Monday, November 21, 2016

Bode Miller for President 2020 (The Physics of Snowmaking)

With the imminent winter upon us, I decided to look at the physics involved with one of my favorite winter activities, downhill skiing. Having skied at Whiteface Mountain in Lake Placid, NY since I was born, I was curious to see what was required in order to operate such a massive resort. Using my new knowledge regarding fluid motion and pressure, I looked particularly at the requirements needed for the snow guns to operate and put down fresh snow on the trails. Using the trail map from Whiteface, I looked specifically at my favorite trail, Skyward. Rising 1337 meters above sea level at the top of the trail and 1006 meters at the bottom, I used the base lodge of the mountain as a reference point as that is where the tubing and water tanks are most likely located. To account for this, I subtracted the lodges elevation of 372 meters from each calculation. I also looked at the tubing used for snow guns on mountains, with the average tubing having a radius of 63.5mm and capability of withstanding up to 800 psi or 5.5e6 Pa. Using the formulas

P₁+½ ρv²+ρgh=P₂+½ ρv²+ρgh

ρ₁A₁v₁=ρ₂A₂v₂

Using the 5.5e6 threshold as my maximum example, I assumed that a velocity of 10 m/s would be sufficient to launch the snow across the trail. Using 1000 kg/m³ for the density of water, and looking at the guns at the bottom of Skyward, h=1006-372=634 meters.

P₁+½ (1000)(10)²+0=5.5e6+½ (1000)(10)²+(1000)(9.8)(634)
P₁=6213200 Pa

Compared to at the top of Skyward:

P₁+½ (1000)(10)²+0=5.5e6+½ (1000)(10)²+(1000)(9.8)(965)
P₁=100086000 Pa
A difference of 4e6;

Even if I assume the fluid is moving at 40 m/s at the base of the mountain, the difference is only 750,000 Pa in terms of the pressure required. These amounts of pressure are incredibly high just to achieve a speed of 10 m/s at the location of the guns. Even then, however, with the tubing going into larger gun barrels themselves, the speed is cut down even more.

Using a density of 750 for slushing or watery snow and the schematics of the snow guns with an opening of 1 meter diameter or 0.5 meters for the radius I looked at the equation of continuity.

(1000)(π(0.0635)²)(10)=(750)(π(0.5)²)(v)
v=0.22m/s

This means that the incredibly large pressure required to get the water up to that height on the mountain is only enough to shoot the snow out at a measly 0.22 m/s. Which would be ridiculously ineffective. Therefore my calculations are too idyllic and don't do justice to the actual mechanism that Whiteface employs. Most mountains have reservoirs of water that are located up the mountain, allowing the weight of the water and gravity to counteract some of the necessary pressure required. There are also pumps throughout the water lines and tubes to help the flow.