A sealing arrangement for a turbine

Abstract

A sealing arrangement, for a turbine with a static casing 221 and a runner 222 that rotates relative to the casing, comprises a plurality of parallel dead-end passages 228a, 228b formed within the run

Description

TECHNICAL FIELD The present invention relates to a sealing arrangement for a turbine used as part of a generation system. More particularly, the present invention relates to a sealing 5 arrangement for a turbine used as part of a high-density pumped energy storage system that uses a suspended solid-in-liquid base. BACKGROUND Turbines are a reliable and efficient way to generate electricity, and are used extensively in hydro-electric projects or systems, with the turbine unit or units forming one part of the 10 overall system. In a hydro-electricity generating system a fluid such as water flows under gravity from one part of the system to another and then into a turbine unit. The fluid flow over the blades of the turbine runner in the turbine unit causes the turbine runner to rotate, spinning the turbine shaft. The mechanical power of the spinning shaft of the turbine can LO then be converted to electric power using a generator. 15 However, renewable energy sources such as wind and solar have highly variable power CM outputs. On-grid energy storage therefore plays a crucial role in smoothing out the 1 electricity supply from these sources and ensuring that the supply of power matches LO demand. Energy storage at grid scale is well established in the form of Pumped Hydro " Storage (PHS) systems. In such systems, during times of low on-grid electricity demand, 20 water is typically pumped from a lower-level reservoir to an upper-level reservoir, thereby gaining potential energy. The water is then stored in the upper-level reservoir until times of high on-grid electricity demand. At such times, the water is allowed to flow from the upper reservoir back to the lower reservoir through a penstock. The water turns a turbine located in the penstock to generate electricity that is then sent to the grid to help meet the 25 high electricity demand. One common type of turbine used for extracting energy from the fluid flow is the Francis turbine. This type of turbine has a rotating component (the runner) and a stationary component (the casing). The runner is connected to a generator via a shaft so that as the runner rotates, the shaft rotates, and energy is extracted from the system. Francis 30 turbines usually use a labyrinth seal as a sealing mechanism between the high and low pressure points. Labyrinth seals are contactless seals which are often used in turbomachinery and, in particular, in Francis turbines. Labyrinth seals generally comprise a series of narrow passages aligned in a first direction, that are interspersed with wider passage portions aligned in a different direction (usually perpendicular to the first direction). The wider passage portions allow rapid expansion of the fluid flowing through the seal as it flows from the narrower passage to the wider passage. The use of labyrinth seals in Francis turbines minimises the flow that bypasses the runner 5 of a turbine, which means that the losses are decreased, and the turbine’s efficiency is increased. However, due to the narrow passages, there are significant torque losses when the system is in use, which in turn acts to lower the efficiency. This allows physical separation - that is, no contact - between the rotating and stationary components, and any leakage through the seal is reduced via a combination of viscous 10 losses, pressure losses due to the sudden expansion, and recirculating flow within the expansion. However, using this type of seal usually leads to increased torque losses. As the fluid in the seal is sheared at a high rate, this applies a moment on the rotating component (the runner) which acts to slow the runner and therefore reduce the power produced by the turbine. Although water is used almost exclusively in these types of i •4 / 15 systems, alternative fluids have also been investigated for use in systems similar to Pumped Hydro Systems. It has been found that the use of high-density fluids (fluids CM having a density greater than that of water at the same temperature and pressure) can be 1 highly beneficial in systems that operate on a similar principle to Pumped Hydro Systems. LO For example, the use of high-density fluids in these types of systems can reduce the 1 20 requirement for there to be a large vertical elevation or separation between the upper and lower-level reservoirs, in comparison to conventional Pumped Hydro Systems (i.e. conventional systems that use water as the working fluid). One issue with these types of systems is that the pressure inside the energy-extraction turbine is usually very high, often 10-20 ATM or higher. This can lead to the leaking of 25 fluids, which leads to a loss of energy recovery, and thus a loss of efficiency. This can also lead to loss of the high-density operating fluid itself, which is highly undesirable as this can be very expensive to replace. Torque loss can also be a particular issue with high-density fluids, as this type of fluid has a higher viscosity than water, and this leads to higher torque losses. 30 In this specification where reference has been made to patent specifications, other external documents, or other sources of information, this is generally for the purpose of providing a context for discussing the features of the invention. Unless specifically stated otherwise, reference to such external documents is not to be construed as an admission that such documents, or such sources of information, in any jurisdiction, are prior art, or 35 form part of the common general knowledge in the art. SUMMARY OF THE INVENTION It is an object of the present invention to provide a sealing arrangement for a turbine used as part of a generation system which goes some way to overcoming the abovementioned disadvantages or which at least provides the public or industry with a useful choice. 5 It is a further object of the invention to provide a sealing arrangement for a turbine used as part of a high-density pumped energy storage system that uses a suspended solid-in liquid base which goes some way to overcoming the abovementioned disadvantages or which at least provides the public or industry with a useful choice. The term “comprising” as used in this specification and indicative independent claims 10 means “consisting at least in part of’. When interpreting each statement in this specification and indicative independent claims that includes the term “comprising”, features other than that or those prefaced by the term may also be present. Related terms such as “comprise” and “comprises” are to be interpreted in the same manner. As used herein the term “and / or” means “and” or “or”, or both. 15 As used herein “(s)” following a noun means the plural and / or singular forms of the noun. Accordingly, in a first aspect the present invention may broadly be said to consist in a sealing arrangement for a pump or a turbine, the pump or turbine comprising: a static casing; a runner configured so that in use the runner rotates relative to the casing, the casing and runner further configured so that there is a gap therebetween, the sealing 20 arrangement further comprising a seal counterpart, the seal counterpart located at least partly within the casing, the seal counterpart comprising a body formed with a plurality of substantially dead-end parallel passages, the passages extending substantially radially; the runner further comprises a plurality of substantially parallel dead-end passages formed within the runner, the passages extending substantially radially; the runner, 25 casing, and seal counterpart configured so that the passages in the runner and seal counterpart interlock to form a tortuous path at a location in the gap between the casing and the runner. In an embodiment, the counterpart component is sealed within the casing via at least one O-ring or lip seal, positioned to avoid leakage of fluid in the gap into the area above the 30 seal counterpart. In an embodiment, the seal counterpart further comprises at least one ring of seal magnets, and the runner further comprises at least one ring of corresponding runner magnets, the magnets positioned so that in use the interacting forces of the magnets on the runner and the seal counterpart act to substantially maintain the relative position of the runner and seal counterpart. In an embodiment, magnets are located on the seal counterpart in a ring on the upper side of the seal counterpart and a ring on the lower side of the seal counterpart, and in a 5 ring on the runner substantially above the seal counterpart and in a ring on the runner substantially below the seal counterpart. In an embodiment, the rings of magnets are arranged so that the rings of magnets are generally aligned in the same vertical plane. In an embodiment, the magnets comprise Neodymium Iron Boron magnets. 10 In an embodiment, the sealing arrangement further comprises a thrust bearing and a spring, the thrust bearing and spring located so as to interact with the runner and seal counterpart, so that any action of the thrust bearing on the seal counterpart is countered by a reactive force from the spring. I / ""S 14 / In an embodiment, the thrust bearing and spring are located at each end of the seal 15 counterpart, above and below the seal counterpart. In an embodiment, the sealing arrangement further comprises first and second thrust 1— bearings, the thrust bearings located so as to interact with the runner and seal counterpart, so that any action of the first thrust bearing on the seal counterpart is countered by a reactive force from the second thrust bearing. 20 In an embodiment, the turbine further comprises at least one sensor configured to monitor the axial position of the runner and the seal counterpart, and an adjusting mechanism configured to adjust the position of the seal counterpart to keep the relative position of the runner and seal counterpart substantially constant in response to data from the at least one sensor. 25 In an embodiment, the at least one sensor comprises an optical sensor configured to measure gap width. In an embodiment, the at least one sensor comprises an inductive sensor. In an embodiment, the at least one sensor comprises a pressure sensor. In an embodiment, the adjusting mechanism comprises space formed within the casing 30 adjacent to the seal counterpart, and a hydraulic positioning system configured to pump fluid to and from the space to increase or decrease the pressure within the space. In an embodiment, the adjusting mechanism comprises at least one linear electrical actuator. In an embodiment, the adjusting mechanism further comprises a controller configured to receive data from the at least one sensor and to send command signals to the adjusting 5 mechanism. In a second aspect the present invention may broadly be said to consist in a sealing arrangement for a pump or a turbine, the pump or turbine comprising: a static casing; a runner; the static casing and runner configured so that a labyrinth seal is formed therebetween, the labyrinth seal comprising a number of substantially parallel dead-end 10 passages formed within the casing and runner, the passages interlocking to form a single passage connected at their free ends by expansion chambers; the seal configured so that the substantially parallel passages where the outer wall is rotating are narrower than the substantially parallel passages where the outer wall is stationary. In an embodiment, the seal is further configured so that one or more of the wider 15 passages have a non-uniform gap width. In an embodiment, the gap width is reduced over part of the length of the passage. In an embodiment, the passage comprises a reduced gap width at either end. With respect to the above description then, it is to be realised that the optimum dimensional relationships for the parts of the invention, to include variations in size, 20 materials, shape, form, function and manner of operation, assembly and use, are deemed readily apparent and obvious to one skilled in the art, and all equivalent relationships to those illustrated in the drawings and described in the specification are intended to be encompassed by the present invention. This invention may also be said broadly to consist in the parts, elements and features 25 referred to or indicated in the specification of the application, individually or collectively, and any or all combinations of any two or more said parts, elements or features, and where specific integers are mentioned herein which have known equivalents in the art to which this invention relates, such known equivalents are deemed to be incorporated herein as if individually set forth. 30 Therefore, the foregoing is considered as illustrative only of the principles of the invention. Further, since numerous modifications and changes will readily occur to those skilled in the art, it is not desired to limit the invention to the exact construction and operation shown and described, and accordingly, all suitable modifications and equivalents may be resorted to, falling within the scope of the invention. Throughout the description and claims of this specification, the words "comprise", "include", "have", and "contain" and variations of these words, for example "comprising" 5 and "comprises", mean "including but not limited to", and do not exclude other components, items, integers or steps not explicitly disclosed also to be present. Moreover, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise. 10 BRIEF DESCRIPTION OF THE DRAWINGS Further aspects of the invention will become apparent from the following description which is given by way of example only and with reference to the accompanying drawings which show an embodiment of the device by way of example, and in which: LO Figure 1 shows a schematic illustration of a power generation system that uses a turbine C\J 15 that uses the seal of the present invention, the system comprising an upper fluid storage unit, a conduit or penstock fluidically connected to the upper fluid storage and extending generally downwards from the storage unit to a lower reservoir so that in use fluid is LO channeled from the upper storage unit to the lower reservoir via the penstock, a pump unit located in / on the penstock and adapted to pump fluid from the lower reservoir to the upper 20 storage unit, flow and safety valves within the penstock, and a turbine unit located towards the lower end of the penstock for extracting energy from the fluid flow, the turbine unit comprising a seal according to an embodiment of the present invention. Figure 2a shows a perspective view from the side and above of a known type of Francis turbine that can be used as the turbine unit in a system substantially similar to that shown 25 in figure 1, the turbine shown partly cut away horizontally to show detail of the inner parts of the turbine. Figure 2b shows a cross-sectional side view of the known type of Francis turbine of figure 2a, the turbine using a labyrinth seal arrangement of the known type. Figure 3 shows a schematic cross-sectional detail view of the known type of labyrinth seal 30 used with the turbine of figure 2b, showing detail of the seal arrangement and the anticipated flow structure. Figure 4a shows a schematic cross-sectional side view of a Francis turbine that has a seal arrangement according to an embodiment of the invention, the seal arrangement configured so that in the passages where the radially outer part or wall is rotating, the passages are narrowed and the passages where the radially outer part or wall is stationary have an increased gap width. Figure 4b shows a schematic cross-sectional side view of a Francis turbine that has a 5 seal arrangement according to a variation of the embodiment shown in figure 4a, the seal configured so that one or more of the wider passages has a non-uniform gap width. Figure 5a shows a schematic side cutaway view of part of a Francis turbine having a seal counterpart located between the runner and casing so as to form a labyrinth seal having narrow passages and wider or larger expansion chambers interposed or alternating with 10 the passages, the narrow passages substantially axially aligned using a mechanical alignment mechanism. Figure 5b shows a side view of a Francis turbine that contains the seal arrangement of figure 5a. LO Figure 5c shows a cutaway side view of the Francis turbine of figure 5b along the plane CM 15 line A-A in figure 5b. CM Figure 5d shows detail of the area designated by circle ‘B’ in figure 5c. 1— Figure 5e shows a partly cutaway exploded perspective view from one side and above of the Francis turbine of figures 5a to 5d. Figure 5f shows detail of the area marked as ‘C’ in figure 5e. 20 Figure 5g shows a non-exploded partly cutaway perspective view from the same angle as figure 5e of the Francis turbine of figure 5e. Figure 5h shows detail of the area marked as ‘D’ in figure 5g. Figure 6a shows a schematic cutaway side view of an embodiment of a Francis turbine having a seal counterpart located between the runner and casing so as to form a labyrinth 25 seal having narrow passages and wider or larger expansion chambers interposed or alternating with the passages, the narrow passages substantially axially aligned, the seal counterpart and the runner fitted with magnets. Figure 6b shows a perspective semi-exploded view of the seal counterpart and magnets of the Francis turbine of figure 6a. 30 Figure 7 shows a schematic cutaway side view of an embodiment of a Francis turbine having a seal counterpart located between the runner and casing so as to form a labyrinth seal having narrow passages and wider or larger expansion chambers interposed or alternating with the passages, the narrow passages substantially axially aligned, the seal counterpart and the runner configured with a hydraulic positioning system. Figure 8a,8b shows the results of modeling Cases 1-3. Figure 9a, 9b shows the results of modeling Cases 1-3 for the large-scale turbine. 5 DETAILED DESCRIPTION The following detailed description illustrates embodiments of the present disclosure and ways in which they can be implemented. Those skilled in the art will recognize that other embodiments for carrying out or practising the present invention are also possible. General Overview 10 A schematic illustration of a power generation system 1 is shown in figure 1. The system 1 comprises main parts as follows: 1. An upper fluid storage unit 4; 2. A conduit or penstock 14 fluidically connected to the upper fluid storage unit 4, the conduit 4 extending generally downwards from the storage unit 4 so as to in use 15 channel fluid downwards under gravity from the upper storage unit 4, the lower end of the penstock 14 branching into two parallel branches 14a, and 14b; 3. A lower reservoir 7, each of the two parallel branches of the penstock 14 fluidically connected to the lower reservoir 7; 4. A pump unit 5 located in / on the second branch 14b of the penstock and adapted to 20 pump fluid from the lower reservoir 7, up the penstock 14, to the upper storage unit 4; 5. A main inlet valve 18a and a pump outlet valve 18b. 6. A turbine unit 2 located in / on the first branch 14a of the penstock so as to receive fluid from the upper storage unit 4 via the first branch 14a, the fluid passing 25 through the turbine unit 2 to drive the turbine unit, and then into the lower reservoir. These parts and their inter-relationship are briefly described below. Fluid Storage Tank The upper fluid storage unit 4 comprises an enclosed fluid storage container or tank 4 that is configured to in use hold / contain / store a fluid. In the preferred embodiment this is a high-density fluid. However, the system could also use water or similar. 5 If using high-density fluid, the tank is fully enclosed to prevent leaking of the fluid into the environment, and to prevent the fluid becoming contaminated - e.g. by absorbing rainwater or similar. However, the tank is not sealed, so the pressure remains the same as the surrounding atmosphere in use. The tank 4 has a fluid outlet 10, which can be generally referred to as a ‘penstock valve’. In the embodiment shown in figure 1, the fluid 10 outlet is in the wall of the tank close to the base. In variations, the fluid outlet could also be located in the base or floor of the tank 4. As shown in figure 1, the tank 4 is located above, and spaced apart from, the turbine unit 2 and lower reservoir 7, so that in use fluid 6 stored in the tank can flow downwards to the turbine unit 2 and lower reservoir 7 under gravity. 15 Conduit / Penstock The conduit or penstock 14 comprises a pipe or series of pipes or similar with an upper end / ends that is / are fluidically connected to the fluid outlet 10 of the upper fluid storage unit 4. The main body of the penstock 14 extends downwards from the storage unit 4 so that in use fluid from the upper storage unit 4 is channeled along the penstock 4 under 20 gravity. As shown in figure 1, the lower end of the penstock 14 branches into two parallel branches 14a and 14b, each of which connects at their lower or outer ends to the lower reservoir 7. A turbine unit 2 is located in / on the first branch 14a of the penstock so as to receive fluid 25 from the upper storage unit 104 via the penstock, and a pump unit 5 is located in / on the second branch of the penstock. Lower Reservoir The lower reservoir 7 acts to receive and store fluid that enters the reservoir 7 from the first branch 14a. In a similar manner to the tank, if using high-density fluid, the lower reservoir is fully enclosed to prevent leaking of the fluid into the environment, and to prevent the fluid becoming contaminated - e.g. by absorbing rainwater or similar. The lower reservoir 7 and the branch 14b are mutually configured so that fluid in the lower reservoir 7 can flow out of the lower reservoir 7 and into the branch 14b in use. 5 Pump Unit The pump unit 5 is located in / on the second branch 14b of the penstock, and is adapted to in use pump fluid from or out of the lower reservoir 7, along the second branch 14b, up the penstock 14, and to the upper storage unit 4. Turbine Unit 10 The turbine unit 2 is configured so as to in use receive a flow of fluid from the conduit or penstock 14 along the passage 14a. This fluid flow passes along the branch 14a and across the blades of the turbine within the turbine unit 2, causing the turbine to spin so as to generate power. On exiting the turbine unit 2, the fluid passes along the rest of the passage 14a and into 15 the lower reservoir 7, where it is stored until it is pumped back to the upper fluid storage unit 4. As can be seen, the system is a closed-loop system that uses the turbine 2 to extract energy. Use 20 In use, in order to commence a power generation cycle, the valve 18a is opened so that fluid flows from the tank 4 along the conduit 14 to the turbine unit 2, to cause the turbine to spin and generate power. At times of low demand, the turbine unit 2 is switched off and the pump unit 5 is switched on so that fluid is pumped out of the lower reservoir 7, along the sub-branch 14b, through 25 the pump, and up along the main body of the penstock 14 to the upper storage unit 4. In the form shown, and as described, the system is intended to be a closed system that uses a high-density fluid in place of water. However, in variations of the system, the system could be used with water or similar. If using water, then the system would be laid out in a more conventional manner, as there would no need to pump water from the lower reservoir to the storage tank at times of low demand. Turbine Sealing Arrangement - General The turbine unit 2 of the power generation system 1 described above comprises a Francis 5 turbine. A typical, known type of Francis turbine is shown in figures 2a, 2b, and 3, designated as ‘turbine 2’. The Francis turbine has main parts as follows: a static casing 21; a runner 22; stay vanes 23; guide vanes 24; and a draft tube 25. The casing 21 has an inlet 26 and an outlet 27. A crown labyrinth seal arrangement 10 generally designated as 28 is shown in figure 3, the labyrinth seal formed as a contactless seal consisting of a single tortuous path from the inside to the outside, the tortuous path consisting of a series of interlocking thin passages 28a alternating with larger and wider end chambers 28b, the tortuous path running between the casing 21 and the runner 22. 14 / Labyrinth seals such as seal 28 are formed so that there is a series of ‘N’ thin passages CM 15 28a alternating with end chambers 28b. The thin passages 28a are shown in figure 3 as CM the generally vertically aligned component of the passage through the seal 28, ‘N’ in the 1 particular arrangement shown in figure 3 being six - six passages 28a. The end LO chambers 28b are located adjacent to the end chambers to form the wider portions of the " tortuous path through the seal, the end chambers 28b shown at the ends of and 20 interspersed between the thinner passages 28a. As fluid flows along the path, it will flow out of a thinner passage 28a and into an end chamber 28b. This causes sudden expansion (expansion of the fluid into the end chamber 28b). Each of the thin passages has a width Hi, (generally designated by numeral 29 on figure 3), and a length L, (generally designated by numeral 30). In the 25 context of Francis turbines, Labyrinth seals are often used as runner seals, reducing the flow that bypasses the runner ( Qs), while introducing a torque loss ( Gs) due to the highly sheared fluid within the small passages of the seal. The flows within the thin passages 28a comprise complex three-dimensional flows that undergo a high degree of shear. Within the thin passages 28a, the fluid is heavily 30 sheared, leading to shear rates of over 10,000 / s. The heavily sheared fluid in the passage leads to torque losses, which, in turn reduces the turbine’s efficiency. The torque losses depend on the fluid’s state within the passage. The exact occurrence and nature of flow instabilities depends on the particular geometry of the seal and the rotational speed of the turbine runner. When flow instabilities do occur, this leads to increased momentum transfer and, therefore, higher torque losses. This can also lead to Taylor-Couette instabilities occurring within the seal (that is, Taylor vortices or Taylor rolls - generally designated by numeral 31 - can be generated within 5 the thin passages 29). The Taylor-Couette instability is a centrifugal instability and the dominant instability mechanism when an inner cylinder rotates in an outer cylinder. This is the typical configuration in turbine seals, particularly for half the passages in a Labyrinth seal. The Taylor-Couette instability can lead to enhanced momentum transfer, which increases the torque losses by a significant factor relative to the simple shear flow in the 10 other half of the passages, which is detrimental for turbine efficiency, as outlined above. Both Qs and Gsare detrimental for turbine efficiency and thus need to be minimized in order to increase efficiency. Embodiments of turbine sealing arrangement within a Francis turbine that assist with LD preventing a decrease in turbine efficiency are described below. 15 Turbine Sealing Arrangement - Asymmetrical Labyrinth Design CM A turbine sealing arrangement according to an embodiment of the invention is shown in 1 figure 4a. Similar numbering is used in this figure as is used for figures 2 and 3 above, LO with similar numbers used to designate the same or similar features within the sealing arrangement - e.g. turbine 2, 102; static casings 21 and 121; runners 22 and 122; 20 labyrinth seal arrangements 28 and 128, narrow / thin passages 28a and 128a, etc. As can be seen with reference to figure 4a, in this embodiment, the seal 128 is configured so that in the passages where the radially outer part or wall is rotating (these passages generally designated 128a(i) in figure 4), the passages are very narrow (narrower than usual for a seal of this type), thereby maximising pressure loss over the passage, while 25 having a relatively small impact on the torque owing to the lack of centrifugal flow instability. In contrast, for the passages where the radially outer part or wall is stationary (generally designated as passages 128a(ii) in figure 4), the gap width is increased. This decreases the impact of the Taylor-Couette instability on the overall torque losses. 30 The dimensions of the passages 128a(i) and 128a(ii) are generally designated by numerals 140 (the passages where the outer part or wall is rotating 128a(i)), and 141 (the passages where the outer cylinder is stationary (passages 128a(ii)). The passages 140 and 141 have width / 7, and Hi+i respectively. This change in seal geometry, as compared to symmetrical labyrinth seal designs as known in the art, leads to a reduction in torque losses and helps to reduce or minimise the impact of the Taylor-Couette instability on the torque losses within the seal (see Case 2, Appendix B). 5 Ina variation of this embodiment, the seal is configured so that one or more of the wider passages 128a(ii) do not have a uniform gap width - the gap width is reduced over a short length of the passage. In the specific form / variation shown in figure 4b, the gap width 141 applies across the majority of the length of the passage, but is reduced at either end. The advantage of this variation is that the reduced gap allows for additional expansion 10 losses (designated by numerals 142 in figure 4b), and this only adds minor additional torque losses. Turbine Sealing - Axially-aligned Labyrinth Designs It is possible to remove the Taylor-Couette instability from the seal flow entirely, by LO orienting the seal so that the narrow passages are substantially perpendicular to the axis CXJ 15 of rotation. This change in seal geometry, as compared to symmetrical labyrinth seal designs as known in the art, leads to a significant reduction in torque losses (see Case 3, -j— Appendix B). LO However, one reason that often excludes this design from being used is that it can be difficult to keep the seal gap(s) at the required width in use. Varying axial loads on the 20 runner lead to axial displacement of the moving runner relative to the static casing, which in turn leads to a change in the seal gap width and potentially even to components coming into contact (the rotating component touching the stationary component). In the description below, and as shown in this figure and the others relating to these embodiments, similar numbering is used as is used for figures 2, 3 and 4 above, with 25 similar numbers used to designate the same or similar features within the sealing arrangement-e.g. turbine 2, 102, 202; static casings 21,121,221; runners 22, 122, 222; labyrinth seal arrangements 28, 128, 228, etc. In the embodiments of turbine described below, the turbines contain a seal counterpart (designated as 250, 350, etc). The seal counterpart in these embodiments is configured 30 and mounted within the turbine so that the narrow passages are substantially perpendicular to the axis of rotation, as outlined above. In the first of these axial-alignment embodiments, the seal counterpart 250 is located between the runner 222 and the casing 221 so that a tortuous path or labyrinth seal arrangement is formed running between the casing 221 and the runner 222. The tortuous path / labyrinth seal arrangement consists of a series of interlocking thin passages 228a alternating with larger and wider end chambers 228b. The thin passages 228a are aligned substantially perpendicular to the axis of rotation of the turbine. The seal 5 counterpart 250 is non-rotating, but can move axially so as to act to adjust the position of the runner 222 in the axial direction. Thereby, the seal gap widths H, (the widths of the narrow / thin passages 228a) remain constant. As shown in figure 5a, the counterpart component 250 is sealed within / against the casing via O-rings 251 to avoid leakage of fluid into the area above the seal counterpart. Lip seals could also be used in place of O-10 rings. Details of specific ways in which the axial alignment can be implemented will now be described. Passive Alignment The axially-moving seal counterpart component can also be passively aligned, as detailed 15 in the description below. Passive Mechanical Alignment The axially moving seal counterpart component can in an embodiment be mechanically aligned. A turbine that contains one specific implementation of an axially-aligned labyrinth design that is mechanically aligned is shown in figures 5b, 5c, and 5d. Figure 5b shows a 20 side view of a Francis turbine 202a that contains an axially-aligned seal arrangement, figure 5c shows a cutaway side view of the Francis turbine 202a of figure 5b along the plane line A-A in figure 5b, and figure 5c shows detail of the area designated by circle ‘B’ in figure 5c. In this embodiment, the counterpart component 250a is located between the casing 221a 25 and the runner 222a. As outlined above, the narrow passages that form the labyrinth seal between the runner 222a and the counterpart component 250a are aligned substantially perpendicular to the axis of rotation of the turbine in order to avoid the occurrence of Taylor-Couette flow. The counterpart component 250a can move parallel to the axis of rotation of the turbine in order to maintain the seal gap at a constant width. The 30 counterpart component 250a is positioned and moved as necessary by a thrust bearing 260 which acts on the counterpart component 250a, and which is balanced by a spring 261 at the opposite end of the counterpart component 250a and which reacts to any force applied to the counterpart component 250a by the thrust bearing 260a. Lip seals 262a are located on the radially outer side of the counterpart component 250a, to seal between the counterpart component 250a and the casing 221a. In a variation of this arrangement, the spring can be replaced by a second thrust bearing, so that the second thrust bearing reacts to any force applied to the counterpart component 5 250a by the thrust bearing 260a in a similar manner to that of the spring described above. In this embodiment, and with reference to figures 5e to 5h, the seal counterpart 250a can be manufactured and assembled as follows: A ring-shaped recess is formed the casing 221a, the recess containing a series of springs 261a located at intervals around the circumference of the seal counterpart 250a. The thrust bearing 260 is mounted to the 10 runner 222a via a press fit and / or fixed in place via screws or glue. Then the seal counterpart 250a (which is formed as a split ring in this embodiment) is mounted on the runner 222a and the two parts of the seal counterpart 250a are joined. Alignment pins (not shown) in both sides of the split ring of seal counterpart 250a ensure accurate alignment of the two components. The lip seals 262a are mounted after this. Finally the LD 15 runner 222a is inserted into the casing 221a - the runner 222a and seal counterpart 250a C\l M are pushed into the casing 221a until the seal counterpart 250a makes contact with the CM springs 260 and compresses the springs 260. 1— Passive Magnetic Alignment m A turbine that contains one specific implementation of an axially moving seal counterpart 20 component that is passively aligned using high-strength magnets is shown in figures 6a and 6b. As shown in figures 6a and 6b, two rings of strong magnets 352a are fixed onto the seal counterpart 350, on the upper and lower sides (each end) of the seal counterpart 350. Two rings of strong magnets 352b are also fixed on the runner 322, the magnets 352b 25 located above and below the ends of the seal counterpart 350. The runner 322, the seal counterpart 350, and the rings of magnets 352a, 352b are configured so that when the turbine is assembled, all of the rings of magnets are generally aligned in the same vertical plane. When assembled, the repulsing forces of the magnets 352a, 352b act on one another so 30 as to keep the relative position of the runner 322 and seal counterpart 350 constant. That is, the runner magnets 352b and seal counterpart magnets 352a are close enough to one another that their magnetic fields interact with one another. The magnets 352a and 352b are positioned so that there is a neutral position, which is set (by designing the geometry of the runner and seal counterpart and by the choice of magnets) to be the ideal operating LO position. In use, the runner rotates freely within the neutral position. However, any axial movement of the seal counterpart component 350 away from this position (movement towards or away from the runner 322) causes interactions in the magnetic fields that produces forces that act on the seal counterpart component 350 to move the seal 5 counterpart component 350 back towards the neutral position / neutral alignment. A suitable category of magnets for this use are Neodymium Iron Boron magnets, or similar. As a first step in the manufacturing process, the runner magnets 352b are mounted to a lower ring on the runner 322. Then, magnets 352a are mounted on the seal counterpart 10 350. In a similar manner to that described above for turbine 202a, the seal counterpart 350 is initially formed in two parts, as a split ring. After the magnets 352a have been mounted on the two parts of the seal counterpart 350, the two parts are brought together and joined around the runner using alignment pins. As a final step, an upper magnet ring is mounted to the runner 322. 15 Active Alignment In other forms of the invention, the position of the counterpart component can be actively controlled. In order to actively control the position of the counterpart component relative to the runner, the axial positions of the runner and the counterpart component are measured continuously in use via sensors. The data from the sensors is sent to a 20 controller, and, based on the data received, the controller sends command signals to a positioning system that adjusts the position of the seal counterpart as required. In a similar manner to that described above, the counterpart component is manufactured in two parts - as two parts of a ring. The two separate parts are brough together and joined around the runner using alignment pins. O-rings can then be mounted into the 25 counterpart component, and sensors into / on the casing. Active Alignment - Hydraulic Alignment In an embodiment, and as shown in figure 7, the positioning system for active alignment comprises a hydraulic positioning system. The turbine is configured so as to comprise a space 454, located above the counterpart 30 component 450. Fluid can be pumped to and from the space 454 via a conduit 455 formed in the casing 421, using a pump (not shown). When fluid is pumped into the space 454 this increases the pressure, and as a consequence, the counterpart component 450 moves downwards. If the action of the pump is reversed, fluid is removed or pumped out of the space 454, and this allows the counterpart component 450 to move upwards. The position of the counterpart component 450 is monitored via sensors, with the data from the sensors transmitted to a controller. In an embodiment, the sensors 453 comprise 5 induction sensors, and the controller comprises a Programmable Logic Controller (PLC) or similar. Sensor 453a is mounted so as to in use monitor the runner position. Sensor 453b is mounted so as to in use monitor the position of the seal counterpart. In use, if the controller receives data from the sensors 453 indicating that the counterpart component 450 is moving out of the required position, the controller sends command 10 signals to the pump to either pump fluid to or from the space 454, in order to correct the position of the counterpart component 450. The sensors could alternatively be optical sensors configured to monitor the gap width, inductive sensors, or pressure sensors. Other types of sensors can also be used as appropriate. 15 Active Alignment - Linear Electrical Actuators In a variation, the positioning system for active alignment can comprise linear electrical actuators. The structure of the turbine in this variation is very similar to that described above for the hydraulic positioning system, but with linear electrical actuators used in place of the hydraulic positioning system described above. 20 The seal configurations and arrangements as described above and shown in the figures are described specifically for use with and configured for a system that utilises high-density fluid - a fluid that is denser than water. Viscous losses increase with increasing fluid viscosity (that is, the viscous losses are higher when the fluid flowing through a turbine is more viscous), and therefore the present invention is more relevant in the 25 context of high-density fluids than for conventional or lower-density fluids. However, it should be noted that the seal configurations and arrangements of the present invention are not limited to higher density / higher viscosity fluids, and can also be used with fluids of different or lower density such as for example water. Appendix A - Seal Modeling The analytical models outlined in this appendix allow the seal losses to be estimated / calculated, for the (classic) symmetrical Labyrinth design, the (novel) asymmetrical Labyrinth design, and the axially-aligning Labyrinth design. 5 References In the calculations below, the following publications are referred to: • [Grossmann et al.,2016] Grossmann, S., Lohse, D., and Sun, C.(2016). High-Reynolds number Taylor-Couette turbulence. Annual Review of Fluid Mechanics, 48(1):53-80. 10 • [Moazzen et aL, 2022] Moazzen, M., Lacassagne, T., Thomy, V., and Bahrani, S.A.(2022). Torque scaling at primary and secondary bifurcations in a Taylor-Couette flow of suspensions. Journal of Fluid Mechanics, 937:A2 • [Yonezawa and Watamura, 2021] Yonezawa, K. and Watamura, T.(2021). Experimental and numerical investigations of erosion on runner seal of a Francis 15 turbine. IOP Conference Series: Earth and Environmental Science,774(1 ):012034. • [Zuk,1976] Zuk, J. (1976). Fundamentals of fluid sealing. Technical report. Seal Modelling The underlying models for the seal flow rate are presented in section A1. The models for the seal torque losses are derived in section A2. 20 The seal is modelled as a series of A / small passages (i) of length ( L,) and width (Hi). A1 - Pressure Drop Within the Seal The pressure drop over a seal is influenced by the flow rate ( Qs), the number of smallwidth sections within the seal (N), the passage length ( Li), width (Hi), the radial position of the passages ( Rsj), the orientation of the passages, as well as the viscosity of 25 the fluid (p ). The overall pressure drop over the seal ( Aps) is modeled as the sum of the pressure drops within the passages ( pi9ap) and expansion losses ( piexp), as shown in equation 1. N APs = £ Apf” + Apr” 4=1 (equation 1) In the following, Aps through the seals is analytically modeled. Following [Zuk,1976] the necessary equations are derived for laminar and turbulent flow within the individual passages (i). 5 The flow velocity within the passages is given by equation 2. mean Qs 2TrRSyiHi (equation 2) This leads to the Reynolds number as shown in equation 3 10 pQs (equation 3) For Re <1400 the flow is considered to be laminar (see section A1.1 - ‘Laminar Poiseuille Flow’). 15 For Re >1400 the flow transitions to a turbulent flow (see section A1.2 -‘Turbulent Poiseuille Flow") A1.1 - Laminar Poiseuille Flow If the flow is laminar, the flow profile within the passage with width H, occurs as shown in equation 4. 20 i dp ((ha2 A 2p ox \ \ 2 / / (equation 4) In equation 4: • p is the dynamic viscosity of the fluid dp 25 • dx Li is the pressure gradient within the seal of length L, • y is the wall-normal coordinate Integrating the above velocity profile allows the derivation of laminar flow rate as shown in equation 5 1 / - / Q \ Q Qlam r / 1 9p f fHA2 2\   _ C J \ \ 2 J y 3 / i dx \ 2 / (equation 5) Where C = 2ttR is the extent of the seal in the circumferential direction, and R is the radius of the seal location. Qslam can be rewritten as Jam __ — s 6p dx ^RH^^RH^ 6 / .1. L 2 / 1 I ox I 10 (equation 6) And this can be rearranged to: ^aP 15 A1.2 - Turbulent Poiseuille Flow (equation 7) For higher flow rates, the flow will transition to turbulence. The equations for turbulent Poiseuille flow are as follows (derived from [Zuk, 1976]): 20 This can be rearranged to calculate pressure loss (Ap,9ap ) as a function of flow rate ( 0stur) A1.3 - Expansion Losses In addition to the pressure losses in the seals, the sudden expansion after thin passages also adds to the pressure losses. Assuming a loss coefficient of K = 1 for sudden expansion, the expansion losses after each individual passage can be estimated as 5 shown in equation 9 (equation 9) A2 - Torque Losses 10 The fluid within the seal is transferring momentum from the rotating part (the runner) to the static component (the turbine casing) and vice versa. This momentum transfer reduces the torque that is available to be converted to electric energy by the turbine’s generator. Similar to the pressure drop modeling, the torque losses are modeled as the sum of the torque losses of each individual passage within the Labyrinth seal as shown in equation 15 10. N i=l (equation 10) For each of the A / passages in the Labyrinth seal, the resulting torque loss is as shown in equation 11: rs,i ^1^8,i a AsRs,i 2?T(JiRg7: wall (equation 11) Where As2TrpRs,iLi\3 the area of the seal, and is the velocity gradient at the wall. A3 - Linear Velocity Gradient The torque losses within the seal depend significantly on the orientation of the passage relative to the axis of rotation, and on which seal surface is rotating. If the passage is 10 parallel to the axis of rotation, and the outer surface is moving (e.g. the left passage in Figure 1 (d)), the shear flow is very stable and the radial velocity gradient within the passage is constant. Similarly, when the seal passage is perpendicular to the axis of rotation (e.g. Figure 5a or figure 5b), a constant gradient within the passage can be assumed. As such, the velocity gradient can be estimated as 15 du l m dy wail (equation 12) where Q = 2nf, and is the rotational velocity of the turbine, and f= RPM / 60 is its rotational frequency. For an outer rotating seal surface or a seal surface that is 20 perpendicular to the axis of rotation, the torque losses can be estimated as: A4 - Taylor-Couette Flow In contrast, if the seal is a passage between an inner rotating cylinder and a static casing (e.g. the right-hand passage in the second break-out box of figure 4), the flow within the passage gets impacted by centrifugal instabilities, leading to the so-called Taylor-Couette flow [Grossmannetal.,2016]. The behavior of the flow depends on the Taylor number (1 + ^(2^ + ¾)2^ 2 Ta = "W----- (equation 14) Where r]=Rs,i / (Rs,i + Hi). As for the present seal configuration Hi «Rs,i Ta can be simplified to 10 n2Q2 n2 I ^L. Cxy (equation 15) For sub-critical Tac^ 104 the velocity gradient remains laminar and equation 13 applies. However, for higher Ta the Taylor-Couette instability leads to a series of axisymmetric 15 helical vortices (Taylor rolls) within the seal passage that enhance the momentum transfer between the rotating and the static seal surface. The scaling of the torque ( GiTC ) as a function of the Taylor number ( Ta ) was estimated using fits on experimental data collected by [Moazzenetal.,2022] for Taylor-Couette flow in dense suspensions. Nu = = O.O4Ta0,33 (equation 16) The ratio Nu = GiTC / Glin is often called pseudo-Nusselt number owing to its (mathematical) similarity to the Nusselt number in Rayleigh-Benard flow 25 [Grossmannetal.,2016]. Note that the onset of Taylor rolls always leads to an increase of momentum transfer ( GiTC >G / In for Ta >104). As such, the impact of Taylor rolls should be minimised or their occurrence should be avoided. 3.5 Numerical Solver In order to solve the equations derived in sections A1 and A2 above, an iterative solver 5 was programmed in Matlab. A flow rate Qs was assumed, and the pressure losses for the runner seal were calculated. The seal flow rate Qs was systematically adjusted until the expected pressure drop ( Ps ) between the runner inlet and the draft tube was matched. Furthermore, the torque losses of Labyrinth-seal passages were aggregated to estimate the seal-torque losses. 10 LD CM Appendix B - Modelled Examples Modelled examples of the two design types of the invention (the asymmetrical design and the axially-aligning design) are outlined in this appendix. The two design types are as detailed above in the main body of the description. 5 The performance of the runner seals for two distinct turbine geometries (‘large scale’ and ‘small scale’) Is estimated for both the known / classic (symmetrical) seal geometries, and the Labyrinth seal geometries of the present invention, so there are four examples for the present invention: 1) small-scale asymmetrical design; 10 2) large-scale asymmetrical design; 3) small-scale axially-aligning design; 4) large-scale axially-aligning design. These are compared to a ‘control’ model for a symmetrical labyrinth seal of the known type, for both the ‘large scale’ and ‘small scale’ turbine geometries, so there a six 15 modelled examples in total. The two distinct turbine geometries considered are: (a) a small 500kW high-density hydroturbine (‘small scale’) and as outlined in example B1 below); and (b) a high-head Francis turbine of similar dimensions as the 110MW Tokke power 20 plant turbines in Norway (‘large scale’) and as outlined in example B2 below. The modelled examples are used to demonstrate the impact of the invention and the reduction of seal losses that are achievable with the present invention. A high-density fluid (density p = 2500kg / m3) with higher viscosity than water(p = 30cP) is used for the examples. Note, however, that similar physics apply to water as a working fluid. 25 An overview of the two turbines and their respective crown seals is provided in Table B1 below. For each of the turbines three distinct scenarios are modeled: 30 • Case 1: The state-of-the-art (SOTA) symmetrical labyrinth seal design. • Case 2: The asymmetrical Labyrinth design with seal passages H'"1 = 0.8Hbase for passages with an outer rotating seal surface, and HiTC = 2.5Hbase for passages with an inner rotating seal surface. • Case 3: An axially aligned seal with seal passages that are oriented perpendicular to the axis of rotation Example 1: Small-scale turbine Example 2: Large-scale turbine Total head; pM pg- 75m pg- 160m Power 530kW 110MW RPM 1000 375 Flow Rate Q 0.34m3 / s 30m3 / s Radial seal position 0.17m 0.9m Seal pressure ps 0.75pf<rf Seal length L, 12mm 50mm Seal passage witdh 0.4 mm 1mm Asymmetric seal width / 1 0.8 Hbase &2.5 H(ya 0.8 &2.5 H / ,^ Number of passages A’ 6 10 Table B1: Overview of two Francis turbines and their respective labyrinth seals 5 Example B1 - ‘small scale’ In this example, modelled examples of the invention are shown for a 530kW high-density hydro turbine. The turbine runs at 1000rpm at its best-efficiency point (BEP) with a flow rate of Qtoi = 3401 / s, and a head of 75m. A high density working fluid (density p = 2500kg / m3) with viscosity (p = 30cP ) is used for the examples. For the sake of 10 simplicity, it is assumed that 75% of the total head (ptot) applies to the Labyrinth seal (Ps). A Labyrinth seal with N = 6 passages was implemented. A base passage width Hbase = 0.4mm was selected and the passages are Li = 12mm long. Figure 8a, 8b - Comparison of seal modeling results for a small-scale turbine. 15 Figure 8a,8b above shows the results of modeling Cases 1-3. • Graph (8a) shows torque losses relative to the known, SOTA seal. The seal passages with Taylor Couette flow ( GSTC ) are responsible for the majority of the torque losses in the known seal. The seal designs of the present invention can be seen to reduce the torque losses significantly. 20 • Graph (8b) shows the distribution of the pressure drop over the seal and the split between expansion losses and pressure drop within the thin passages. The flow rate Qs is 0.69% of Qtot for all cases. The torque losses caused by the crown seal are 0.73% of the shaft torque ( Gshaft )■ The asymmetric seal design achieved a 13% reduction of the torque losses. The axially aligned perpendicular seal design achieves a 25 46% reduction of torque losses. The reduction of torque losses is achieved by reducing the impact of the Taylor vortices ( Gjc} Example B2 - ‘ large scale’ In this example, modelled examples of the invention are shown for a 110MW high-density 5 hydroturbine with similar dimensions as the Tokke power plant in Norway. The turbine runs at 375rpm at its best-efficiency point (BEP) with a flow rate of Qtot = 30,000l / s, and a head of 160m. A high-density working fluid ( density p = 2500kg / m3 ) with viscosity (p = 30cP ) is used for the examples. For the sake of simplicity, it is assumed that 75% of the total head (ptot) 10 applies to the Labyrinth seal (ps) A Labyrinth seal with N = 10 passages was implemented. A base passage width Hbase = 1 mm was selected and the passages are L, = 50mm long. Figure 9a, 9b - Comparison of seal modeling results for a large-scale turbine. 15 Figure 9a, 9b above shows the results of modeling Cases 1-3 for the large-scale turbine. Qualitatively, the results are similar. However, the achieved torque loss reductions are significantly higher. The asymmetric seal design achieved a 20% reduction of the torque losses. The axially aligned perpendicular seal design achieves a 77% reduction of torque losses.

Claims

1. A sealing arrangement for a pump or a turbine, the pump or turbine comprising:a static casing;a runner configured so that in use the runner rotates relative to the casing, the casing5 and runner further configured so that there is a gap therebetween,characterised in thatthe sealing arrangement further comprises a seal counterpart, the seal counterpart located at least partly within the casing, the seal counterpart comprising a body formed with a plurality of substantially dead-end parallel passages, the passages extending10 substantially radially;the runner further comprises a plurality of substantially parallel dead-end passages formed within the runner, the passages extending substantially radially;the runner, casing, and seal counterpart configured so that the passages in the runner and seal counterpart interlock to form a tortuous path at a location in the gap 15 between the casing and the runner.

2. A sealing arrangement for a turbine as claimed in claim 1 wherein the counterpart component is sealed within the casing via at least one O-ring or lip seal, positioned to avoid leakage of fluid in the gap into the area above the seal counterpart.

3. A sealing arrangement for a turbine as claimed in claim 1 or claim 2 wherein the seal20 counterpart further comprises at least one ring of seal magnets, and the runner further comprises at least one ring of corresponding runner magnets, the magnets positioned so that in use the interacting forces of the magnets on the runner and the seal counterpart act to substantially maintain the relative position of the runner and seal counterpart.

4. A sealing arrangement for a turbine as claimed in claim 3 wherein magnets are25 located on the seal counterpart in a ring on the upper side of the seal counterpart and a ring on the lower side of the seal counterpart, and in a ring on the runner substantially above the seal counterpart and in a ring on the runner substantially below the seal counterpart.

5. A sealing arrangement for a turbine as claimed in claim 3 or claim 4 wherein the rings 30 of magnets are arranged so that the rings of magnets are generally aligned in the same vertical plane.

6. A sealing arrangement for a turbine as claimed in any one of claims 3 to 5 wherein the magnets comprise Neodymium Iron Boron magnets.

7. A sealing arrangement for a turbine as claimed in claim 1 or claim 2 further comprising a thrust bearing and a spring, the thrust bearing and spring located so as to 5 interact with the runner and seal counterpart, so that any action of the thrust bearing on the seal counterpart is countered by a reactive force from the spring.

8. A sealing arrangement for a turbine as claimed in claim 7 wherein the thrust bearing and spring are located at each end of the seal counterpart, above and below the seal counterpart.10 9. A sealing arrangement for a turbine as claimed in claim 1 or claim 2 further comprising first and second thrust bearings, the thrust bearings located so as to interact with the runner and seal counterpart, so that any action of the first thrust bearing on the seal counterpart is countered by a reactive force from the second thrust bearing.LO 10. A sealing arrangement for a turbine as claimed in claim 1 or claim 2 wherein theCXJ 15 turbine further comprises at least one sensor configured to monitor the axial position of the runner and the seal counterpart, and an adjusting mechanism configured to adjust the position of the seal counterpart to keep the relative position of the runner and sealLO counterpart substantially constant in response to data from the at least one sensor.

11. A sealing arrangement for a turbine as claimed in claim 10 wherein the at least one 20 sensor comprises an optical sensor configured to measure gap width.

12. A sealing arrangement for a turbine as claimed in claim 10 wherein the at least one sensor comprises an inductive sensor.

13. A sealing arrangement for a turbine as claimed in claim 10 wherein the at least one sensor comprises a pressure sensor.25 14. A sealing arrangement for a turbine as claimed in any one of claims 10 to 13 wherein the adjusting mechanism comprises space formed within the casing adjacent to the seal counterpart, and a hydraulic positioning system configured to pump fluid to and from the space to increase or decrease the pressure within the space.

15. A sealing arrangement for a turbine as claimed in any one of claims 10 to 13 wherein 30 the adjusting mechanism comprises at least one linear electrical actuator.

16. A sealing arrangement for a turbine as claimed in any one of claims 10 to 15 wherein the adjusting mechanism further comprises a controller configured to receive data from the at least one sensor and to send command signals to the adjusting mechanism.

17. A sealing arrangement for a pump or a turbine, comprising:a static casing;a runner;the static casing and runner configured so that a labyrinth seal is formed5 therebetween, the labyrinth seal comprising a number of substantially parallel dead-end passages formed within the casing and runner, the passages interlocking to form a single passage connected at their free ends by expansion chambers;characterised in thatthe seal is configured so that the substantially parallel passages where the outer wall 10 is rotating are narrower than the substantially parallel passages where the outer wall is stationary.

18. A sealing arrangement as claimed in claim 17 wherein the seal is further configured so that one or more of the wider passages have a non-uniform gap width.LOC\j 19. A sealing arrangement as claimed in claim 18 wherein the gap width is reduced over15 part of the length of the passage.“ 20. A sealing arrangement as claimed in claim 18 or claim 19 wherein the passageLf) comprises a reduced gap width at either end.

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