 So, let's consider the concept of measuring the amount of flow that's coming down a river or down some sort of channel. And it's easy when you have water flowing in a channel to measure the depth of the water in the channel. You take a large stick, you stick it down to what you believe is the deepest point, and then you see the height of the free surface. All right, that's fairly straightforward. And often that particular piece of equipment is some variation on that piece of equipment is used to sort of get the idea of the water height as the water's flowing by. However, it's a little bit harder to get the other sort of details about how fast the water's flowing by. And we might have some information about the cross section itself, the area of the cross section. We probably do have that information. We probably have measured it. But it still requires a little bit more information to figure out how much flow is going by. Okay? And there's other information that's part of it. So, the speed of water in an open channel can be estimated using an empirical equation called Manning's equation. Let's see here, so let's use Manning's equation. What I mean by empirical, it means that this equation was generated by looking at relationships. It wasn't built up from sort of a series of physical principles like f equals ma, okay, or some other relationships that we know in physics. It was built by looking at some relationships between depth and area and some measurable relationships and the flow, which was also measurable. However, it was unclear how these things would be related until they were plotted on some graph somewhere, and you could see that there was either some form of curve that actually fit fairly well. Okay? So Manning's relationship is one that was generated from experiments, and we call it an empirical relationship or an empirical equation. So here's the first sort of part of Manning's equation, where v equals 1.49 over n, and we'll leave a little space there, rh to the two-thirds s to the one-half. Well, now we have some very interesting exponents involved here, plus a few other things that are balanced in here. So let's talk about what each piece of this equation actually is. The first piece here, this is the velocity, okay? That's the water velocity as it's moving downstream. So if we think about the stream, if I look at the stream from the side, okay, it's the average speed at which the water is moving. Yes, this is the same speed as in our continuity equation, q equals vA, the same sort of speed there. So there's our velocity, okay? The rh here is our hydraulic radius. The hydraulic radius basically being a measure of the geometry of the cross section. And you may recall that the hydraulic radius is the ratio between the cross-sectional area, same cross-sectional area, and this thing we call the wetted perimeter, which is the area that's in contact with the water and the surface it's flowing through. And notice both of those things will vary with the depth of the water, okay? In addition, this letter s here is a slope. In order for the water to get moving through this cross-sectional area, it must be flowing, it must be having some sort of gradient. It's basically flowing downhill, water primarily flows downhill. So let me redraw this sort of flow and think about it as the bottom. If we have the bottom of our channel, the water is flowing roughly following the bottom of that channel and keeping a depth that's roughly about the same, okay? But what we care about in this case is the slope of the bottom of the channel. And that slope is a relationship between, well, in your algebra class you learn slope is equal to rise over run, it's a relationship between rise and run. This change in height over this change in distance will say change in length here, okay? So that slope is the relationship between those two things, okay? Slope is change in height over change in channel length. And notice if the channel is kind of winding or moving like that, you actually do measure along the entire distance, you don't make a straight linear distance for that, you actually follow the channel in that measurement, okay? So that's the third piece. And the last unidentified piece here is this letter N, which is called Manning's Roughness. This is a coefficient that represents how much the sides of the channel are slowing the water down. And if it's a nice, smooth, maybe channel made out of metal or something, it's going to flow really quickly through that as a raceway. But if it has rocks in it, if it's a natural thing with dirt and rocks or even grass and weeds, then it's going to actually slow the water down substantially. So this Manning's Roughness value, okay, ranges from, and basically ranges something from around, I believe the lowest numbers I've seen are 0.01 to 0.30. And this is where the measurement, the empirical part came through, that these numbers were generated and often recorded in charts by lots and lots of different experiments to see how the fluid interacted with the different sides of different types of channel roughness, okay? So for some examples, as an example, if we have the main channel, like it's the main channel of a river and it's clean and straight, those are the kypo descriptors that you might use to apply here, you might have a value of 0.030, 0.03 as a point to 0.30, okay? If you had a mountain stream with large boulders and cobbles, cobbles basically being large stones, usually round stone, cobbles, boulders, that might have a Manning's coefficient of around 0.05, okay? It might slow the water down more substantially than it's the straight track of a main channel. If you're talking about a floodplain, and now instead of us thinking about it, this is just being a channel, like a thin channel, it's basically a large flat region where the water's flowing, we call it sheet flow. It flows like a big sheet of water. Okay, if you had a floodplain and it had lots of dense brush in it, so you're flooding some area that is not normally flooded, it's usually a field that has lots of brush that might resist a lot of the flow with a Manning's of 0.10, okay? Whereas if you went in the other direction and you had some lined cement, you had a very nice cement surface, raceway, very minimal 0.011 might be your Manning's roughness. So you can see that there's a variety of these roughness values that are used here, these coefficients, okay? Now one thing about this equation here is that the units don't balance. The units here, hydraulic radius, are going to be in length units. The slope units are going to disappear. Like it's usually length per length, so that's actually going to be a unitless value, okay? But this over here, our velocity is going to be length per unit time. So right now things don't, oh, and our Manning's roughness is also a unitless value. It actually doesn't have any units to it. It's just a coefficient. It's just a value or a number. So however, these don't balance. So if you look, we end up with length to the two-thirds, it has to be a little length over time. So there's actually a set of units hidden in this equation. And I'm going to write this here, that this is actually feet to the one-third per second. There are units that are assumed here, that this 1.49 value corresponds with measurements in feet, that if you're going to use the number 1.49, you actually have to measure your radius in feet, and you will get an answer in feet per second, okay? Notice, where does this number 1.49 come from? Okay, well it turns out that 1.49 feet to the one-third power, feet to the one-third, is equal to 1.0 meters to the one-third power. So that 1.49 is related to the value for meters. Meters, there are 3.28 feet in a meter, and if you take that to the one-third, you end up getting this value of 1.49. Okay, so that's important there, that if you, there's two ways to remember this. You can either remember this, there's the foot to the one-third. The other way to remember it is that there's basically two equations. The English equation uses the 1.49, whereas the metric system equation simply uses 1.0 in the Manning's equation. Okay, so this is our relationship for our velocity here, our velocity relative to these different values. Notice we also have our continuity equation, q equals va over here. Well if I plug that in, if I put that in over here, I can write this same version of the Manning's equation, but for flow. If q equals va and I plug in the v, I just basically have to add an a in here. We'll notice our hydraulic radius was equal to a over the wetted perimeter. So now I have a in two places. I'm going to multiply this entire equation by a, and then I'm also going to substitute in for the hydraulic radius. If I do that, I get a version of q, 1.49 feet to the one-third per second over n. There's the Manning's coefficient piece. Okay, then I can say a to the five-thirds power over wetted perimeter, the two-thirds power. In other words, there's the two-thirds, the a to the two-thirds, plus you're multiplying by a, so you add three-thirds and you get to the five-thirds power. And then that's times the slope to the one-half power. So that's another commonly seen version of Manning's equation where we're talking about the flow rate instead of talking about the velocity. And in this case, we've eliminated the hydraulic radius by replacing it, and now we just have versions of our area and our wetted perimeter. Note that all of these things, our area, our wetted perimeter, okay, both of those and our Rh over here, all three of these have a dependency upon the depth. So if we're able to calculate them from the depth or we somehow have them from reading the gauge, we can use that information plus the known information about the area, the known information about the roughness and the slope near where the gauge is to help us establish, to help us establish the flow rate. Okay, and in fact, most places will have established the N and the S and have it in some constant value that's already applicable to that location. And then they can recognize that they have some function of the depth in the area and some function of the depth in the wetted perimeter. Five thirds and two thirds, and they can use that to calculate the flow as some function of the depth.