Happy Monday! This is probably the most MATH-Y #mathematicalmodel post to date, so get real excited! This week's #mathematicalmodel is water resources engineer Andrea, who takes us DEEP into the mathematics involved with her job. She helps design weirs, which are water relief structures that transport wastewater to a wastewater treatment plant. Part of her job involves using mathematical equations and mathematical software (like MathCAD, InfoWorks, and even Excel!) to help model or calculate the flow of the water in the weir given certain constraints, so that she can determine the most efficient way to design the weir. I'm not sure if I explained that correctly...let's just let Andi take it away...:)
Even though I'm an engineer, not a mathematician, there have been a handful of times where I've sat down and completed some hardcore math problems. I got excited, I geeked out, I spent way more hours than I should have thinking about it and going through the best way to set up the equations for different scenarios. And...I told everyone I know that I got paid to use MathCAD!
For a wastewater conveyance system for a very large city in central Texas, we were scoped to create water relief structures and additional parallel pipes to transport wastewater during high flows to the wastewater treatment plant. Right now, the pipes are huge, 42-inch diameter and we needed to design a parallel 60-inch diameter to meet high flow demand scenarios. (This design is being constructed as I type this!!)
So with the design of a "parallel interceptor" we wanted water to transfer over to the second pipe during low flows. A weir is a weird structure over which the water will pass.
In this case, the weir is a section of the two pipes where they meet tangentially to let water flow from one to the other.
If you are interested, there's a super detailed write-up to tour one of the sewer systems in London, the River Fleet CSO. That's where these pictures come from to give you a bit of an idea about how a weir would look underground, used in sewage/stormwater conveyance.
So our senior conveyance engineer ordered a book from the UK, all in metric units and intended for square channel, irrigation side weirs. I was tasked with taking the concept and designing how long this weir needed to be to let water spill into the second channel. The reason it gets designed by engineers, and not just cut into a piece of pipe by the contractor, is because we want the water to only flow into the second pipe during high flow events. During low flow, that water should stay in the smaller pipe so that "solids" (haha that's poop) don't deposit on the bottom of the pipe when water velocities might be lower than 2 feet/second. We also don't want too much turbulence in the water which can release smelly hydrogen sulfide gas and provides resisting forces on the direction of flow.
Whew, that was a lot of information. This might help explain what deposition at low velocities looks like:
For tasks where there are small pieces of design that need to be done, a lot of the time the senior engineer will do a back-of-the-napkin calculations and then look at the overall picture of the system to find where it makes sense to put in the weir. To look at the scale of the entire system, engineers use a computer program called Infoworks RS which does a lot of math calculations in the background and then displays it using GIS into a spatial-graphical view. By looking at the graphs and cross sections displayed, they can figure out where it makes sense to place the weir. Where is it easiest to construct? What place doesn't disrupt downtown traffic? What place has the right slope of the existing pipe to support movement of water by gravity to the new pipe? Where is the weir needed before pressures are too high? Are there other relief structures that will perform better?
The design of the side weir is more finite and has to be done by hand to make it's accurate at that scale. They then check it against the model to make sure that the pressures and flows they are getting are similar to the pressures and flows that the simulation is getting for that area.
So now we need to design the exact length of that side weir which will be placed at Manhole 30212! (The exclamation mark was our modeler's way of saying that it's a new manhole near manhole 30212).
I started off with many rough drawings which in the end, turned into a rough sketch. I needed to take the equations from the book and get them to fit a rounded pipe that was not completely full. I also had to convert between the UK metric units displayed in the book and the US customary units.
I ended up with these equations:
For this particular method, you need to iterate (run the calcs over and over again) until you can narrow in and get the flow in your equation to match the flow you want to go through the pipe. So basically, we calibrate the equation until it spits out our desire length of the weir. Pretty cool!
The only problem? I didn't want to do calculations by hand that many times! Generally, that's where I'd use Excel. This one is a bit trickier because every time you run the calculation, you have to change the coefficients K and J based on the answers to some of the formulas. Which means the simplest thing to do would be to write some code. For me, writing code would take more time and has the potential for more errors. I like to use MathCAD. It's a super easy program once you know its idiosyncrasies and displays the equations and answers which can be PDFd and attached to a report. It's more difficult to retool an algorithm than Excel, but a more straightforward way to look at math problems. And my company has a license for it so...it's free for my project!
Putting variables into the program is easy.
And then I found myself putting everything into the program. Including all the checks and double checks. Here's a typical flow channel check. The Froude number is a celebrity in open channel flow dynamics because it can tell you how reliable your flow will be.
We're all nerds here.
I know because I have a little slice of the internet where math is okay. It's fun sometimes. (It really really is). When you have a problem, there's an answer. There's usually one or two paths to that answer. When has life ever been so black & white? When in life have you been able to flip to the last part of the story and look up all the answers to the odds? Or go online and download ALL the answers from the internet?
I sometimes explain that math is not all engineering is. I don't sit in a cubicle with protractors and engineering scales doing differential equations all day long. But I know them. I know their purpose and their theory even if I'd need to pull out a few resources to make sure I knew what I was doing if I ever attempted one.
But it should be said I do this probably <1% of my job. I'm not always sitting in my cubicle spitting out problems. Generally I'm writing reports, coordinating with everyone else on the project, or just responding to emails. Sometimes, I'm not even in the office but hanging out on a field site watching the weir that I designed be constructed! Do you follow my IG account? I post lots of pictures of the stuff I do outside there. Check it out at @dumontandi.