The physics of a subway tunnel plug

The physics of a subway tunnel plug: an analysis for the skeptic

By Erin Krysinski

http://www.nytimes.com/2012/11/20/science/creating-a-balloonlike-plug-to-hold-back-floodwaters.html?pagewanted=all&_r=0

Following the flooding that threatened NYC subway systems after hurricane Sandy, scientists are now looking for new ways to block flood waters from entering the city’s transit systems in case of emergency.  One option is a new inflatable 16 ft radius “plug”, which may be used to block flood waters within the subway tubes themselves.  Like most physicists, I’m skeptical if the plug could actually work, and if so, how much pressure it would have to be able to withstand from the incoming water flow.  To find the pressure in the subway tunnels from the water, we must use Bernoulli’s equation to find the difference in pressure between the water entering the stairwells(P1) and the water flowing through the tunnel itself(P2).  Because the plug will be in the tunnel, we will be searching for P2.  However, because there are several factors we don’t know, specifically the speed of water flowing through the tunnel, we must first make some estimations about the speed of the water entering the stairwell(v1), and then use the equation of continuity to determine the speed through the tunnel (v2).  If the water is entering the stairwell at an average of 4.02 m/s (v1), and the radius of the stairwell is 1.83m, the Area of the stairwell(A1)=πr² or A1=10.52 m².  We can use the same equation to find the Area of the subway tunnel(A2), which has a radius(r2) of 4.88m, and thus A2=74.82 m².  Now, with the equation of continuity (A1v1=A2v2), we can find v2, or the speed of the water in the tunnel.  10.52m²x4.02m/s=74.82xv2, v2=0.57 m/s.  We now know v1, v2, as well as that the stairwell is about 1m below the surface (y1), the tunnel is about 9.14 m below the surface(y2), and the pressure of the stairwell(P1) is 1013x10⁵ N/m²  since it’s open to the atmosphere.  Using Bernoulli’s equation (P1+1/2pv1²+pgy1=P2+1/2pv2²+pgy2à1.013x10⁵N/m²+1/2(1x10³kg/m³)(4.02m/s)²+(1x10³kg/m³)(9.8)(1m)=P2+1/2(1x10³kg/m³)(0.57m/s)²+(1x10³kg/m³)(9.8)(9.14m).  Therefore P2=29,445.75 N/m².  Though the plug would have to with stand this pressure, in reality, the pressure of the water held back may be far greater, since water is probably entering from various other areas other than the stairwell.  Generally, there would be a lot of water to hold back, so if the plug were to break, the force of water behind could be quite dangerous.  Despite the fact that flood walls or other types of barriers may be expensive to build, they may be a safer option in the long run.