# just water over the dam

Some months ago, I posted an engineering challenge: figure out the water rate over an non-standard V-weir dam during a high flow period (a storm) based on measurements made on the weir during a low flow period. My solution follows. Weir dams of this sort are erected mostly to prevent flooding. They provide secondary benefits, though by improving the water and providing a way to measure the flow.

A series of compound, rectangular weir dams in Maine.

The problem was stated as follows: You’ve got a classic V weir on a dam, but it is not a knife-edge weir, nor is it rectangular or compound as in the picture at right. Instead it is nearly 90°, not very tall, and both the dam and weir have rounded leads. Because the weir is of non-standard shape, thick and rounded, you can not use the flow equation found in standard tables or on the internet. Instead, you decide to use a bucket and stopwatch to determine the flow during a relatively dry period. You measure 0.8 gal/sec when the water height is 3″ in the weir. During the rain-storm some days later, you measure that there are 12″ of water in the weir. Give a good estimate of the storm-flow based on the information you have.

A V-notch weir, side view and end-on.

I also gave a hint, that the flow in a V weir is self-similar. That is, though you may not know what the pattern will be, you can expect it will be stretched the same for all heights.

The upshot of this hint is that, there is one, fairly constant flow coefficient, you just have to find it and the power dependence. First, notice that area of flow will increase with height squared. Now notice that the velocity will increase with the square root of hight, H because of an energy balance. Potential energy per unit volume is mgH, and kinetic energy per unit volume is 1/2 mv2 where m is the mass per unit volume and g is the gravitational constant. Flow in the weir is achieved by converting potential height energy into kinetic, velocity energy. From the power dependence, you can expect that the average v will be proportional to √H at all values of H.

Combining the two effects together, you can expect a power dependence of 2.5 (square root is a power of 0.5). Putting this another way, the storm height in the weir is four times the dry height, so the area of flow is 16 times what it was when you measured with the bucket. Also, since the average height is four times greater than before, you can expect that the average velocity will be twice what it was. Thus, we estimate that there was 32 times the flow during the storm than there was during the dry period; 32 x 0.8 = 25.6 gallons/sec., or 92,000 gal/hr, or 3.28 cfs.

The general equation I derive for flow over this, V-shaped weir is

Flow (gallons/sec) = Cv gal/hr x(feet)5/2.

where Cv = 3.28 cfs. This result is not much different to a standard one  in the tables — that for knife-edge, 90° weirs with large shoulders on either side and at least twice the weir height below the weir (P, in the diagram above). For this knife-edge weir, the Bureau of Reclamation Manual suggests Cv = 2.49 and a power value of 2.48. It is unlikely that you ever get this sort of knife-edge weir in a practical application. Be sure to measure Cv at low flow for any weir you build or find.

Robert Buxbaum, vote for me for water commissioner. Here are some thoughts on other problems with our drains.

# A plague of combined sewers

The major typhoid and cholera epidemics of the US, and the plague of the Al Qaeda camps, 2009 are understood to have been caused by bad sewage, in particular by the practice of combining sanitary + storm sewage. Medieval plagues too may have been caused by combined sewers.

A combined sewer system showing an rain-induced overflow, a CSO.

A combined system is shows at right. Part of the problem is that the outfall is hard to contain, so they tend to spew sewage into the lakes and drinking water as shown. They are also more prone to back up during rain storms; separated systems can back up too, but far less often. When combined sewers back up, turds and other infected material flows into home basements. In a previous post, “follow the feces,” I showed the path that Oakland county’s combined sewers outfalls take when they drain (every other week) into Lake St. Clair just upstream of the water intake, and I detailed why, every few years we back up sewage into basements. I’d like to now talk a bit more about financial cost and what I’d like to do.

The combined sewer system shown above includes a small weir dam. During dry periods and small rain events, the dam keeps the sewage from the lake by redirecting it to the treatment plant. This protects the lakes so that sometimes the beaches are open, but there’s an operation cost: we end up treating a lot of rain water as if it were sewage. During larger rains, the dam overflows. This protects our basements (usually) but it does so at the expense of the drinking water, and of the water in Lake St. Clair and Lake Erie.

As a way to protect our lakes somewhat Oakland county has added a retention facility, the George W. Kuhn. This facility includes the weir shown above and a huge tank for sewage overflows. During dry periods, the weir holds back the flow of toilet and sink water so that it will flow down the pipe (collector) to the treatment plant in Detroit, and so it does not flow into the lake. Treatment in Detroit is expensive, but nonpolluting. During somewhat bigger rains the weir overflows to the holding tank. It is only during yet-bigger rains (currently every other week) that the mixture of rain and toilet sewage overwhelms the tank and is sent to the river and lake. The mess this makes of the lake is shown in the video following. During really big rains, like those of August 2014, the mixed sewage backs up in the pipes, and flows back into our basements. With either discharge, we run the risk of plague: Typhoid, Cholera, Legionnaires…

Some water-borne plagues are worse than others. With some plagues, you can have a carrier, a person who can infect many others without becoming deathly sick him or herself. Typhoid Mary was a famous carrier of the 1920s. She infected (and killed) hundreds in New York without herself becoming sick. A more recent drinking water plague, showed up in Milwaukee 25 years ago. Some 400,000 people were infected, and 70 or so died. Milwaukee disinfected its drinking water with chlorine and a bacteria that entered the system was chlorine tolerant. Milwaukee switched to ozone disinfection but Detroit still uses chlorine.

Combined sewers require much larger sewage treatment plants than you’d need for just sanitary sewage. Detroit’s plant is huge and its size will need to be doubled to meet new, stricter standards unless we bite the bullet and separate our sewage. Our current system doesn’t usually meet even the current, lower standards. The plant overflows and operation cost are high since you have to treat lots of rainwater. These operation costs will keep getter higher as pollution laws get tougher.

In Oakland county, MI we’ve started to build more and more big tanks to hold back and redirect the water so it doesn’t overwhelm the sewage plants. The GWK tank occupies 1 1/2 miles by about 100 feet beneath a golf course. It’s overwhelmed every other week. Just think of the tank you’d need to hold the water from 4″ of rain on 900 square mile area (Oakland county is 900 square miles). Oakland’s politicians seem happy to spend money on these tanks because it creates jobs and graft and because it suggests that something is being done. They blame politics when rain overwhelms the tank. I say it’s time to end the farce and separate our sewers. My preference is to separate the sewers through the use of French drains or bio-swales, and through the use of weir dams. I’m running for drain commissioner. Here’s something I’ve written on the chemistry of sewage, and on the joy of dams.

Dr. Robert E. Buxbaum, July 1-Sept 16, 2016.