Apparatus for treating cellulose pulp with intermeshing disks and asymmetrically pulp moving means

Abstract

An apparatus for treating cellulose pulp having a consistency above the flowage limit, provided with two shafts (1, 2) rotating in the same axial plane and each carrying working disks (8) the cylinders of rotation of which are in mutually intermeshing engagement in a working zone and which are driven within a housing (3) which conforms to the common cylinder-of-rotation space of the disks carried by the shafts and is provided with a pulp inlet (4) and a pulp outlet (5). The working disks on the shafts (1, 2) are constituted of a number of radially directed disks (8) in mutually cooperative positions between the pulp inlet (4) and the pulp outlet (5) for working of the pulp between the opposed surfaces (11, 12) of mutually intermeshing disks (8) the pulp being moved from the inlet (4) to the outlet (5) by members positioned asymmetrically in relation to such opposed surfaces (11, 12) carried by the housing (3) or the disks (8).

Description

FIELD OF THE INVENTION

The invention relates to an apparatus for treating cellulose pulp having a consistency above the flowage limit, said apparatus being provided with two shafts rotating in the same axial plane and each carrying working means mutually intermeshing in their cylinders of rotation in a working zone and driven in a housing which closely fits to the common cylinder-of-rotation space of the means carried by the shafts and is provided with a pulp inlet and a pulp outlet.

DESCRIPTION OF THE PRIOR ART

A presently generally used apparatus of the above indicated type comprises two intermeshing rotary screws coupled for synchronized rotation in mutual interaction, the material intended to be treated, for example pulp supplied at the inlet, being conveyed by the co-operation of the screws towards the outlet and being treated during passage through the space bounded by the thread portion of the screws and the surrounding housing towards the outlet. In this case the treatment is performed between opposed surfaces of the two co-operating screws.

An apparatus of the type as indicated above, but provided with working means in the form of disks mutually intermeshing in their cylinders of rotation rather than mutually intermeshing screws has been proposed for about 80 years ago in Swedish Pat. No. 21 004. However, this apparatus has been explicitly designed for treating a pulp of low concentration enabling the pulp to be conveyed from the inlet to the outlet under the action of gravity. The use of such an apparatus for treating high-concentration pulp having a consistency above the flowage limit has never been proposed, obviously for the reason that the problem to perform an even and uniform transport of the pulp from the inlet through the working zone to the outlet has been considered insurmountable.

SUMMARY OF THE INVENTION

The present invention is based on the surprising discovery that also pulp having a consistency above the flowage limit by means of feeding members not taking part in the working can be forced to pass in an even flow and without clogging through a working zone between working means carried by two shafts and mutually intermeshing in their cylinders of rotation, said working means being in the form of radially directed disks. By the apparatus construction according to the invention, the characteristic features of which appear from the attached claims, a construction is obtained which in many respects is cheaper, more easily handled and more efficient than the double-screw apparatus while at the same time the working effect not only is equivalent to the working principle of the old type of apparatus but rather considerably superior due to the possibility to perform several types of adjustment of the depth of engagement and the width of the treating gap between co-operating working means permitted by the novel type of apparatus and not feasible in an apparatus in which the working is performed between mutually intermeshing screw threads. Thus, said parameters may be controlled easily and at short notice when, for example, a change in the physical properties of the pulp occurs during operation.

The new apparatus construction also eliminates several of the most difficult problems of the double-screw apparatus, in particular due to the fact that the novel apparatus with particular advantage is combined with a disposition of the pulp outlet in a lateral direction at an angle to the general direction of feed of the pulp through the apparatus.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be explained in detail in connection with several embodiments shown in the attached drawings in which

FIG. 1 is an end view towards the outlet end of an apparatus according to the invention for treating cellulose pulp,

FIG. 2 is a section of the apparatus substantially along line 2--2 in FIG. 4,

FIG. 3 is a plan view of the apparatus with the upper portion of the housing removed,

FIG. 4 is a side view of the apparatus with the housing shown in section and the shaft shown below in FIG. 3 together with its bearings, couplings and driving means removed,

FIG. 5 is a section corresponding to FIG. 2 of an apparatus with a modified embodiment of the working disks,

FIG. 6 is a partial view to an enlarged scale of the range of intermesh between two disks supported by one of the shafts each,

FIG. 7 is a section along line 7--7 in FIG. 6,

FIG. 8 is a radial partial section of the range of intermesh between two disks showing a second embodiment of the means serving to move the pulp in the direction towards the outlet,

FIG. 9 is a side view of the same range with the side wall portions of the housing cut-away,

FIG. 10 is a radial section of the range of intermesh between two disks each supported by one of the shafts showing a third embodiment of means for shifting the pulp towards the pulp outlet,

FIG. 11 is a partial section along line 11--11 in FIG. 10 and

FIG. 12 is a partial view of the bottom of the housing with the shaft and disks removed and the housing sectioned along line 12--12 in FIG. 10.

DESCRIPTION OF THE PREFERRED EMBODIMENT

FIGS. 1 to 4 show an apparatus for treating cellulose pulp, the essential portions of which are two shafts 1, 2 supported in the same axial plane and surrounded by a housing 3 having a pulp inlet 4 at the upper side close to the end wall to the left in FIG. 3 and a pulp outlet 5 on the lower side close to the opposite end wall. A mounting bed 25 carries schematically shown upright bearing supports 26 for supporting the bearings 27 of at least one 2 of the shafts.

While in the simplest possible embodiment of the apparatus also the other shaft 1 might be suported in the same way by correspondingly supported bearings, FIGS. 1 to 4 show an advantageous modification in which the bearings 28 of the shaft 1 situated below in FIG. 3 are supported on a cradle 29 which permits an adjustment of the spacing between shafts 1 and 2 by parallel shift of shaft 1. The cradle 29 itself comprises two end walls 30 which are pivotally journalled at 31 in the bearing supports 26 supporting bearings 27 of the second shaft 2 at a position below the part of housing 3 in which shaft 1 operates. On each end wall 30 an arm 32 extends in an upward direction from the journal 31 to carry one of the bearings 28 of shaft 1. Approximately at right angles to the arm 32 carrying the bearing 28 and on the same level as the journals 31 the end wall 30 comprises a horizontal arm 33. The two arms 33 horizontally extending from journals 31 on either end wall 30 are mutuallly connected by a cross beam 35. The cradle formed by cross beam 35 and end walls 30 as well as the bearings 28 carried by end wall arms 32 form a stiff unit that may be swung about a common axial line of journals 31. A hydraulic or pneumatic adjusting device 36 is at its one end pivotally connected to the lower side of cross beam 35 by a pivotal bearing 37 and at the other end to the mounting bed 25 by a pivotal bearing 38. During extension and retraction respectively of the moving part of the pneumatic or hydraulic device 36 the unit comprising cradle 29, bearings 28 and shaft 1 carried by the bearings will be swung respectively towards and away from shaft 2 for adjustment of the working distance between the shafts.

Obviously, it is important to prevent excessive swinging movements of shaft 1 to avoid contact between the working means carried by shaft 1 with respectively corresponding means on shaft 2 or the housing. For this reason the end wall 30 at the right-hand end of the housing as seen in FIG. 3 is extended in a downward direction from the journal 31 by an arm 40 the free end of which is positioned between adjustable abutments 41 limting the amplitude of the swinging movement of the cradle.

As the amplitude of swinging movement is rather small it is sufficient that the passage openings for shaft 1 in the end walls of housing 3 are widened to a corresponding extent and sealed by gaskets (not shown) which are resilient or adapted to be shifted in the swinging direction of the shaft.

Shafts 1 and 2 might be coupled for synchronous operation with the same or different speed with the aid of gears arranged in a gearbox 41, but in the embodiment shown the shafts are provided with oppositely directed driving connections. On the part of shaft 1 extending between bearing 28 and gearbox 41 and the driving motor (not shown) there are provided two universal joints 43 serving the purpose to absorb any lateral shift of the portion of shaft 1 between bearings 28 during swinging movement of the cradle 29 without exposing the bearings in gearbox 41 or the coupling to the motor for undue stresses.

The housing 3 conforms closely to the common cylinder-of-rotation space of the means carried by the shafts and leaves only so much free space around the working means, in particular disks 8, as is required in respect to the adjustability of the one shaft or both shafts in a lateral direction in relation to their axial lines for changing the mutual depth of intermesh of the disks.

Within housing 3 shafts 1 and 2 carry working means 8 in the form of a number of radially directed disks 8 in mutually co-operating positions between the end walls of housing 3 for compressive and kneading working of the pulp within a working zone between opposed disk surfaces 11, 12 on disks 8 mutually intermeshing in their cylinder of rotation and carried by one of the shafts 1, 2 each.

Suitably the thickness of disks 8 decreases towards the periphery. As depicted in FIGS. 3 and 7, the peripheral edge of the disks might be acute but suitably each disk 8 has a peripheral edge 15 of some extension in the axial direction and spaced from an opposed bottom surface 16 between two disks 8 at a distance suitable for the treatment of the pulp. Experience has shown that such a compressing treatment between the circumferential edge 15 on one disk on the one shaft and the bottom 16 between two opposed disks on the other shaft yields a particularly efficient treatment of the pulp enclosed in the interspace.

Suitably the cross-section of disks 8 is symmetrical. However, it is sufficient that disks 8 each provided on one of shafts 1, 2 have identical bevel angle on mutually opposed surfaces, which bevel angle may be different on the two sides of the same disk. The main point is that opposed disk surfaces on disks belonging to different shafts enclose a working zone having substantially uniform thickness.

In the embodiment according to FIG. 5 the peripheral edge on at least some of the disks 8 on each shaft is provided with indentations 20 for forming cogs, teeth or arcuate recesses in which separate portions of the treated material are exposed to local, radial compression against the opposed groove bottom 21.

As previously described, such a radial compression also occurs between flat peripheral disk edges and the opposed groove bottom but the amount of pulp treated at each such interaction between a peripheral section on a disk and the opposed groove portion and thereby the working effect are considerably increased by providing such indentations 20.

The working effect achieved by the indentations 20 may be additionally increased by adapting the profile of the groove bottom 21 to the profile of the periphery of an opposed disk 8 provided with indentations 20, such adaptations, for example, being obtained by the provision of beads 22 positioned opposite to the indentations 20.

For conveyance of the pulp through the apparatus means are provided according to the invention which are asymmetricallly positioned in relation to opposed surfaces on mutually intermeshing disks 8 and which are adapted to move pulp enclosed between opposed disk surfaces 11, 12 in the direction towards the pulp outlet.

In a first embodiment which is generally indicated in FIGS. 2 to 4 and shown in detail in FIGS. 6 and 7 disks 8 carry said means on their sides 11 facing the pulp outlet 5 in the form of pulp-engaging projections, profiles or edges. In FIGS. 2 to 4 these sides on the disks facing outlet 5 are shown provided with some kind of rugged structure which increases the friction of the disk side in relation to the treated material whereas opposed disk sides 12 are substantially plain. Already a rather small difference in the engagement between respectively rugged and plain disk sides produces a stepwise advance of pulp portions in the direction towards the lesser frictional resistance on the plain disk sides 12 facing away from the outlet and thereby a feeding movement from the inlet 4 to the outlet 5. The effect will appear at any difference in the frictional engagement of opposed disk sides in relation to the pulp but may be increased by a particular design of the forward-feeding sides 11 which may be provided with projections, profiles or edges in contrast to the opposed sides 12 which are plain or possibly profiled in such a way that a shifting of the pulp in the desired direction is promoted.

As an example of a suitable profile FIGS. 6 and 7 shows an embodiment of sides 11 of disks 8 comprising concentric edges 45 in a stair-step like succession.

It is important to prevent sticking of worked pulp between the end wall of the housing 3 at the outlet 5 and the disks 8 positioned closest to said end wall. For this reason, as shown in FIGS. 3 and 4, the last disks on the shafts have the distribution of rugged and plain surfaces reversed in relation to the disks 8 positioned between the opposite end wall and the outlet. While the two last disks 8 on shaft 2 thus have two mutually opposed rugged surfaces, the intermediate disk 8 on shaft 1 has plain surfaces on both sides. Thus, there will be no forward feed of the pulp by differential frictional action but the pulp will be diverted straight downwardly in the direction of the outlet.

A second embodiment of the means shifting the pulp in the direction towards the pulp outlet 5 is shown in FIGS. 8 and 9 where these means are guide rails 56 extending from the inner surface of the housing 3 at a distance from the range of intermesh 55 between disks 8 in the interspace between adjacent disk 8' on respectively shaft 1 or shaft 2, said guide rails 56 due to their asymmetric position in relation to adjacent disk 8' on the same shaft causing the pulp to move in a path leading towards the pulp outlet. In FIG. 8 the direction of rotation of shaft 1 and 2 is indicated by arrows 57, 58. It appears from the side view of FIG. 9 that pulp taking part in the rotation between two adjacent disks 8' on shaft 1 when moving past guide rails 56 will be shifted in a direction towards the outlet 5 to the right in FIG. 9 and will fall down into the interspace between the meeting pair closest to the right of adjacent disks 8' on shaft 2. This shifting movement towards the outlet 5 continues in consecutive steps so that a great portion of the pulp rotating together with the disks all the time will be shifted closer to the outlet.

In the same way as in the feeding profile of the embodiment according to FIGS. 1 to 4, also in this embodiment according to FIGS. 8 and 9 guide rails 56 having an asymmetric position opposed to the rest of the guide rails may be provided at the outlet and of housing 3 to perform a counter-pressure guiding of the pulp for the purpose of its discharge through the pulp outlet 5. Hereby the risk is avoided that a compact accumulation of pulp is formed in the zone bounded by the end wall of the housing on the discharge side and the disks postioned adjacent this end wall.

It is to be observed that the position of the guide rails 56 is not only asymmetric in relation to two adjacent disks 8' on either of shaft 1 or 2 but also in relation to co-operating pairs of disks 8 on each of shafts 1 or 2. Thus, the guide rails 56 are not asymmetrical in relation to the axial lines of two such co-acting disks but extend from the wall of housing 3 in parallel with the common axial plane of the two shafts to form an end edge 56' which in the embodiment shown in disposed exactly above the axial line of adjacent shaft 1.

As the guide rails are exposed to comparatively strong wear, they are supported in the embodiment shown in FIGS. 8 and 9, by a particular, easily detachable and exchangeable part of the housing wall extending over about a fourth of the perimeter of the disks.

In FIGS. 10 to 12 a third embodiment is shown of means disposed asymmetrically in relation to opposed surfaces on mutually intermeshing disks thereby causing pulp enclosed between opposed disk surfaces to be shifted in the direction towards the pulp outlet. In this embodiment said means are in the form of guide fins extending from the inner surface of housing 3 adjacent the periphery of either disk 8 on respectively shaft 1 or 2 into the interspace between adjacent disks 8 on the other shaft 2 and 1 respectively, said guide fins 65 diverting the pulp against that side of the disk 8 entering into the intermeshing space which is facing the pulp outlet.

In the embodiment shown the guide fins 65 are attached to the central part 67 of the bottom 66 of the housing, which central part 67 due to the shape of the bottom in agreement with the common cylinder-of-rotation space of the two disk sets forms an edge 68 extending in parallel to the axial lines of the shafts. An edge 71 on every guide fin 65 positioned opposite the peripheral edge 70 of a disk 8 merges at the longitudinal bottom edge 67 in a soft transition into the inner surface of bottom 66 that follows the periphery of disk 8, whereas the opposite free edge 72 of the guide fin 65 extends straight down in the interspace between two adjacent disks 8" opposed to the first-mentioned disk 8. Pulp enclosed between two disks 8" which in the range of intermesh meets the edge surface 71 of guide fin 65 will thus be diverted into the interspace closer to the outlet between opposed disk pairs 8 and so on.

FIG. 12 illustrates the shape of a set of guide fins 65 when the disks 8 are removed.

Also in this embodiment of the means for moving pulp enclosed between opposed disk surfaces towards the pulp outlet it is possible to provide beyond the outlet the guide fins 65 in an asymmetric position which is contrary to the position of the rest of the guide fins thereby to produce a counter-pressure shifting of the pulp away from the end wall of housing 3 on the outlet side and into the pulp outlet 5.

Obviously several of the means provided for moving the pulp into the direction towards the pulp outlet may be combined in the same apparatus in order to increase the shifting effect. It is also possible to use as counter-pressure means, in place of oppositely asymmetric means of the above described type, counter-pressure screw threads on shafts 1, 2, such screw threads preferably being in mutual intermeshing engagement and being supported on shafts 1, 2 beyond the last disks 8 in the direction of feed on both shafts 1 and 2.

In the shown embodiment of the apparatus the pulp outlet 5 extends laterally in relation to the common axial plane of shaft 1 and 2 adjacent the last disks 8 in the feeding direction of the pulp on both shafts 1, 2. Obviously the outlet might also be provided in a different way, for example in the form of an adjustable gap between the housing 3 and either of shafts 1 and 2 and in concentric position in relation to the shafts.

In the embodiment shown the outlet is a tube downwardly projecting centrally between shafts 1 and 2 and, for example, having rectangular cross-section and preferably outwardly increasing cross-sectional area. The outlet is closed by a flap 50 which is pivotally journalled in the tube wall and extends in an obliquely downward direction to offer a resistance against the discharge which is adjustable by hydraulic or pneumatic means 51.

In the embodiment according to FIGS. 1 to 4 shaft 1 is adjustable by parallel shift in relation to shaft 2 with the aid of cradle 29 whereas bearings 27 for shaft 2 are stationary on the bearings support 25. However, it is possible to provide for mutual parallel shift of both shafts 1 and 2 for adjustment of the depth of engagement between the disks on both shafts. A fully serviceable apparatus may also be obtained without providing for parallel shift of one of the shafts in relation to the other. Certain adjustment of the depth of engagement between the working means can also be obtained by providing one of the shaft angularly adjustable in relation to the other shaft in the common axial plane of the shafts. This may be brought about by cardanically connecting the one shaft to its driving means while the opposite end of the shaft on the other side of the housing is supported for lateral shifting movement.

In the embodiment according to FIGS. 1 to 4 the shafts are shown to have mutually independent driving means M1, M2 to enable them to be driven with different speeds. The same result may also be obtained when the shafts are coupled for common operation by suitable choice of the gear rate.

While the shafts in the described embodiment are driven in mutually opposite directions, it is obviously also possible to let the shafts rotate in the same direction. While the working in the first case is predominantly kneading, it is rather of a rubbing character in the other case.

The constructions of the apparatus according to the invention as described permit also an adjustment of at least one of the shafts 1, 2 in the longitudinal direction for adjustment of the spacing between mutually opposed disk surfaces.

All the above-mentioned adjustments between shafts by parallel shift, angular adjustment and longitudinal shifting may be used separately or in combination to obtain the intially mentioned advantages of the apparatus according to the present invention.

For repair and maintenance purposes the housing 3 is composed of easily detachable parts. FIGS. 1 to 4 show a horizontal subdivision in the level of the common axial plane of shafts 1, 2. An additional subdivision of the housing has been described in connection with FIG. 8. In another suitable embodiment (not shown) the housing may have a central portion in firm connection with the bearing support and provided with inlet and outlet as well as side portions that may be removed to expose the shafts and working means.

The embodiments which have been described herein are but some of several which utilize this invention and are set forth here by way of illustration but not of limitation. It is apparent that many other embodiments which will be readily apparent to those skilled in the art may be made without departing materially from the spirit and scope of this invention.

Claims

1. Apparatus for treating cellulose pulp having a consistency above the flowage limit, comprising two spaced shafts rotatable in the same axial plane, each shaft carrying means for working said pulp in a working zone, said working means defining respective cylinders of rotation which are in mutually intermeshing engagement to form said working zone and which are driven with a housing enclosing in substantial conformity the space occupied by the intermeshing cylinders of rotation of said working means, a pulp inlet at one longitudinal end of said shafts and a pulp outlet at the other longitudinal end of said shafts, said inlet and outlet communicating with said working zone, said working means comprising a plurality of radially directed axially spaced disks in mutually cooperative positions between said pulp inlet and said pulp outlet for compressive and kneading working of said pulp in said working zone between opposed inlet facing surfaces and outlet facing surfaces of mutually intermeshing of said disks, the thickness of said disks diminishing towards the periphery thereof, means for rotating said shafts, and means positioned asymmetrically in relation to said opposed surfaces for causing pulp enclosed between said opposed surfaces to move in a direction towards said pulp outlet, said pulp-moving means comprising a plurality of projections of said outlet-facing surfaces increasing the friction in relation to the treated material to a greater extent than any structure on said opposed inlet-facing surfaces.

2. Apparatus as claimed in claim 1 wherein each disk on one of said shafts comprises a peripheral edge extending in the axial direction and located at a spacing from an opposed bottom surface between two disks on the other of said shafts suitable for the working of the pulp.

3. Apparatus as claimed in claim 1 wherein said pulp outlet extends laterally in relation to the common axial plane of said shafts at a position adjacent the last disks on the end of said shafts in the direction of pulp feed.

4. Apparatus as claimed in claim 3, wherein means are provided beyond the last disk on both of said shafts seen in the direction of the feed of the pulp for diverting the pulp back towards said outlet.

5. Apparatus as claimed in claim 4, wherein said diverting means are counter-pressure screw threads on each of said shafts, said screw threads preferably being arranged for mutually intermeshing cooperation.

6. Apparatus as claimed in claim 1, wherein said disks on at least one of said shafts are bevelled at an identical bevel angle on mutually opposed surfaces.

7. Apparatus as claimed in claim 1 wherein the peripheral edge of at least one of said disks on one of said shafts is provided with indentations forming recesses in which separate portions of the treated material are exposed to local, radial compression against an opposed groove bottom between adjacent disks on the other of said shafts.

8. Apparatus as claimed in claim 7, wherein said groove bottom has a profile adapted to the periphery of said disk.

9. Apparatus as claimed in claim 1 wherein at least one of said shafts is adjustable in the longitudinal direction for adjustment of the spacing between mutually opposed disk surfaces.

10. Apparatus as claimed in claim 1 wherein said shafts have mutually independent driving means.

11. Apparatus as claimed in claim 1 wherein said shafts are mutually coupled for synchronous operation by common driving means.

12. Apparatus as claimed in claim 11 wherein said shafts are mutually coupled for operation at different speeds.

13. Apparatus as claimed in claim 1, wherein the body of each disk apart from any differentiated surface pattern has a symmetrical cross-section.

14. Apparatus for treating cellulose pulp having a consistency above the flowage limit, comprising two spaced shafts rotatable in the same axial plane, each shaft carrying means for working said pulp in a working zone, said working means defining respective cylinders of rotation which are in mutually intermeshing engagement to form said working zone and which are driven within a housing enclosing in substantial conformity the space occupied by the intermeshing cylinders of rotation of said working means, a pulp inlet at one longitudinal end of said shafts and a pulp outlet at the other longitudinal end of said shafts, said inlet and outlet communicating with said working zone, said working means comprising a plurality of radially directed axially spaced disks in mutually cooperative positions between said pulp inlet and said pulp outlet for compressive and kneading working of said pulp in said working zone between opposed surfaces of mutually intermeshing of said disks, the thickness of said disks diminishing towards the periphery thereof, means for rotating said shafts, and means positioned asymmetrically in relation to said opposed surfaces for causing pulp enclosed between said opposed surfaces to move in a direction towards said pulp outlet, said pulp-moving means comprising a plurality of guide rails each extending from an inner surface of said housing at a distance from said working zone and into the interspace between adjacent disks of the same shaft, said guide rails extending in a radial plane in relation to said shafts and being arranged in an asymmetric position in relation to opposed disks of the same shaft thereby to divert the pulp into a path leading to said pulp outlet.

15. Apparatus as claimed in claim 14 wherein each disk on one of said shafts comprises a peripheral edge extending in the axial direction and located at a spacing from an opposed bottom surface between two disks on the other of said shafts suitable for the working of the pulp.

16. Apparatus as claimed in claim 14 wherein said pulp outlet extends laterally in relation to the common axial plane of said shafts at a position adjacent the last disks on the end of said shafts in the direction of pulp feed.

17. Apparatus as claimed in claim 14, wherein said disks on at least one of said shafts are bevelled at an identical bevel angle on mutually opposed surfaces.

18. Apparatus as claimed in claim 14 wherein the peripheral edge of at least one of said disks on one of said shafts is provided with indentations forming recesses in which separate portions of the treated material are exposed to local, radial compression against an opposed groove bottom between adjacent disks on the other of said shafts.

19. Apparatus as claimed in claim 14 wherein at least one of said shafts is adjustable in the longitudinal direction for adjustment of the spacing between mutually opposed disk surfaces.

20. Apparatus as claimed in claim 14 wherein said shafts have mutually independent driving means.

21. Apparatus as claimed in claim 14, wherein the body of each disk apart from any differentiated surface pattern has a symmetrical cross-section.