Tuesday, October 28, 2014

A Leafy Curse: The Physics of Leaves on the Track

Photo by author  

It's that time of year again. The colors of the trees are beautiful and vivid oranges, reds, and purples.

But autumn leaves are a nightmare for train operators, affecting anywhere with heavy deciduous tree growth — places like New England, the mid-Atlantic states, and the United Kingdom.
In the UK, delays due to leaves are so disruptive that the colloquial phrase of 'leaves on the line' has emerged, often jokingly referred to as a fictional excuse for delays. But the innocuous leaf is in fact no laughing matter.

Here's the scenario. Leaves fall to the ground and some inevitably land on the train tracks. A train runs over the leaves and compresses them to the tracks where they stick fast due to leaf oils. More leaves fall, or get blown onto the track in the wake of the train, and the process of leaf compression and build up continues. The residue black slime that results is remarkably resilient and rainy weather only adds to the problem.

The surface becomes so slick that trains have to accelerate and decelerate much more slowly than normal in order to avoid slipping, which could cause catastrophic accidents. Many trains, if they detect some acceleration due to 'spinning' wheels, can automatically apply breaks which lock the wheels, only making the problem worse. A locked wheel sliding along a track will wear and deform along a particular side, creating a 'flat'. Damage repair can run into the tens of millions for a single season, along with the frustration of delays.
Tracks of the Canadian Pacific Railway. Credit: Adapted from Mark Stevens 

Commuter trains, with their frequent stops, are affected by this problem much more than freight trains. Often slower travel times are built into autumn timetables, much to the consternation of commuters.

So what's happening here? Time for a bit of mid-week physics!

Trains rely on maintaining a rolling, not sliding, movement along the tracks, in order to avoid uneven wear on the wheels. To prevent the wheels from sliding, static friction between the steel train wheel and the steel track is needed. Normally as the train moves forward, the point at which the wheel touches the track at any given moment is stationary and not slipping, balanced exactly by the static friction force and the force applied to the wheel.
A wheel rolling forwards experiences a static friction force in the same direction of motion, which counters the force driving the base of the wheel backwards, such that static contact is maintained. A slippery surface means that the applied force might exceed the maximum static friction force and the wheel will start to 'spin'.
Diagram by author
The strength of friction between any two surfaces is characterized by the coefficient of friction, μ, and this is different depending on whether that friction is static or kinetic (sliding). Some surfaces are 'stickier' than others and have higher coefficients of friction.

If a train driver wants to avoid 'spinning' the wheels, then she or he must not accelerate or decelerate too quickly. The maximum force a rolling wheel can withstand before sliding at the base is given by this equation:
This says that the maximum static friction force depends on how sticky the interface is between the wheel and the track and, on flat terrain, the weight of the train (mass times acceleration due to gravity).
What happens when slippery leaves are added to the surface of tracks? The coefficient of friction between the wheels and the track surface decreases dramatically. Determining the value of the coefficient of friction is something that has to be done experimentally, and depends on the exact material composition, amount of leaves, level of moisture, temperature, and so on. 
But we can do a rough estimate:
The British train company, South West Rail, likens the slick compressed leaves on the track to a layer of teflon, creating a 'non-stick' surface. 
Following this analogy, we can turn to a look-up table of friction coefficients. For dry steel on steel, as would be the case on an ideal train track, μ is between 0.5 and 0.8. But for steel on teflon, this value falls to between 0.05 and 0.2. 
Silverliner V. Credit: John Corbett via Wikimedia Commons
Let's say we have a railcar with a mass of 80,000 kg (this is roughly the mass of a fully loaded Pennsylvania regional Silverliner V). How quickly can it stop from an initial velocity of, say, 80 miles an hour without sliding? 
The work required to stop the railcar is equal to its initial kinetic energy:
The kinetic energy is related to the mass of the train and its velocity, while the work done on the train is force times distance traveled:
In our case, we want to apply has much braking force as possible without actually causing the wheels to slip, so we plug in the force from the first equation: 
Now we simply cancel out the mass of the train on both sides, and solve for 'd', the distance required to stop a train without sliding.
And that's it! We can already see that if a train is traveling twice as fast, then it would take four times as long to safely bring it to a halt. 
(Side note: this assumes that all of the wheels of the railcar are equally capable of braking and that the weight of the train is evenly distributed over the wheels. If only some cars or engines can brake, then we cannot simply cancel our masses on either side of the equation.)
Photo by author  

Now let's consider our leaf scenario:
On a dry, summer day with clean tracks, the coefficient of static friction might be 0.5 while on an autumnal day with lots of wet, compressed leaves on the track, the coefficient of friction might be as low as 0.05. 
If the mass and initial velocity of the train are the same in both scenarios, then the stopping distance for a slippery, leafy track would be 10 times greater than on a dry, clean track. So if the train on a dry track only takes 130 meters to stop, then a train on a leafy track will take 1.3 kilometers! That's a huge difference and means train drivers tend to travel much more slowly rather than risk missing their next platform altogether.
A Network Rail (UK) 'leaf buster' train clearing the tracks from leaves.
Credit: Geof Sheppard via Wikimedia Commons
This seasonal problem is not going away anytime soon and rail companies have tried a number of different techniques to combat the foliage problem. Of course, trimming back the actual source of the leaves along the track is an obvious solution. In the days of steam-powered trains, this was done routinely by large teams of men since grass and leaves could easily catch fire from stray sparks and coals. Today the fire risk from diesel and electric engines is minimal and the large amount of manual labor required would be hugely expensive.

Alternative methods of clearing the track tend to be more temporary. Some companies employ 'leaf buster' trains which blast the tracks with high-power jets of water, sometimes in addition to 'sandite' trains which apply a sandy paste to the tracks to increase friction. There are also passenger trains that automatically release sand when they detect wheel slippage. Even intense laser beams have been shown to effectively disintegrate and blow away the leaves, but at a high cost.

All this from a simple autumnal leaf. Next time your train is delayed by 'leaves on the line' you'll have much to ponder.

By Tamela Maciel, also known as "pendulum"

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