JPH0134445B2 - - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention relates to an apparatus for continuously producing a polymer.

Emulsion polymerization, suspension polymerization, solution polymerization, and bulk polymerization are known methods for producing polymers, but these polymerization methods have slightly different properties of the resulting polymers, so they are difficult to manufacture. They are selected and adopted as appropriate depending on the target polymer. When viewed as a polymerization reaction process, a continuous bulk polymerization method is pointed out as a preferable method because it saves resources and energy, and can solve the problem of pollution by making it a closed process. However, at present, continuous bulk polymerization methods have various problems related to the instability of the polymerization system, the viscosity that increases as the polymerization progresses, and the heat removal area that decreases relative to the reaction volume as the scale increases. remains to be resolved.

Generally, in bulk polymerization, the viscosity within the reaction system increases exponentially as the polymerization reaction progresses. In such a case, a so-called abnormal stagnation part that does not move forever tends to grow in a certain part of the reaction system. Since this abnormal retention area remains at high temperature for a long time, the polymer produced in this area is likely to deteriorate or gel, and if this is mixed into normal polymer, the quality of the produced polymer will be significantly impaired.

Various methods have been proposed in the past to eliminate such abnormal retention areas. One method is to terminate the polymerization in a state where the viscosity of the polymerization solution is low without increasing the final polymerization rate, or to carry out the polymerization by mixing a certain amount of solvent. Since the polymerization liquid handled by this method has a low viscosity, abnormal stagnation areas are less likely to occur, but there is a drawback that the operating rate of the apparatus becomes poor.

Another method is to use a screw-type stirring blade or the like that can apply shear to the polymerization liquid and increase the shear rate of the liquid near the wall of the reactor as much as possible. but,
In this case, not only is power consumed significantly, but the heat of stirring results in an increase in the temperature within the system. Further, depending on the polymer, the physical properties of the product obtained may deteriorate when subjected to strong shearing.

Continuous polymerization apparatuses generally include a complete mixing tank reactor, which is a differential reactor, and a tube or tower reactor, which is an integral reactor. When performing continuous bulk polymerization using a complete mixing tank reactor, it is necessary to make the reaction system homogeneous, so vigorous stirring is required in the highly viscous liquid, and the above-mentioned power consumption is required. As it increases, it becomes more susceptible to shear, and the residence time distribution of the liquid in the system becomes wider. Therefore, when carrying out continuous bulk polymerization using a tubular or tower reactor, if measures can be taken to prevent abnormal stagnation, it is not necessary to make the entire reaction system homogeneous, and there is no need to stir so vigorously. Furthermore, since the residence time distribution of the liquid within the system is extremely narrow and similar to a piston flow, such a tubular or tower reactor can be said to be a reactor suitable for continuous bulk polymerization.

However, conventionally used tubular or tower reactors have problems such as the presence of abnormal stagnation, piston flow characteristics, and equipment manufacturing. For example, in the case of using a tower reactor as described in JP-A-53-99290, there are no problems in terms of the presence of abnormal stagnation parts and equipment fabrication, but the piston flow is poor and the residence time distribution is narrow. I can't say that.

In view of this situation, the present inventors conducted intensive research to develop a column-type reactor suitable for continuous bulk polymerization that does not produce abnormal retention areas, has a narrow residence time distribution, and is easy to manufacture, and as a result, the present invention was achieved. This is what I did.

That is, the present invention comprises a cylindrical reaction vessel having a structure elongated in the direction of liquid flow and having a liquid inlet and a liquid outlet, and a rotating shaft attached to the inside of the reaction vessel. Double helical band type stirring blades are arranged so that most of them face the same direction, and between the stirring blades there are baffles that partition the inside of the reaction vessel into a plurality of cells. The continuous bulk polymerization apparatus is characterized in that an auxiliary stirring blade is attached to the rotating shaft inside at least one of the cells facing the liquid inlet and the cell facing the liquid outlet.

Monomers that can be bulk polymerized in the continuous polymerization apparatus of the present invention include styrene, α-methylstyrene, styrene in which the benzene ring is substituted with alkyl,
For example, O-, m-, p-methylstyrene, O-,
m-, P-ethylvinylbenzene and styrene in which the benzene ring is halogenated, such as O-, m-,
Examples include alkenyl aromatic compounds such as p-chlor or bromstyrene. These can be used alone or in mixtures as monomers. Further, a copolymerizable monomer such as acrylonitrile or methacrylic acid ester may be added to these alkenyl aromatic monomers. Furthermore, rubbery polymers such as polybutadiene, butadiene and styrene, acrylonitrile,
Various copolymers such as methyl methacrylate, natural rubber,
Solutions in which polychloroprene, ethylene-propylene copolymer, ethylene-propylene-diene monomer copolymer, etc. are dissolved in one or more of the above-mentioned monomers can also be used.

Monomers that can be subjected to bulk polymerization reactions in the continuous polymerization apparatus of the present invention are as described above, but they can also be applied to those that cause addition polymerization reactions and those that cause condensation polymerization reactions such as nylon and polyester. obtain. The term "bulk polymerization" as used herein also includes solution polymerization using 30% by weight or less of a solvent.

Polymerization can be initiated thermally or by known initiators that release free radicals upon decomposition, such as azo compounds such as azodiisobutyronitrile or peroxides such as benzoyl peroxide. be able to.

An example of a continuous bulk polymerization apparatus according to the present invention is shown in FIGS. 1, 3, and 4, and the effects of the present invention will be explained with reference to FIG.

In the figure, reference numeral 1 denotes a cylindrical polymerization reaction vessel that is long in the direction of flow of the liquid and is equipped with a jacket 2, which allows heating, heat retention, or cooling as appropriate. The jacket may be one, or may be divided into several pieces. 8 and 9 are the inlet and outlet of the heating medium.

Reference numeral 3 denotes a rotating shaft, and a stirring blade 4, a baffle plate 5, and an auxiliary stirring blade 11 are attached to the rotating shaft.

The interior of the reaction vessel is divided by baffles into chambers (cells) each containing one stirring blade.

6 and 7 are a liquid inlet and a liquid outlet, and the liquid enters from 10 and exits from 11.

The stirring blades in each cell are all attached in the same direction, and the rotation of the blades makes the liquid flow state in each cell almost the same. Note that the circulation flow indicated by the arrow in the stirring blade portion in the figure is the partial flow direction of the liquid. The rotation direction may be either direction, but preferably, as shown in Figure 1 or Figure 5, when the blades rotate, the flow of liquid near the wall surface of the reaction vessel is opposite to the direction of flow of liquid throughout the reaction vessel. It is better to rotate it accordingly. If the reaction vessel is rotated so that the flow of the liquid near the wall surface is the same as the flow direction of the liquid throughout the reaction vessel, short passes of the liquid are likely to occur even if a baffle plate is present, and the piston flow properties are deteriorated.

It is preferable that the stirring blades in each cell are all oriented in the same direction, and at least 80% or more of them must be oriented in the same direction. For example, Tokukai Akira
53-99290, if the stirring blades are installed in alternating or irregular orientations, the movement of liquid between each cell will be hindered to some extent and the entire interior of the reactor will be uniform. Although this is not the case, the residence time distribution becomes wider than when they are attached in substantially the same direction, and this cannot be adopted for the purpose of the present invention.

FIG. 2 shows a residence time distribution curve measured using the apparatus shown in FIG. 1 to examine the piston flow properties of fluid. θ is time, average residence time is φ=θ/
θ is an elementalized time, and E(φ) is a residence time distribution function. In the figure, curve A is the residence time distribution curve when the stirring blades are all attached in the same direction, and curve B is the residence time distribution curve when the stirring blades are attached in alternate directions. Compared to curve B, curve A has a narrower residence time distribution and improved piston flow properties. Curve C is 8
This is a residence time distribution curve when only one of the stirring blades is attached in a different direction, and is almost the same as curve A.

The stirring blades of the apparatus of the present invention are preferably half-pitch double helical band blades, but other double helical band blades such as one-pitch double helical band blades may also be used. Alternatively, a screw or the like may be combined with the double helical band blade.

A schematic diagram of a double helical band type stirring blade used in the continuous bulk polymerization apparatus of the present invention is shown in FIG.

Regarding the blade width b of the double helical band stirring blade, its ratio to the inner diameter D of the reaction vessel is 0.05≦b/D≦
It is preferable to use one that satisfies the relationship of 0.3. If b/D is smaller than 0.05, the amount of liquid sent out by the blades is small, which causes abnormal stagnation. On the other hand, if b/D is larger than 0.3, the amount of liquid sent out by the blades is too large and each cell This is undesirable because the movement of the liquid between the stirring blades increases, making it impossible to obtain a narrow residence time distribution, and at the same time, the power required to rotate the stirring blades increases.

The outer diameter d of the double helical band type stirring blade is preferably set so that the clearance (δ=D−d/2) with the inner wall of the reaction vessel satisfies 1 mm<δ<30 mm. If δ is less than 1 mm, it is undesirable because it is difficult to manufacture the device and the stirring blade may come into contact with the reaction vessel. If δ is more than 30 mm, the area near the wall of the reaction vessel may be abnormally stagnant. This is undesirable as it causes generation.

Regarding the axial length of the double helical band stirring blade, 8h,
For the axial length L of each cell with stirring blades, h/
It is preferable that L≧0.5. If h/L is less than 0.5, the space between the stirring blade and the baffle plate becomes wide, which is not preferable because it causes an abnormal stagnation part to occur there.

There is no particular restriction on the rotation speed of the stirring blade when producing a polymer using the apparatus of the present invention, and it is 1 rpm.
If it is above, there is an effect of preventing abnormal retention near the inner wall of the reaction vessel. However, if the rpm is higher than 30 rpm, the movement of liquid between each cell will be rapid, making it impossible to obtain a narrow residence time distribution, and the power required for stirring will increase.
Generally undesirable.

In the device of the present invention, the presence of a baffle plate having a partitioning effect between the stirring blades prevents the free movement of the liquid between each cell, resulting in a residence time distribution close to that of a piston flow. If there is no baffle plate having a partitioning effect, the flow of liquid in the reaction container will not be separated within each cell, and the entire reaction container will be uniform, resulting in an extremely wide residence time distribution. Curve D in Figure 2
Figure 1 shows the residence time distribution curve for the device shown in Figure 1, with all stirring blades mounted in the same direction but no baffle plates, and shows a residence time distribution close to that of a complete mixing tank type reactor. A very wide range of results have been obtained.

Regarding the baffle plate that has a partitioning effect and is installed between the stirring blades, the opening area ratio thereof is 5 to 40%, preferably 7 to 30%, of the cross-sectional area inside the tank.
It is best to keep it within the range of %. Opening area ratio is 5%
If less than 40% is used, the viscosity of the polymerization liquid will increase, causing the polymer to adhere to the baffle plate, which is undesirable, and if more than 40% is used, the movement of the liquid between adjacent cells may become unfavorable. The number of partitions increases, and the partitioning effect disappears.

The opening area ratio referred to here is the ratio of the sum of the area of the opening on the baffle plate and the area of the clearance between the baffle plate and the inner wall of the reaction vessel to the cross-sectional area of the polymerization reaction vessel perpendicular to the rotation axis. This is the value shown by . However, when using a multi-tubular heat exchanger as a baffle plate, the opening area ratio is the sum of the area of the opening on the baffle plate and the clearance between the baffle plate and the rotation axis perpendicular to the rotation axis. This is a value expressed as a ratio to the cross-sectional area of the reaction vessel.

As the baffle plate having a partitioning effect, a baffle plate of a disc-shaped perforated plate having the above-mentioned opening area ratio that rotates with the shaft is suitable, but the baffle plate is not particularly limited thereto. For example, other baffles include shell-and-tube heat exchangers or other heat exchangers having the above-mentioned opening area ratio. When using these heat exchangers, if the opening area ratio is kept within the above range, it is possible to have a partitioning effect, and at the same time, a large amount of polymerization heat generated when monomers are polymerized can be removed, and the polymerization equipment This also solves the problem of removing polymerization heat when scaled up.

FIG. 5 shows an example of a continuous bulk polymerization apparatus in which a disk-shaped perforated plate rotating with the aforementioned shaft and a multi-tubular heat exchanger are simultaneously used as baffles having a partitioning effect.

Reference numeral 1 denotes a cylindrical polymerization reaction vessel which is long in the direction of flow of the liquid, and two of them are stacked on top of each other at a flange 13 for use.

In this device, a rotating disc-shaped perforated plate 5 and a multi-tubular heat exchanger 12 are used as baffle plates having a partitioning effect. The shell-and-tube heat exchanger 12 is
In addition to having a partitioning effect, the heat of polymerization can also be removed by flowing a heat medium from 8 to 9. That is, in this apparatus, the heat of polymerization can be removed by the jacket 2 and the shell-and-tube heat exchanger 12. When the equipment is scaled up, jacket 2 alone will not be sufficient to remove polymerization heat, but the insufficient polymerization heat removal area will be
This problem can be solved by appropriately adjusting the size of the multi-tubular heat exchanger 12.

14 in FIG. 1 or 5 is an auxiliary stirring blade. The auxiliary stirring blade is attached to the rotating shaft in the cell facing the liquid inlet of the polymerization reaction vessel, the cell facing the liquid outlet, or both. At the liquid inlet or liquid outlet of the reactor, since the liquid inlet or the liquid outlet does not cover the entire cross section of the reaction vessel, it may be partially damaged, especially in cases such as those shown in Figure 1 or Figure 5. Abnormal stagnation tends to occur near the flange around the rotating shaft. The auxiliary stirring blades are provided to prevent the formation of such abnormal stagnation portions, and are preferably installed at both the liquid inlet and the liquid outlet.

FIG. 4 shows a schematic diagram of the auxiliary stirring blade used in the present invention.

The auxiliary stirring blades are preferably a plurality of paddle blades, for example two or four paddle blades, but other stirring blades such as curved paddle blades or inclined paddle blades may also be used.

Regarding the size of the auxiliary stirring blade, the blade length d′ is d′<
D-2b-2δ, and it is preferable that the wing span b' be such that b'<h/3. If d' is larger than D-2b-2δ, it is not preferable because it cannot be installed inside the double helical band stirring blade. Also, b′ is h/
If it is larger than 3, the amount of liquid discharged by the auxiliary stirring blade becomes too large, which disturbs the liquid flow pattern by the double helical band stirring blade and increases the required power, which is not preferable.

The position of the auxiliary stirring blade is not particularly limited, but it is preferably as close to the flange as possible, and more preferably the distance δ' from the flange satisfies δ'<b'.

When performing continuous polymerization using the continuous bulk polymerization apparatus according to the present invention, the viscosity of the polymerization solution is 1 poise to 1 poise.
30000 poise is appropriate. There is no need to use this device for low-viscosity polymerization liquids of 1 poise or less, but for high-viscosity polymerization liquids of 30,000 poise or more, there are problems such as increased stirring power and the occurrence of abnormal stagnation. Even when using this device, measures other than those according to the present invention are required.

As described above, by using the apparatus of the present invention, continuous bulk polymerization with extremely narrow residence time distribution close to piston flow can be carried out without producing abnormal stagnation portions and with a small amount of power. At the same time, this device is extremely simple to manufacture in that double helical band blades and the like used in general industrial equipment can be used as they are.

In addition, compared to a complete mixing tank type continuous bulk polymerization device, this device requires less rotational speed of the stirring blades, so it does not require as much power, and it can be operated with a high viscosity polymerization solution, which reduces the amount of solvent used. This makes it possible to reduce the amount of polymerization or increase the final polymerization rate, resulting in an efficient polymer production process.

Furthermore, this apparatus can be used either horizontally or vertically, and a side feed port can be provided in the middle of the reaction tank to continuously inject and mix monomers, solvents, or various additives.

From these points, the present invention is extremely versatile and is not limited to the limited illustrations in the specification, and any device that satisfies the contents described in the claims is included in the present invention. It is.

Examples are shown below.

Example 1 95% by weight styrene monomer, 5% by weight commercially available polybutadiene (e.g. Diene 55 from Asahi Kasei)
After mixing and dissolving the monomer composition, the mixture was continuously fed at 3.0/Hr into a 3.0 complete mixing tank reactor equipped with a screw and a draft tube, and prepolymerization was carried out at 135°C. . This polymerization liquid was continuously taken out from the reactor and continuously supplied to the main polymerization reactor for subsequent polymerization.

As the main polymerization reactor, the cylindrical reactor of the present invention shown in FIG. 1 was used. This reactor has an inner diameter of 10
A cylindrical reactor with a length of 40 cm and a jacket and a liquid inlet and outlet. This reaction vessel has a rotating shaft in the center, and on the rotating shaft are eight half-pitch double helical band stirring blades, and between the stirring blades are disc-shaped porous holes that rotate with the shaft. A plate is attached to divide the inside of the reaction vessel into eight cells. The size of the stirring blade is blade width 2
cm, wing axis length 4 cm, and wing outer diameter 9.5 cm.
The clearance between the inner wall of the reaction vessel and the stirring blade is 2.5mm.
It is. The stirring blades are all oriented in the same direction, and the direction of rotation is such that the liquid flows as shown in FIG.
The disc-shaped perforated plate has a diameter of 9.5 cm and a thickness of 2 mm, and has a total of 24 holes of 4 mm in diameter arranged radially in eight directions around the axis (opening area ratio: 14%). Also,
Four paddle blades with a blade length of 4 cm and a blade width of 0.5 cm are installed on both the cells facing the liquid inlet and liquid outlet for auxiliary stirring, and the clearance between each flange is 2.
It is attached to the rotating shaft so that the diameter is 1 mm. Three such reactors were connected in series to carry out the main polymerization reaction.

The prepolymerized polymer solution described above was continuously supplied to the first main polymerization reactor and heated to 130°C through a jacket.
Heat to 10 rpm and stir at 10 rpm to polymerize.
Furthermore, it was supplied to the second main polymerization reaction vessel. Before the second main polymerization reactor, a 2:1 mixture of ethylbenzene and white mineral oil was continuously fed as a solvent and an additive at a rate of 0.3/hr, and mixed in the second main polymerization reactor. In the second main polymerization reactor, the jacket was heated to 135° C. and stirred at 10 rpm to effect polymerization, and was then continuously fed to the third main polymerization reactor. In the third main polymerization reactor, the jacket was heated to 155°C and the polymerization was completed with stirring at 5 rpm.

The monomer conversion rate in the polymerization liquid exiting the third main polymerization reactor was 86% by weight. The temperature of the polymerization solution was 165°C.

The polymerization liquid continuously discharged from the third main polymerization reactor is subjected to a conventionally known devolatilization method to remove unreacted monomers and solvents, and is then turned into pellets using an extruder to improve impact resistance. A polystyrene product was obtained.

The final product thus obtained exhibits the following properties.

Rubber content 5.8% by weight Intrinsic viscosity of the soft phase 0.74 (measured in toluene at 30°C) Melt flow index (190°C)
0.91g/10min Izot impact strength 9.5Kg・cm/cm (with notch) Tensile strength 240Kg/cm ² Tensile elongation 64% Example 2 This example uses the multi-tube thermal reactor shown in Figure 5 as the main polymerization reactor. This experiment was carried out using a reactor of the present invention equipped with an exchanger, and the same conditions as in Example 1 were used except for the following points.

a. Use 20 complete mixing tank reactors for prepolymerization.

b. A monomer composition in which polybutadiene rubber is dissolved is supplied at a rate of 20/Hr.

c. Flow a 120°C heat medium through the jacket and heat exchanger of the first main polymerization reactor. (Rotation speed: 10 rpm) d. Flow a heat medium at 130°C through the jacket and heat exchanger of the second main polymerization reactor. (Rotation speed is 10 rpm) e. Flow a heat medium at 150°C through the jacket and heat exchanger of the third main polymerization reactor. (Rotation speed is 5 rpm) f A 2:1 mixture of ethylbenzene and white mineral oil is fed at 3.0/Hr before the second main polymerization reaction vessel.

The following main polymerization reactor was used.

The total length of the reaction vessel is 80 cm, and the inner wall is 20 cm in cell section.
cm, and are divided into two parts that are stacked on top of each other. The reaction vessel was equipped with a rotating shaft at the end, and six half-pitch double helical band type stirring blades were attached to the rotating shaft. The entire reaction vessel is partitioned into six cells by a disk-shaped perforated plate rotating with three shafts and two shell-and-tube heat exchangers. The stirring blade has a blade width of 2cm, a blade axis length of 7cm, and an outer diameter of the blade.
A 19.5 cm one was used, and the clearance with the inner wall of the reaction vessel was 2.5 mm. The stirring blades are all oriented in the same direction, and the direction of rotation is such that the liquid flows as shown in FIG. The disc-shaped perforated plate has a diameter of 19.5cm and a thickness of 2.
18 mm size holes with a diameter of 1.4 cm in an equilateral triangular array
There are enough pieces. (Opening area ratio 14%). As for the shell-and-tube heat exchanger, we use a shell-and-tube type with 18 tubes with an inner diameter of 1.4 cm and a length of 10 cm arranged in an equilateral triangle, and one tube with an inner diameter of 4 cm and a length of 10 cm in the center (opening area ratio 11%). Further, one auxiliary stirring blade consisting of four paddle blades is attached to each cell facing the liquid inlet and the liquid outlet. The size of the four paddle wings is 10cm long and 1.5cm wide.
The distance from each flange is 3 mm.

When carried out under these conditions, the final product obtained was almost the same as in Example 1.

Reference Example 1 Same conditions as Example 1, but continuous polymerization using a 9.0 cm diameter disc with 6 3 cm diameter holes (opening area ratio 73%) instead of the disc-shaped perforated plate in the main polymerization reactor. Polymerization was carried out.

As a result, the temperature distribution within the reaction vessel was disturbed, and the structure and physical properties of the resin were unstable and inferior to those of Example 1.

Comparative Example 1 Same conditions as Example 1, but with a blade width of 3 cm and a blade length instead of the double helical band type stirring blade in the main polymerization reactor.
Continuous polymerization was carried out using four 9.5 cm paddle-type stirring blades.

As a result, a high-temperature region was generated in the reaction tank, and stable operation was not possible.

[Brief explanation of drawings]

1 and 5 are schematic diagrams of a continuous bulk polymerization apparatus according to the present invention. 1: Polymerization reactor main body, 2: Jacket, 3: Rotating shaft, 4: Stirring blade, 5: Baffle plate, 6: Liquid inlet,
7: Liquid outlet, 8: Heat medium inlet, 9: Heat medium outlet, 1
0: liquid inflow direction, 11: liquid outflow direction, 12: multi-tubular heat exchanger, 13: flange, 14: auxiliary stirring blade. FIG. 2 shows residence time distribution curves measured using the apparatus shown in FIG. 1 with various specifications changed. FIG. 3 is a schematic diagram of a double helical band type stirring blade used in the continuous bulk polymerization apparatus of the present invention. D: inner diameter of polymerization reaction container, b: blade width of stirring blade,
δ: clearance between the inner wall of the polymerization reaction vessel and the outer periphery of the stirring blade, h: axial length of the stirring blade, L: axial length of the cell with the stirring blade. FIG. 4 is a schematic diagram of an auxiliary stirring blade used in the continuous bulk polymerization apparatus of the present invention. d′: Wing length, b′: Wing span, δ: Distance between wing and flange.

Claims (1)

[Scope of Claims] 1. Consists of a cylindrical reaction vessel having a structure elongated in the direction of flow of the liquid and equipped with a liquid inlet and a liquid outlet, and a rotating shaft attached to the inside of the reaction vessel, the rotating shaft being attached to the rotating shaft. A plurality of double helical band type stirring blades are arranged so that most of them face the same direction, and a baffle plate is provided between the stirring blades to partition the inside of the reaction vessel into a plurality of cells. A continuous bulk polymerization apparatus characterized in that an auxiliary stirring blade is attached to the rotating shaft inside at least one of the cells facing the liquid inlet and the cell facing the liquid outlet. 2. The apparatus according to claim 1, wherein the ratio of the blade width b of the double helical band stirring blade to the inner diameter D of the reaction vessel is 0.05≦b/D≦0.3. 3 The clearance δ between the inner wall of the reaction vessel and the outer periphery of the double helical band stirring blade is 1 mm<δ<
3. The device according to claim 1 or 2, which has a diameter of 30 mm. 4 The ratio of the axial length h of the double helical band type stirring blade to the axial length L of the cell in which the stirring blade exists is h/
The device according to claim 1, 2 or 3, wherein L≧0.5. 5. The apparatus according to claim 1, 2, 3, or 4, wherein the opening area ratio of the baffle plate to the cross-sectional area of the reaction vessel is in the range of 5 to 40%. 6. The device according to claim 1, 2, 3, 4 or 5, wherein the auxiliary stirring blades are a plurality of paddle blades. 7 In the auxiliary stirring blade, the blade length d′ satisfies d′<D−
2b-2δ and the blade span b' is b'<h/3.