The Physics of Mississippi Flood Control

Louisiana's Bonnet Carré Spillway diverts some of the Mississippi's floodwaters. But it also offers up a wealth of good physics questions.Rhett Allain
It's amazing how you can live close to something for such a long time, but never really see it. In my case, it's the Bonnet Carré Spillway. It's basically an overflow valve for the Mississippi River just upriver from New Orleans. When the river level gets dangerously high, the spillway is opened. This allows some of the river water to be diverted and instead of flowing downstream, it goes into Lake Pontchartrain. The spillway doesn't open all that often, but this year it was opened twice, the first time in its history that that has happened.

It's kind of a big deal when they open up this spillway. Oh sure, I've seen the water rushing through the trees as I drive over it on the interstate—but that was about it. Well, until recently. I decided to make the trek down there with my younger son. It was fun. Now I can check this spillway off my "to do" list.

But wait! This is a great opportunity for some real-world physics homework and estimation problems. Don't worry. I'm going to get you started. I'll give you some basic data on the spillway and then do two estimations for you. After that, you are on your own. Actually, I will start off with this video showing the spillway. It's not perfect, but this is from our trip to the spillway—also, you can see what this thing looks like.

Oh, you can even see a jumping asian carp in this short clip. From that video, you could get a rough estimate of the flow rate—but I'm going to give it to you anyway along with some other important facts.

  • The spillway is 7000 feet long (2134 meters for non-Imperials).
  • There are 350 "bays" that can be opened with each bay 20 feet wide (6.1 meters).
  • The maximum flow capacity is 250,000 cubic feet per second of water flow (you can convert this to m3/s as a homework question).
  • The floodway (the part of land that becomes a temporary river) is 5.7 miles long. At the Mississippi river side, the floodway is 7700 feet wide. When it gets to Lake Pontchartrain, the floodway is 12,400 feet wide.
  • Finally, here is a nice table that shows the number of spillway bays open on different days. For the video above (when I visited), there were 138 bays open.

How much power could you get from the spillway?

Suppose this spillway was some type of hydroelectric generator? How much power could you get from it? Let me start with the physics. If you take some water with a mass of m and drop it a height h, it will have a change in gravitational potential energy. This change in energy can be expressed as:

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In this expression, g is the local gravitational field with a value of about 9.8 Newtons per kilogram. If you just drop some water, this change in gravitational potential energy will make the water increase in speed to give it an equal amount of change in kinetic energy. However, if you have some type of generator you can use this energy to power stuff—like your phone or some other electric device. But what about the power? Power is a measure of how quickly something produces energy. That means it not only depends on energy, but also time. Here is an expression for the power.
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Now let's say that this water is pouring over the spillway at a flow rate of f in cubic meters per second (yes, the spillway page gives this in cubic feet per second). With the density of water (ρ), I can find the mass of water falling per second as ρf. Using this with the change in gravitational energy and the definition of power, I get:

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Wait! What is that e? I added that in there—it's the efficiency. Surely you don't think that all of the energy from that falling water will go into useful stuff, do you? Let's say that this hydroelectric generator is just 25 percent efficient (so e = 0.25). That can account for the amount of water that doesn't pass through the generator as well as the energy lost in the power conversion process. I'll put all these values together in a python calculator (that way you can change the values and re-run it). Here is the result (just click the pencil icon to edit the code).

With my estimates, that's a power output of 11.9 megawatts. Just for comparison, most nuclear power plants produce at least 100 times more power than this spillway generator. Oh, and it will probably be open for about a month. But it's still a fun estimate.

How much rain is in this spillway?

Just to be clear, the spillway is open because of rain and snow melt (but mostly rain in this case). It rains somewhere in the Mississippi river basin (which is huge) and some of this rain water ends up in Louisiana. So, the real question. Suppose it rains in Ohio. How much rain would need to fall on the whole state in one day such that you get the current flow rate at the spillway (86,000 cubic feet per second)?At first pass, this is a flow rate problem. Imagine this: A bunch of water enters Ohio and then leaves through the spillway. What's the difference between the entry and exit? The cross-sectional area is the difference. In Ohio, this area is the size of the whole state. So, let's approach this problem from the backend. I already know the flow rate at the spillway. I can use this to calculate the total volume of water that passes in one day. This volume of water will then be spread over the state of Ohio to calculate the rainfall.Step one is to convert the flow rate in units of cubic feet per second to cubic feet per day. Yes, I'm using imperial units—but only because I'm dealing with a state in the US. I would do the same for the only other two imperial unit countries: Liberia and Myanmar. So, this is really just a unit conversion problem. The key to unit conversions is to just multiply by the fraction (1/1). Here, let me show you.
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Notice the fraction of 3600 seconds divided by 1 hour? Since 3600 seconds is equal to 1 hour, this fraction is just 1. If you multiply any number by one, you get the same thing. This is important. Although the numerical value will change, the actual value doesn't change. It's just a different number in different units, but the same actual amount. That's the key to unit conversions.

In order to calculate the rainfall in Ohio, it's just one more unit conversion. What if I convert volume units from cubic feet to inches per Ohio? Yes, I can do that by multiplying by "1" again. I'm not going to write the whole thing out, I'm just going to list my conversions.

  • 1 Ohio = 208x230 square miles (approximately).
  • 1 mile = 5280 feet. Note, you have to do this conversion twice to go from square miles to square feet.
  • 12 inches = 1 foot.
This gives a rain rate of 0.067 Ohio-inches per day. Yes, Ohio-inches is a unit of volume. If you convert the time part to months (assume 28 days), this would be 1.87 Ohio-inches per month. That seems fair. Oh! But remember that much of the rain that falls in Ohio wouldn't make it to Louisiana. Some of it would evaporate, some goes into the ground, and some gets used by humans. I don't know the percent of rain that makes it to the river, so I will just leave my answer alone.

Homework

Now it's your turn. Here are some questions for you to consider.

  • If the spillway is fully open (all bays open), what percent of the Mississippi river would be diverted through the spillway? Yes, you will either need to estimate the flow rate of the Mississippi, or look it up.
  • Suppose Lake Pontchartrain was completely empty. How long would it take to fill up the lake using just the Bonnet Carré spillway?
  • How deep is the floodway? Hint: use the dimensions of the spillway along with the flow rate. Double hint: it should be deeper on the side of the river.
  • If the spillway is open for 30 days, estimate the change in salt water content (the salinity) of Lake Pontchartrain. Here—this might be useful.