming device 300 Cutting device MG Molten glass SG Sheet glass

Claims (5)

a tubular member made of a material containing a platinum group metal, the tubular member having a circumferential wall for processing molten glass inside the wall;
a refractory support positioned to surround and support the wall;
a refractory protective layer formed in a gap between the wall and the refractory support so as to be in contact with the wall and the refractory support;
The wall portion has a first wall region located in a partial region on the circumference and a second wall region located below the first wall region,
The size of the second gap S2 between the second wall region and the support is 1 of the size of the first gap S1 between the first wall region and the support. .1 times or more , the glass substrate manufacturing apparatus characterized by the above-mentioned. 2. The glass substrate manufacturing apparatus according to claim 1, wherein the size of said second gap S2 is 10 times or less the size of said first gap S1. the outer peripheral surface of the wall portion has a circular shape in a cross section perpendicular to the axial direction of the tubular member;
Of the inner peripheral surface of the support that faces the outer peripheral surface, the inner peripheral region of the support that faces the wall region has an arc shape in the cross section,
A second inner peripheral region of the support facing the second wall region has a larger radius of curvature in the cross section than a first inner peripheral region of the support facing the first wall region. The glass substrate manufacturing apparatus according to claim 1 or 2, comprising: The first wall region and the second wall region are regions above and below the central axis of the pipe member in a cross section perpendicular to the axial direction of the pipe member. 4. The glass substrate manufacturing apparatus according to any one of 1 to 3. A method for manufacturing a glass substrate,
A treatment step in which molten glass is treated while flowing through a tubular member made of a material containing a platinum group metal and having a circumferential wall;
and a preparation step performed before the treatment step,
The preparation step includes
an arrangement step of arranging a refractory support so as to surround the wall; and a monolithic refractory in a gap between the wall and the refractory support so as to be in contact with the wall and the refractory support. a filling step of filling the
After the construction step, a temperature raising step of raising the temperature of the pipe member,
The wall portion has a first wall region located in a partial region on the circumference and a second wall region located below the first wall region,
The size of the second gap S2 between the second wall region and the support is 1 of the size of the first gap S1 between the first wall region and the support. .A method for producing a glass substrate, characterized in that the ratio is 1 or more .

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