Ruling gradient
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The ruling gradient (or ruling grade) of a section of railway line is the steepest section of that line. The ruling gradient is important in assigning locomotives to trains; the locomotives must have enough power to pull the train over the ruling gradient of a line.
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[edit] Compensation for curvature
The ruling grade of a train on curved track is steeper than the same train on straight track because the wagons follow the chord of the curve and not the arc of the curve. To compensate for this, the gradient should be a little less steep the sharper the curve is. The required compensation for curvature is a simple equation.
In addition, the friction of the wheels against the curved rails also increases the effective load that the locomotive must haul.
[edit] Compensation for gradients in tunnels
Tunnels on steep gradients can present problems for locomotives that emit fumes, both steam locomotives and diesel locomotives. If this is not done, the tunnel can become a rathole tunnel. The solution is analogous to compensation for curvature and requires the gradient in the tunnel and for some distance on either side to be greatly reduced compared to the ruling grade. Unfortunately, the necessary compensation for gradient is not a simple equation, but is rather a trial and error process. As happened on the first Cascade Tunnel, too little compensation for tunnel gradient was made.
Moisture from exhausts and springs can also make the rails slippery, and allowance may need to be taken for that as well.
[edit] First Cascade Tunnel
Because the First Cascade Tunnel gave so much trouble because of its steep grade, it is worthwhile even in hindsight to consider how else the problem might have been tackled.
[edit] Tunnel 1 as built
the tunnel joins the top of the more gently graded approaches, but had a steep climb.
The lines approaching the first cascade tunnel had a ruling gradient of 2.2% (1 in 45.5), while the tunnel itself had a gradient of 1.7% (1 in 58.8) facing eastbound trains. This gradient appears to be chosen by the expedient of building the shortest possible tunnel that connected with the original switchback route connecting Tye (Wellington) at 955 m and Cascade Tunnel Station at 1031 m. The higher portal of the 13,873 feet long (4228 m) tunnel was 305 feet (93 m) higher than the lower portal. Even though the ruling gradient in the tunnel was (slightly) less than that on the tracks on either side, the first Cascade tunnel soon proved to be a rathole tunnel, a problem which was later tackled by electrification.
[edit] Tunnel 1 as might have been
If the engineer had had more time to study the problem (and he was in a hurry) then the profile should have been arranged to make the first summit tunnel almost level, with a very steep section in the open air on the western approaches.
The gradient problem in the first tunnel is the result of the western portal being 100 metres lower than the eastern end, assuming that the tunnel joins the switchback route at the first available opportunity. If the 4% gradients at the western end had started several kilometres earlier, then the western portal could have been 100 m higher, making the first tunnel practically level and free of smoke problems. Strictly speaking, the level gradient should continue for a suitable distance beyond the portal, so that eastbound trains crawling up the hill can accelerate to line speed by the time that they enter the tunnel, so that the passage through whole of the tunnel uses the minimum throttle and generates minimum smoke.
[edit] Open air grade
Since in the open air, as many engines as needed can be added to a train to overcome a steep gradient, a better solution would have been to have a level tunnel from "New Tye" to Cascade Tunnel Station, with a 2.6 mile stretch of say 4% gradient ( 1 in 25 ) on the western approaches from say Windy Point Tunnel. The New Tye would have been about 250 feet higher in elevation. Levelling the main tunnel at the Cascade Tunnel Station is likely to require extra tunnel because the elevation of the valleys are higher.
Should the new steep approaches ever become operationally inconvenient, then it might be replaced by say, a spiral tunnel. Unlike the main 2.6 mile tunnel which can only be replaced on an all or nothing basis, an inadequate approach can be replaced in dribs and drabs.
A level main tunnel would have avoided the need for electrification of the main tunnel to eliminate the smoke problem.
[edit] Lapstone Zig Zag deviation tunnel
The original zig zag at Lapstone was built in 1869 to avoid having to build a long tunnel.
The zig zag became a bottleneck because amongst other reasons the length of the top and bottom roads restricted train lengths. A single track tunnel was built in 1892 to replace the Lapstone Zig Zag. Unfortunately, this tunnel was on a steep gradient, indeed the ruling grade of 1 in 33, which caused problems with the smoke from hard working locomotives, classifying this tunnel as a rathole tunnel.
By 1910, the situation had become unbearable, and with the need to duplicate the line, the old single track tunnel was replaced by an almost tunnel-free alignment at a reduced ruling grade on 1 in 60.
These problems predated the fume problems with the first Cascade Tunnel at the opposite side of the world.
[edit] Lithgow Zig Zag deviation tunnel
The original Lithgow Zig Zag was built in 1869, when the colony of New South Wales had limited tunnel building resources, and the Zig Zag became something of a wonder of the world.
By 1910, the single track Zig Zag had become a bottleneck, and duplication of the line became essential. By careful alignment of the new line to exploit a large number of clefts in the terrain, a double track deviation with 10 short tunnels was possible rather than one long one. These tunnels have a relatively gentle gradient of 1 in 90 (1.11%) while the western approaches are steeply graded at 1 in 42 (2.5%), while the frequent gaps between the tunnels provide good natural ventilation. In the latter days of steam, a train with 4 locomotives might haul a train to the entrance of the ten tunnels, with only 2 locomotives having to proceed through the tunnels themselves. Thus the fume problems in these tunnels was moderate, and these tunnels would not be classified as rathole tunnels.
[edit] Woodhead tunnel (1845)
The Woodhead Tunnel was summit tunnel of 3 mile length and suffered from poor working conditions as the gradient of 1 in 200 in the tunnel was not sufficiently reduced compared to the 1 in 100 ruling grade of the approaches as a whole. The tunnel actually consisted of two separate bores, of minimum cross-section.
[edit] Woy Woy tunnel (1888)
The Woy Woy tunnel in New South Wales is approximately 2000 m long. It is straight and level, and double track. There are no intermediate ventilation shafts. There are short rising grades of 1 in 40 and 1 in 80 on either side of the tunnel. The ruling grade of the line is 1 in 40, and taking all these factors into account, the tunnel has never suffered heavily from smoke. In all, a reasonably well designed tunnel. The line was electrified in 1961, but since the 1990s freight trains have reverted to diesel haulage.
[edit] Earlier tunnels with gradient problems
The degree to which the gradient in a tunnel must be eased to avoid a fume problem is not an easy thing to determine. The best answer is to study existing tunnels with steep gradients. The list of rathole tunnels shows some of these.
