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Abstract： Straight- and curved-bar refining plates are two important types of plates commonly used in disc refiners in the papermaking industry. Theoretically， the curved-bar refining plate has a relatively uniform bar interaction angle， which indicates uniform refining effects. The bar angle of the curved bar was proposed and two typical curved-bar plates， the three-stage radial curved-bar plate and isometric curved-bar plate， were designed in this paper. The arc equations of the curved-bar center line and curved-bar edges were established and finally， the specific edge load （SEL） of the curved-bar plate was derived. The determination of bar parameters was discussed， which provides a theoretical basis for the design of curved-bar plates.
Keywords： disc refiner; refining plates; curved bar; design method
1 Introduction
The disc refiner is an important piece of equipment to improve the properties of fiber and pulp in the pulp and paper industry. During the refining process， the pulp is fed into a groove-type rotating refining area composed of a stator and a rotor. It is subjected to complex mechanical actions， such as shearing and compression of the bar as well as friction between fibers， so that the fiber morphology and the properties of pulp are changed， and finally， the desired properties of the target paper are achieved.
The plate is the core working part of the disc refiner， and its bar profile directly affects the refining quality， efficiency， and energy consumption. Plates are usually assembled by many segments with the same bar structure， which includes the bar width， groove width， bar height， bar angle， and dams[1]. Many plates with different bar parameters are used in mills， so it is important to understand how the bar profile affects the changes in the pulp and fiber.
According to the form of the bar distributed in the refining area， the plate type can be roughly divided into straight-bar and curved-bar plates. The conventional bar angle of a straight-bar plate is 15°～20°. It is the most widely used type of refining plate at present， but has a disadvantage in that the bar-interaction angle changes during the interaction between the rotor and stator. If the straight-bar plate has a bar angle of 10°， the bar-interaction angle will vary between 15° and 40°， with an average angle of 30°[2]. The pulp flow in the refining area is unstable， or the pulp layer would be blocked because of the change in the bar-interaction angle; as a result， the quality of the fibers refined at different positions is uneven， which affects the paper properties. Theoretically， there is a certain curvature on the curved bar， so that using curved-bar plate in refining process will solve the problem of excessive changes in the interaction angle to a certain extent， especially for the logarithmic spiral-bar plate， which creates a constant bar-interaction angle. The quality of the fiber or pulp will be approximately uniform based on theoretical analysis， but the mathematical analysis and experimental research of the curved-bar plate need to be further developed. Many researchers hold different views on the refining characteristics of the curved-bar plate. The medium-density fiberboard （MDF） spiral plate and LemaxX series plate proposed by Andritz[3] adopt a spiral-bar design to ensure the stability of the bar-interaction angle and uniformity of the pulp quality. Some studies expressed that the cutting effect of the curved-bar plate is weaker than that of the straight bar[4-6]， but the actual refining effect has not been tested and verified experimentally.
As a kind of ordinary bar plate， the curved-bar plate has many applications in the pulp and paper industry， which can be divided into radial type and isometric type according to the distribution of bars[7]. The arc of the radial curved bar is distributed along the radial direction and along the circumference of the center of the circle， as shown in Fig.1. Moreover， there are more bars in the inner part of the refining zone and fewer in the outer part， which will reduce the effective refining area when only one level of curved bars is placed in the refining zone. To solve this problem， the bars are usually arranged again at a larger groove width. The isometric curved-bar plate， which has the same groove width in the refining zone， can also solve the problem mentioned above， as shown in Fig.2. The isometric-bar plates are usually manufactured with several identical segments， and their refining effects are uniform compared to those of radial curved-bar plates， theoretically.
At present， the design of refining plate is based on specific edge load （SEL） theory[8]， which has limitations in some degree because it gives less consideration to the bar parameters， such as bar angle and bar width. The design of the plates should consider the refining intensity， residence time of fibers in the refining area， and hydraulic performance， and the bar parameters， such as bar width， groove width， bar angle， and dams， should be selected reasonably to achieve specific requirements while ensuring production. However， there is no clear basis for the design of curved-bar plates， which have complex bar edges， unlike the straight bar. In this study， the design of multi-stage radial curved-bar plates and isometric curved-bar plates was conducted.
2 Definition of bar angle of curved bar
Leider et al designed a curved-bar plate for the pulp and paper industry[9]， as shown in Fig.3， using the angle （a+90°） between AB and OB and b between the tangent of the curved-bar end point C and the radius OC to indicate the curved bar， in which the definition of the angles at starting point B and end point C are different. Hackl et al[10] designed a non-paper refining plate in which the curved bar is represented by the angles a and b at starting point B and ending point C， as shown in Fig.4， and the drawing and measuring of the tangent are complicated. The above two methods define only one arc angle of the curved bar， which cannot express the full arc of the curved bar. Through analysis of predecessors， a new definition of the curved bar is proposed in this paper. The arc that passes through point C， the intersection of the center line of the segment and the center circle of the refining area， is called the curved-bar center line and is denoted by two angles a （the starting angle of the curved bar） and b （the bar angle of the curved bar）， as shown in Fig.5. Therefore， the curved-bar center line is determined by the location B and the bar angle that measures the curvature of the curved bar.
3 Design of curved-bar refining plates
3.1 Design of radial curved-bar plate
The groove width of radial-bar refining plates is gradually increased from the inner part to the outer part of the refining area. If only a one-stage grinding tooth is used; the effective refining area is reduced， which yields a lower utilization efficiency of the refining zone. Therefore， multi-stage refining plates have been developed. Fig.6 shows a three-stage radial curved-bar plate， which inserts more bars between the adjacent bars where the bar angle of the different-stage curved bar is different. In this part， a three-stage radial curved-bar plate is designed as follows.
3.1.1 Design of the 1st-radial arc of curved bar
The main technical parameters of the refining plate are shown in Fig.7. If the 1st-stage curved-bar angle is b， the center line of the curved bar is arc AC， which has a bar angle of b and radius of Ra.
The origin O1 of the center line of the 1st-stage curved bar was selected as the pole， and the function of the circle of arc AC is expressed by Eq.（1）.
If the bar width of the 1st-stage curved bar is b， the functions of its inner and outer arc， as shown in Fig.8， can be expressed by Eq.（2）.
Where the inner arc of the 1st-stage curved bar is obtained by subtraction and outer one is obtained by addition.
1st-stage curved bar
At last， the number of 1st-stage curved bars， n， should be determined by the size of the refining plate and the process requirements. After this， the circumferential array of curved bars should be completed.
3.1.2 Design of the 2nd- and 3rd-radial arc of curved bar
The design of the 2nd- and 3rd-stage curved bars of the three-stage radial curved-bar plate is slightly different from that of the 1st-stage curved bar. According to the size of the refining plate， process requirements， and refining intensity， the refining zone was divided reasonably into the breaking zone， coarse refining zone， and refining zone. The values of R1 and R2 should be selected correctly， as shown in Fig.9 and Fig.10. The starting circle with radius Ri intersects the two center arcs of the adjacent 1st-stage curved bars. The radius of the center circle of area through which the 2nd-stage curved bar is （R1+Ro）/2， and the starting point and intermediate position of the 2nd-stage curved bar can be determined by the equal points of arcs FG and HI. In this paper， the starting point and intermediate position of the center arc of the 2nd-stage curved bar are determined by their two equal points， as shown in Fig.9. The design of the 3rd-stage curved bar is the same as that of the 2nd-stage curved bar. It should be noted that the starting point is the equal points of arc KJ， which is part of the starting circle of the 3rd-stage curved bar cut by the adjacent curved-bar center arc of the 2nd-stage curved bar. The two equal points of KJ is defined as the starting point in this paper， as shown in Fig.10.
If the bar widths of the 2nd- and 3rd-stage curved bars are a and c and the bar angles are b1 and b2， the function of the center arc of the 2nd- and 3rd-stage curved bar can be obtained by Eq.（3）.
Thus， the function of the 2nd- and 3rd-stage curved bar is
For the 2nd-stage curved bar， X=b and Y=a， and for the 3rd-stage curved bar， X=c and Y=c.
The circumferential array of the 2nd- and 3rd-stage curved bar is arranged according to the number of 1st-stage curved bars. Thus， the number of 2nd-stage curved bars is n（m-1） and that of 3rd-stage curved bars is nm（z-1）.
Multi-stage radial curved-bar plates have the same bar angle in the same-stage curved bars， which are evenly distributed in the whole circumference. Therefore， the division of the plate does not affect the orientation of the curved-bar distribution. The plates with large diameters were manufactured by segments because of the complex overall manufacturing. Therefore， the center angle of the segment should be designed reasonably to equally divide the plate.
3.2 Design of isometric curved-bar plate
Compared with the radial curved-bar plates， the distribution of the bar width and groove width is more uniform from the inner part to the outer part of the refining zone. Theoretically， the isometric curved-bar plates have a uniform bar-interaction angle， which is conducive to uniformity of refining. The design of the isometric curved-bar plates is briefly introduced by using the bar angle of the curved bar proposed above. This article takes the right-hand rotational curved bar as an example.
3.2.1 Design of the center arc of isometric curved bar
The design of the center arc of the isometric curved bar is the same as that mentioned above. However， the intersection point B of the curved-bar center line， the center circle of the refining zone， and the center line of the segment are defined based on the design of the bar angle， as shown in Fig.11. The center arc of the center curved bar is determined when the bar angle a and starting point A are selected， and its function is similar to Eq.（1） and Eq.（3）. After this， the function of the inner and outer arc can be determined considering the bar width b and groove width g， which are consistent with Eq.（2） and Eq.（4）. 3.2.2 Design of curved bar on both sides of the center bar
If the groove width of the isometric curved-bar plate is g and the bar width is b， the equation of the 1st- stage curved-bar arc， as shown in Fig.12， on the left side （the left-hand bar is on the right side） is
And the functions of the 2nth arc， when n≥1， on both sides of the center curved bar can be expressed as
Similarly， the equations of the （2n+1）th arc， when n≥1， can be determined by Eq.（7）.
When determining the function of curved-bar edges on the left side of the center bar with Eq.（6） and Eq.（7）， the arc can be obtained by addition. That of the right side can be obtained by subtraction. However， the choice of symbol is reversed for the left-hand bar.
The curved bars are arranged by the above arc equation. When the arc is full for the entire refining segment， the design of the isometric arc in a segment can be completed by trimming the arc out of the refining zone.
4 Bar profile of curved-bar plates
4.1 SEL of curved-bar plates
The SEL is a common way of quantifying refining intensity[8]， which denotes the net energy applied to each meter of the bar crossing （J/m） and is calculated by Eq. （8）.
where SEL is the refining intensity （J/m）， Pnet is the net refining power （kW）， n is the rotation speed （r/min）， and CEL is the cutting edge length （km/r）.
The SEL is proposed for straight-bar plates， and the calculation of CEL[11] （Eq.（9））， the common contact length of the opposite bars per revolution， is also for straight-bar plates， and cannot be applied to curved-bar plates.
Where r1 is the inner radius of the plate （mm）， r2 is the outer radius of the plate （mm）， nr is the total bar number of the rotor， ns is the total bar number of the stator， and a is the bar angle of the plate （°）.
The calculation principle of the curved bar’s CEL is the same as that of the straight bar， as shown in Fig.13; the bar segments are divided into several zones and the number and length of the bars in each zone are counted. CEL of the curved-bar plate is calculated by the following equation.
Where gi is the center angle of the curved-bar center line at zone i and Ri is the circle radius of the curved-bar center line at zone i.
SEL is affected by Pnet， n， and CEL of the refining process， and it should be selected reasonably for the refining process of different pulps. When refining softwood pulp， SEL should be 1.5～4.5 J/m， for hardwood pulp， it should be 0.5～1.5 J/m[12]. The refining plates are the core component of the refining process. Under the condition of constant Pnet and n， CEL can be adjusted by reasonable design of the bar parameters， through which SEL can reach the range mentioned above for effective refining. Through the analysis of Eq.（10）， CEL is directly related to the configuration of the bar width， groove width， bar angle， etc.， and the corresponding SEL can be matched by rationally designing the bar profile.
4.2 Bar-parameter determination of curved-bar plates
As a direct acting component of the refining process， the bar parameters of the refining plates have a direct influence on the refining effects. This paper introduces the design methods of two curved bars， but the selection of bar parameters is not provided， and will be briefly introduced below.
The selection of the bar profile， such as the bar width， groove width， and bar height， can be referred to that of the straight-bar plates. According to the literature[12]， for the refining of softwood pulp， with a consistency of 3.5%～4.5%， the bar width of the refining plates can be chosen from 3.0 mm to 5.5 mm， and the groove width is generally between 5 mm and 7 mm. For refining hardwood pulp， the bar width is generally from 2.0 mm to 3.5 mm， and the width of the groove is generally from 3 mm to 4 mm. Generally， the width of the groove is usually 2～3 times the average fiber length.
The bar angle is one of the key bar parameters of refining plates， and is usually 15°～20° for straight-bar plates[13]. The bar angle of curved-bar plates can be selected as （q+b/2）， in which q （15°～20°） is the bar angle of the straight-bar plate and b is the center angle of the segment， based on the different definition of the bar angle compared to that of the straight bar. Further， there is another angle that should be noted for the multi-stage radial curved-bar plate. The starting angle of a curved bar is usually between 0° and 45°， preferably between 15° and 30°. Here， it will be advantageous if a is larger than a1 and a1 is larger than a2[10].
5 Conclusions
As a common type of refining plate in the pulp and paper industry， curved-bar plates are weaker than straight-bar plates in fiber cutting， and theoretically， the refining performance provided by curved-bar plates is relatively uniform. However， the design of curved-bar plates is limited by a lack of theoretical basis. The work done by this study is described below. The representation method of the curved-bar refining plates at home and abroad was analyzed and a new method to represent the bar angle of the curved bar was proposed in this paper to clearly describe the curved bar.
The multi-stage radial curved-bar plate and isometric curved-bar plate were designed based on the bar angle of the curved bar as described in this paper. Subsequently， the mathematical expressions of the curved-bar edges and the center arc of the curved-bar plate were established， which will provide a theoretical basis for parametric modeling of curved-bar plates and enhance the efficiency of their design.
The SEL of curved-bar plates was deduced compared to that of straight-bar plates， which can provide a theoretical basis for design of curved-bar plates. Furthermore， the determination of the bar profile was discussed based on that of straight-bar plate.
Acknowledgments
The authors gratefully acknowledge the funding by the National Natural Science Foundation （Grant No. 50745048）.
References
[1] LU Qianhe. Principles and Engineering of Papermaking[M]. Beijing： China Light Industry Press， 2004： 48-50.
[2] Kenichi Ito， Yosuke Takeshita， Antensteiner P. Low consistency refining technology： LemaxX Spiral-Nature Applied[J]. Japan TAPPI Journal， 2006， DOI： 10.2524/jtappij.60.718.
[3] Mikko Pfaffli. Rethinking the Art of Refining： Improving the Efficiency and Quality of Refining[C]//CIPTE， Beijing： 2015.
[4] JIANG Simeng， YAN Zhen， JIANG Xiaojun. High consistency refiner curved bar refining plate [J]. East China Pulp and Paper Industry， 2016， 47（1）： 27-29.
[5] WANG Chengkun， WANG Ping. Logarithmic Spiral and Its Application in the Design of Refiner Plate[J]. China Pulp and Paper， 2015， 34（9）： 37-41.
[6] Liang Qianhua. A novel design of a refiner plate [J]. China Pulp & Paper Industry， 2014， 35 （24）： 33-34.
[7] Han Lubing. Research on refining zone flow field simulation and optimization the bar structure of the disc refiner[D]. Xi’an： Shaanxi University of Science and Technology， 2017.
[8] Brecht W， Siewert W. Zur theoretisch-technischen Beurteilung des Mahlprozesses Moderner Mahlmaschinen[J]. Das Papier， 1966， 20（1）： 4-14.
[9] Leider P J， Rihs J. Spiral groove pattern refiner plate， US：4023737[P]. 1997-5-17.
[10] Hackl M， Feichtinger K， Wendelin G. Rotor disk， US： 20120294725A1[P]. 2012.
[11] Technical Association of the Pulp and Paper Industry. TAPPI standard TIP 0508-05：Refiner Plate Intensity[S]. USA： Technical Association of the Pulp and Paper Industry， 2001.
[12] Hannu Paulapuro. Papermaking Part 1 Stock Preparation and Wet End[M]. 1st edition. Finland： Finnish Paper Engineers’ Association/Paperi ja Puu Oy， 2001： 47-48.
[13] He Beihai. Papermaking Principle and Engineering [M]. Beijing： China Light Industry Press， 2013： 32-34. .
Keywords： disc refiner; refining plates; curved bar; design method
1 Introduction
The disc refiner is an important piece of equipment to improve the properties of fiber and pulp in the pulp and paper industry. During the refining process， the pulp is fed into a groove-type rotating refining area composed of a stator and a rotor. It is subjected to complex mechanical actions， such as shearing and compression of the bar as well as friction between fibers， so that the fiber morphology and the properties of pulp are changed， and finally， the desired properties of the target paper are achieved.
The plate is the core working part of the disc refiner， and its bar profile directly affects the refining quality， efficiency， and energy consumption. Plates are usually assembled by many segments with the same bar structure， which includes the bar width， groove width， bar height， bar angle， and dams[1]. Many plates with different bar parameters are used in mills， so it is important to understand how the bar profile affects the changes in the pulp and fiber.
According to the form of the bar distributed in the refining area， the plate type can be roughly divided into straight-bar and curved-bar plates. The conventional bar angle of a straight-bar plate is 15°～20°. It is the most widely used type of refining plate at present， but has a disadvantage in that the bar-interaction angle changes during the interaction between the rotor and stator. If the straight-bar plate has a bar angle of 10°， the bar-interaction angle will vary between 15° and 40°， with an average angle of 30°[2]. The pulp flow in the refining area is unstable， or the pulp layer would be blocked because of the change in the bar-interaction angle; as a result， the quality of the fibers refined at different positions is uneven， which affects the paper properties. Theoretically， there is a certain curvature on the curved bar， so that using curved-bar plate in refining process will solve the problem of excessive changes in the interaction angle to a certain extent， especially for the logarithmic spiral-bar plate， which creates a constant bar-interaction angle. The quality of the fiber or pulp will be approximately uniform based on theoretical analysis， but the mathematical analysis and experimental research of the curved-bar plate need to be further developed. Many researchers hold different views on the refining characteristics of the curved-bar plate. The medium-density fiberboard （MDF） spiral plate and LemaxX series plate proposed by Andritz[3] adopt a spiral-bar design to ensure the stability of the bar-interaction angle and uniformity of the pulp quality. Some studies expressed that the cutting effect of the curved-bar plate is weaker than that of the straight bar[4-6]， but the actual refining effect has not been tested and verified experimentally.
As a kind of ordinary bar plate， the curved-bar plate has many applications in the pulp and paper industry， which can be divided into radial type and isometric type according to the distribution of bars[7]. The arc of the radial curved bar is distributed along the radial direction and along the circumference of the center of the circle， as shown in Fig.1. Moreover， there are more bars in the inner part of the refining zone and fewer in the outer part， which will reduce the effective refining area when only one level of curved bars is placed in the refining zone. To solve this problem， the bars are usually arranged again at a larger groove width. The isometric curved-bar plate， which has the same groove width in the refining zone， can also solve the problem mentioned above， as shown in Fig.2. The isometric-bar plates are usually manufactured with several identical segments， and their refining effects are uniform compared to those of radial curved-bar plates， theoretically.
At present， the design of refining plate is based on specific edge load （SEL） theory[8]， which has limitations in some degree because it gives less consideration to the bar parameters， such as bar angle and bar width. The design of the plates should consider the refining intensity， residence time of fibers in the refining area， and hydraulic performance， and the bar parameters， such as bar width， groove width， bar angle， and dams， should be selected reasonably to achieve specific requirements while ensuring production. However， there is no clear basis for the design of curved-bar plates， which have complex bar edges， unlike the straight bar. In this study， the design of multi-stage radial curved-bar plates and isometric curved-bar plates was conducted.
2 Definition of bar angle of curved bar
Leider et al designed a curved-bar plate for the pulp and paper industry[9]， as shown in Fig.3， using the angle （a+90°） between AB and OB and b between the tangent of the curved-bar end point C and the radius OC to indicate the curved bar， in which the definition of the angles at starting point B and end point C are different. Hackl et al[10] designed a non-paper refining plate in which the curved bar is represented by the angles a and b at starting point B and ending point C， as shown in Fig.4， and the drawing and measuring of the tangent are complicated. The above two methods define only one arc angle of the curved bar， which cannot express the full arc of the curved bar. Through analysis of predecessors， a new definition of the curved bar is proposed in this paper. The arc that passes through point C， the intersection of the center line of the segment and the center circle of the refining area， is called the curved-bar center line and is denoted by two angles a （the starting angle of the curved bar） and b （the bar angle of the curved bar）， as shown in Fig.5. Therefore， the curved-bar center line is determined by the location B and the bar angle that measures the curvature of the curved bar.
3 Design of curved-bar refining plates
3.1 Design of radial curved-bar plate
The groove width of radial-bar refining plates is gradually increased from the inner part to the outer part of the refining area. If only a one-stage grinding tooth is used; the effective refining area is reduced， which yields a lower utilization efficiency of the refining zone. Therefore， multi-stage refining plates have been developed. Fig.6 shows a three-stage radial curved-bar plate， which inserts more bars between the adjacent bars where the bar angle of the different-stage curved bar is different. In this part， a three-stage radial curved-bar plate is designed as follows.
3.1.1 Design of the 1st-radial arc of curved bar
The main technical parameters of the refining plate are shown in Fig.7. If the 1st-stage curved-bar angle is b， the center line of the curved bar is arc AC， which has a bar angle of b and radius of Ra.
The origin O1 of the center line of the 1st-stage curved bar was selected as the pole， and the function of the circle of arc AC is expressed by Eq.（1）.
If the bar width of the 1st-stage curved bar is b， the functions of its inner and outer arc， as shown in Fig.8， can be expressed by Eq.（2）.
Where the inner arc of the 1st-stage curved bar is obtained by subtraction and outer one is obtained by addition.
1st-stage curved bar
At last， the number of 1st-stage curved bars， n， should be determined by the size of the refining plate and the process requirements. After this， the circumferential array of curved bars should be completed.
3.1.2 Design of the 2nd- and 3rd-radial arc of curved bar
The design of the 2nd- and 3rd-stage curved bars of the three-stage radial curved-bar plate is slightly different from that of the 1st-stage curved bar. According to the size of the refining plate， process requirements， and refining intensity， the refining zone was divided reasonably into the breaking zone， coarse refining zone， and refining zone. The values of R1 and R2 should be selected correctly， as shown in Fig.9 and Fig.10. The starting circle with radius Ri intersects the two center arcs of the adjacent 1st-stage curved bars. The radius of the center circle of area through which the 2nd-stage curved bar is （R1+Ro）/2， and the starting point and intermediate position of the 2nd-stage curved bar can be determined by the equal points of arcs FG and HI. In this paper， the starting point and intermediate position of the center arc of the 2nd-stage curved bar are determined by their two equal points， as shown in Fig.9. The design of the 3rd-stage curved bar is the same as that of the 2nd-stage curved bar. It should be noted that the starting point is the equal points of arc KJ， which is part of the starting circle of the 3rd-stage curved bar cut by the adjacent curved-bar center arc of the 2nd-stage curved bar. The two equal points of KJ is defined as the starting point in this paper， as shown in Fig.10.
If the bar widths of the 2nd- and 3rd-stage curved bars are a and c and the bar angles are b1 and b2， the function of the center arc of the 2nd- and 3rd-stage curved bar can be obtained by Eq.（3）.
Thus， the function of the 2nd- and 3rd-stage curved bar is
For the 2nd-stage curved bar， X=b and Y=a， and for the 3rd-stage curved bar， X=c and Y=c.
The circumferential array of the 2nd- and 3rd-stage curved bar is arranged according to the number of 1st-stage curved bars. Thus， the number of 2nd-stage curved bars is n（m-1） and that of 3rd-stage curved bars is nm（z-1）.
Multi-stage radial curved-bar plates have the same bar angle in the same-stage curved bars， which are evenly distributed in the whole circumference. Therefore， the division of the plate does not affect the orientation of the curved-bar distribution. The plates with large diameters were manufactured by segments because of the complex overall manufacturing. Therefore， the center angle of the segment should be designed reasonably to equally divide the plate.
3.2 Design of isometric curved-bar plate
Compared with the radial curved-bar plates， the distribution of the bar width and groove width is more uniform from the inner part to the outer part of the refining zone. Theoretically， the isometric curved-bar plates have a uniform bar-interaction angle， which is conducive to uniformity of refining. The design of the isometric curved-bar plates is briefly introduced by using the bar angle of the curved bar proposed above. This article takes the right-hand rotational curved bar as an example.
3.2.1 Design of the center arc of isometric curved bar
The design of the center arc of the isometric curved bar is the same as that mentioned above. However， the intersection point B of the curved-bar center line， the center circle of the refining zone， and the center line of the segment are defined based on the design of the bar angle， as shown in Fig.11. The center arc of the center curved bar is determined when the bar angle a and starting point A are selected， and its function is similar to Eq.（1） and Eq.（3）. After this， the function of the inner and outer arc can be determined considering the bar width b and groove width g， which are consistent with Eq.（2） and Eq.（4）. 3.2.2 Design of curved bar on both sides of the center bar
If the groove width of the isometric curved-bar plate is g and the bar width is b， the equation of the 1st- stage curved-bar arc， as shown in Fig.12， on the left side （the left-hand bar is on the right side） is
And the functions of the 2nth arc， when n≥1， on both sides of the center curved bar can be expressed as
Similarly， the equations of the （2n+1）th arc， when n≥1， can be determined by Eq.（7）.
When determining the function of curved-bar edges on the left side of the center bar with Eq.（6） and Eq.（7）， the arc can be obtained by addition. That of the right side can be obtained by subtraction. However， the choice of symbol is reversed for the left-hand bar.
The curved bars are arranged by the above arc equation. When the arc is full for the entire refining segment， the design of the isometric arc in a segment can be completed by trimming the arc out of the refining zone.
4 Bar profile of curved-bar plates
4.1 SEL of curved-bar plates
The SEL is a common way of quantifying refining intensity[8]， which denotes the net energy applied to each meter of the bar crossing （J/m） and is calculated by Eq. （8）.
where SEL is the refining intensity （J/m）， Pnet is the net refining power （kW）， n is the rotation speed （r/min）， and CEL is the cutting edge length （km/r）.
The SEL is proposed for straight-bar plates， and the calculation of CEL[11] （Eq.（9））， the common contact length of the opposite bars per revolution， is also for straight-bar plates， and cannot be applied to curved-bar plates.
Where r1 is the inner radius of the plate （mm）， r2 is the outer radius of the plate （mm）， nr is the total bar number of the rotor， ns is the total bar number of the stator， and a is the bar angle of the plate （°）.
The calculation principle of the curved bar’s CEL is the same as that of the straight bar， as shown in Fig.13; the bar segments are divided into several zones and the number and length of the bars in each zone are counted. CEL of the curved-bar plate is calculated by the following equation.
Where gi is the center angle of the curved-bar center line at zone i and Ri is the circle radius of the curved-bar center line at zone i.
SEL is affected by Pnet， n， and CEL of the refining process， and it should be selected reasonably for the refining process of different pulps. When refining softwood pulp， SEL should be 1.5～4.5 J/m， for hardwood pulp， it should be 0.5～1.5 J/m[12]. The refining plates are the core component of the refining process. Under the condition of constant Pnet and n， CEL can be adjusted by reasonable design of the bar parameters， through which SEL can reach the range mentioned above for effective refining. Through the analysis of Eq.（10）， CEL is directly related to the configuration of the bar width， groove width， bar angle， etc.， and the corresponding SEL can be matched by rationally designing the bar profile.
4.2 Bar-parameter determination of curved-bar plates
As a direct acting component of the refining process， the bar parameters of the refining plates have a direct influence on the refining effects. This paper introduces the design methods of two curved bars， but the selection of bar parameters is not provided， and will be briefly introduced below.
The selection of the bar profile， such as the bar width， groove width， and bar height， can be referred to that of the straight-bar plates. According to the literature[12]， for the refining of softwood pulp， with a consistency of 3.5%～4.5%， the bar width of the refining plates can be chosen from 3.0 mm to 5.5 mm， and the groove width is generally between 5 mm and 7 mm. For refining hardwood pulp， the bar width is generally from 2.0 mm to 3.5 mm， and the width of the groove is generally from 3 mm to 4 mm. Generally， the width of the groove is usually 2～3 times the average fiber length.
The bar angle is one of the key bar parameters of refining plates， and is usually 15°～20° for straight-bar plates[13]. The bar angle of curved-bar plates can be selected as （q+b/2）， in which q （15°～20°） is the bar angle of the straight-bar plate and b is the center angle of the segment， based on the different definition of the bar angle compared to that of the straight bar. Further， there is another angle that should be noted for the multi-stage radial curved-bar plate. The starting angle of a curved bar is usually between 0° and 45°， preferably between 15° and 30°. Here， it will be advantageous if a is larger than a1 and a1 is larger than a2[10].
5 Conclusions
As a common type of refining plate in the pulp and paper industry， curved-bar plates are weaker than straight-bar plates in fiber cutting， and theoretically， the refining performance provided by curved-bar plates is relatively uniform. However， the design of curved-bar plates is limited by a lack of theoretical basis. The work done by this study is described below. The representation method of the curved-bar refining plates at home and abroad was analyzed and a new method to represent the bar angle of the curved bar was proposed in this paper to clearly describe the curved bar.
The multi-stage radial curved-bar plate and isometric curved-bar plate were designed based on the bar angle of the curved bar as described in this paper. Subsequently， the mathematical expressions of the curved-bar edges and the center arc of the curved-bar plate were established， which will provide a theoretical basis for parametric modeling of curved-bar plates and enhance the efficiency of their design.
The SEL of curved-bar plates was deduced compared to that of straight-bar plates， which can provide a theoretical basis for design of curved-bar plates. Furthermore， the determination of the bar profile was discussed based on that of straight-bar plate.
Acknowledgments
The authors gratefully acknowledge the funding by the National Natural Science Foundation （Grant No. 50745048）.
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