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On a related note, I drive up Woodruff St. from 51 into Mt. Washington on a regular basis. It looks like a smaller scale of the drainage issues on Washington Blvd, i.e. it seems like it's a watershed down into the Saw Mill Run creek. Yesterday when it was raining, you could see that the water was running down the hill glancing off of blocked storm drains and running across the road. I doubt that area will ever flood, but if the city doesn't do something about the drainage issues the road, which they just paved, is going to be buckled and damaged over the winter or if we get another really heavy rain like last week..
But on the chance that you're waiting for a response to your post responding to my post....
That's why I was critical of that map. It doesn't give you enough detail to tell you what really goes on, and might actually be misleading.
Does the flow in the 102" pipe run south, cross the street, and run north again in the 108" pipe. I don't know. But if it does, that would negate any discussion about how much capacity the two together have, because effectively, that means one discharges into the other.
As far as ARB goes, I know they did some work there in the not too distant past (I dunno. 10-20 years?), that involved some pretty big pipe, cause that's just the kind of stuff I notice when it's going on, but I don't have any idea what they added or changed.
Well, whatever the case, you're going about it wrong.
You just can't add the dimension of those pipes together and express the capacity in inches. You need to calculate the area of the circles described by those sizes, and add those results together. You know, the whole "Pi R squared" thing.
A 48" pipe has a circular area of about 12 and a half square feet.
A 132" pipe has a circular area of about 95 square feet.
So that's about 107 square feet of discharge capacity.
A 102" pipe has a circular area of about 57 square feet.
A 108" pipe has a circular area of about 63 and a half square feet.
So that's about 120 square feet of capacity.
So, yeah, there's a little bit more going in, than coming out, in terms of ultimate capacity.
But even that doesn't really matter. What really matters is the ultimate capacity of the pipe(s) coming out, vs. what's being fed in.
On your street (if it's typical of most city or suburban streets), there's literally dozens of 4" house laterals feeding into one 8" sewer main. When something happens upstream that puts more water into that main than it can carry, whether it's a lot from one source, or a little from each of them, that's when the sewer backs up into the lowest basements. And that's essentially what happened here.
As to the other posts...
I agree with MLNorth's assesment of what the decision making process should look like.
And to the question of "over-reaction"--Well if you react in such a way as to prevent issues without creating more issues than you've prevented, that would be within the parameters of appropriate reaction, wouldn't it? Otherwise, over-reaction is over-reaction.
I've looked at this problem a little, and my theory is that the sewer sizing is not necessarily the issue. I built a simple hydraulic model to simulate the sewer system performance. Here's a schematic:
It's hard to model rainfall, but fortunately it's easy to get around this by assuming a very large mass flow that far exceeds the storm that we experienced. I came up with a reasonable number by calculating a mass flux based on 4 inches of rain per hour and then multiplying it by a factor of 1000 to account for error and runoff. As long as we're using a number that's large enough that it floods the system, it doesn't really matter what it is since we're mainly interested in the performance characteristics but not necessarily the magnitude of those parameters. I set the water temperature equal to 60 degF and atmospheric temperature and pressure equal to 75 degF and 14.7 PSI.
So we're looking at the sizing of the sewer lines and discharge lines. In the post gazette article it is reported that the sewer lines are 108 inches and 102 inches and that the discharge lines are 48 inches and 132 inches. As ditchdigger has shown, this difference results in there being a slightly larger inlet flow area than there is outlet flow area. My model is showing that despite these differences, the inlet mass flow rates (actual amount of water passing through the pipe during a given time) and outlet mass flow rates are nearly equal after an initial "surge." (We'll talk about the surge later).
How can the same amount of water pass through two differently sized pipes? Well, in the small pipes water must flow faster. The linear velocity, or the distance that a fixed particle of water travels during a given time is greater in the smaller pipes than it is in the larger pipes (This follows from the Conservation of Mass, or Continuity Equation). In fact, it's often the case that in extremely small pipes at high velocities the flow can be dominated by friction along the pipe wall, however this effect quickly becomes less important as pipe sizes increase, relative roughness decreases, and velocity decreases (frictional losses are proportional to the square of velocity).
Well if the difference in size of the inlet and outlet pipes is not the reason for the back up then what could it be? Look at the "Mass Flux" and "Line Velocity" charts. Notice how there is flow in the sewer lines (inlet lines) before there is flow in the discharge. In my model this occurs because how the flow paths are connected to the volume that represents the sewer system. In reality this would occur due to restrictions in the system which could include debris, pipe blockage, restrictions in the flow path (elbows, connectors, etc.), and most importantly obstructions that prevent water from entering the sewer system. In this specific scenario, the blockage and source of the "surge" is due to the "intersection of Washington and Allegheny River Boulevard, which currently forms "a dam" at one end of a vast topographic bowl," as is stated in the Post Gazette article.
After the surge, the system will begin to drain, but it can't work fast enough to catch up. After the rain stops, the system is able to catch up and entirely drain, but as long as the rain persists there will be some level of flooding.
So, it was a long way of making a simple argument (though I enjoyed it!), but I think you could increase the discharge lines dramatically without significantly reducing the risk and magnitude of floods. Topology is the issue, not hydraulics.
Hot darn! I took a Fluid Mechanics course a few semesters ago and thought, "what practical applications does any of this stuff have in Urban Planning (my ideal career)?"
Thank you ML North for reminding me just why fluid mechanics is so important for planning in the Pittsburgh region.
Location: About 10 miles north of Pittsburgh International
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Quote:
I've looked at this problem a little, and my theory is that the sewer sizing is not necessarily the issue. I built a simple hydraulic model to simulate the sewer system performance. Here's a schematic: .....
Yeah, but I think you're missing an important part of the equation.
(And we'll keep in mind that I have no educatuion in fluid dynamics, and math is far from my strong suit.)
We'll also keep in mind that any restrictions in the discharge portion of the pipe, be it sediment, or the ALCOSAN diversion chamber, or just bends, will have a negative impact on capacity.
Quote:
How can the same amount of water pass through two differently sized pipes? Well, in the small pipes water must flow faster. The linear velocity, or the distance that a fixed particle of water travels during a given time is greater in the smaller pipes than it is in the larger pipes (This follows from the Conservation of Mass, or Continuity Equation).
For the water to flow faster in a smaller pipe, it needs to be under greater pressure.
In chemistry, thermodynamics, and other chemical engineering, a steady state is a situation in which all state variables are constant in spite of ongoing processes that strive to change them. For an entire system to be at steady state, i.e. for all state variables of a system to be constant, there must be a flow through the system (compare mass balance). One of the simplest examples of such a system is the case of a bathtub with the tap open but without the bottom plug: after a certain time the water flows in and out at the same rate, so the water level (the state variable being Volume) stabilizes and the system is at steady state.
(Bold added.)
I think we can agree that the bathtub analogy fits perfectly with the situation we're discussing.
The thing that makes the difference is "hydraulic head". When the tap is opened it delivers water, under pressure, to the tub. Lacking that pressure, the drain cannot pass the same amount of water that's flowing in. Once the tub fills to a certain point, there's enough head pressure to move more water down the drain. That's the steady state, and your continuity equation works then.
(Of course the nuts and bolts of why it works this way is that the faucet is delivering water through a 1/2" pipe, and the drain is a 1 1/2" pipe. If the faucet were fed by a 1 1/2" pipe, or the drain was 1/2", you'd end up flooding the bathroom, unless you had a very deep bathtub.)
Similarly, the discharge part of the Washington Blvd system wasn't up to the task of getting rid of that water until it had built up enough head (nine feet over the surface of the road, plus whatever the depth of the sewer is below the road), to move the water faster. That was the point at which it was in a steady state. (Or, maybe it just stopped raining so hard that the inflow was reduced so it didn't go any higher. We here, are not in a postion to know.)
Quote:
So, it was a long way of making a simple argument (though I enjoyed it!), but I think you could increase the discharge lines dramatically without significantly reducing the risk and magnitude of floods. Topology is the issue, not hydraulics.
Nice work, but unfortunately, I have to disagree.
Yes, the problem stems from the topography of the watershed, but the upshot of all of this is that a bigger discharge pipe would make a difference. (And again, we're back to the question of what it costs, vs. how frequently a potentially lethal situation presents itself.)
Yeah, but I think you're missing an important part of the equation.
(And we'll keep in mind that I have no educatuion in fluid dynamics, and math is far from my strong suit.)
We'll also keep in mind that any restrictions in the discharge portion of the pipe, be it sediment, or the ALCOSAN diversion chamber, or just bends, will have a negative impact on capacity.
For the water to flow faster in a smaller pipe, it needs to be under greater pressure.
I think we can agree that the bathtub analogy fits perfectly with the situation we're discussing.
The thing that makes the difference is "hydraulic head". When the tap is opened it delivers water, under pressure, to the tub. Lacking that pressure, the drain cannot pass the same amount of water that's flowing in. Once the tub fills to a certain point, there's enough head pressure to move more water down the drain. That's the steady state, and your continuity equation works then.
(Of course the nuts and bolts of why it works this way is that the faucet is delivering water through a 1/2" pipe, and the drain is a 1 1/2" pipe. If the faucet were fed by a 1 1/2" pipe, or the drain was 1/2", you'd end up flooding the bathroom, unless you had a very deep bathtub.)
Similarly, the discharge part of the Washington Blvd system wasn't up to the task of getting rid of that water until it had built up enough head (nine feet over the surface of the road, plus whatever the depth of the sewer is below the road), to move the water faster. That was the point at which it was in a steady state. (Or, maybe it just stopped raining so hard that the inflow was reduced so it didn't go any higher. We here, are not in a postion to know.)
Nice work, but unfortunately, I have to disagree.
Yes, the problem stems from the topography of the watershed, but the upshot of all of this is that a bigger discharge pipe would make a difference. (And again, we're back to the question of what it costs, vs. how frequently a potentially lethal situation presents itself.)
The model I made solves a nodal form of the mass, momentum, and energy equations which take into account the static head in each volume.
The bathtub analogy is a good one, as well as a fine example of a steady state system, however there are a couple differences. First, the flow in the drain line of a bath tub is dominated by friction. The cross section of the drain line and the cross section of the tub are different by one of two orders of magnitude. As you have mentioned, if the faucet is wide open the tub will fill to a certain level and then reach steady state. This will happen whenever the water reaches such a height that the potential energy (of the static head) equals the frictional head loss in the drainage line. The height of water necessary for reaching this point can be calculated using Bernoulli's equation. Further, from from Bernoulli's equation, you can show that the pressure in the high-velocity flow of the drain line will be lower than that in the low-velocity flow in the tub. For the following equation to remain constant, if velocity increases, pressure must decrease proportionally.
(1.)
A frictional loss term can be added to this equation, which would look something like this:
(2.)
Equations 1 and 2 can be combined to calculate the steady state level of the bath tub.
The sewer problem is going to be slightly different because the difference in frictional losses between the sewer lines and the discharge lines are going to be negligible. Therefore, I think that any flow accepted by the sewer lines should be able to be handled by the discharge lines so be it there aren't other obstructions. The "other obstructions" in this case are topological issues.
This does leave open the possibility that the sewer lines (inlet) themselves are too small, however that's different than saying that the sewer lines are acceptable but the system is being choked by the discharge lines. So maybe the topology is the bath tab, and the sewer lines are the drain. Regardless, increasing the sewer lines to any reasonable size still cannot close the difference between the cross section of the "tub" (topological bowl) and the "drain" (sewer line). So the solution would be to eliminate the "tub."
Quote:
Originally Posted by ditchdigger
(And we'll keep in mind that I have no educatuion in fluid dynamics, and math is far from my strong suit.)
Eh, I think that's pretty much irrelevant. You obviously have a fine understanding of the basic principles, which has made this a good discussion.
Since I was actually in one of the Washington Blvd floods, the bathtub analogy doesn't make sense to me except for how it takes time to drain after-the-fact.
The valley starts to flood after the manholes blow. The manholes don't blow after the bathtub fills. The manholes blow first and then fill the bathtub from the bottom.
I've seen it with my own eyes. The manholes blow and then water starts rushing down Leech Farm Hill Road, like a waterfall. In that order.
Then the bathtub fills from the manhole geisers shooting 15 feet into the air and the overflow coming down Leach Farm Hill Road.
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You're too nice...
