 So today, let's talk a little bit about dispersion modeling. What exactly is dispersion modeling? Well, basically, the idea is that if you have a pollutant coming out of something like a smokestack, the pollutant's not going to stay in one place. The entire reason for releasing it from the smokestack is to actually have it disperse and spread the pollutant out widely and thinly to a point that it's not really harmful to anybody, because if its concentration is low enough, it won't be harmful anymore. And we sort of release the pollutant and remove it from where we are. Well, if we do so, we count upon the fact that the pollutant itself, which will be in a high and probably dangerous concentration at the top of the smokestack, will disperse and end up spread out into the atmosphere in much lower concentrations. Well, we're interested in being able to figure out where that goes. For science purposes, we'd like to be able to predict where it goes. And for our engineering purposes, we'd like to be able to, if possible, control where it goes or make some design decisions that will allow us to maximize the effectiveness of our dispersion. For example, we might need to decide how high to make the smokestack in order to make our dispersion efficient. So in order to consider this idea of dispersion, we're actually going to think about three primary processes here. And you can see here, there's a picture of a number of smokestacks, but let's consider the tall one there. There are three major transport processes that we're interested in here. The first one we're interested in is advection, or the motion due to the wind and the motion of the air. When you actually have the motion of a fluid, this is true for water sources as well, whatever is inside the fluid is also going to be carried with the fluid. And that's going to generally create a motion of the pollutant. Within the fluid itself, within the air or the water, you're also going to get diffusion. That's a tendency to move out from some sort of central location and spread out. If you drop a little bit of dye into water, you'll notice that the dye, even if the water's not moving, will start spreading out and eventually change the color of the entire volume of water that you have. So that tendency to spread out is caused somewhat by small molecular motions of both the pollutant, whatever it is, and of the fluid itself as they bump into each other and move around something often called brownian motion. And there are other sort of things that we consider. And the third important piece here is the activity of gravity or deposition. That there's going to be a tendency over time for the particles, if they're heavier than air, to slowly settle out and end up landing on the ground again. And so those three sort of primary drivers of the motion we need to consider as we're considering how the air is distributed. Now, one of the things here that occurs here is the gravity piece can often be a very different piece than the others just because it tends to act on a very different scale. You'll see here that you actually have quite a bit of the air rising up from the smokestack. In this particular case, it's rising up from the smokestack less due to the actual wind in the environment, but also do somewhat to the heating of the air itself, which is a form of wind. In this case, it's the heating of the air that's coming out of the smokestack, usually warmer. Coming out of the smokestack will rise of its own volition. And usually, so you see rising at the beginning in very little effect of the actual gravity until much later in the life cycle of the pollutant. So let's talk a little bit about how, again, both diffusion and advection react. First of all, in diffusion, the tendency is to move from a very compact sort of form out in pretty much all directions. Notice I'm drawing this two-dimensionally, but you can also think about this in three dimensions as things going directly into the page and coming out of the page. And after you've had some diffusive processes, you reach a point where the particles are distributed relatively evenly throughout whatever fluid you have. There's a tendency to do that. In addition, when you have advection, you have some motion. Usually, the motion of the air is not perfect. There's usually different speeds at different locations. And those speeds will tend to take an otherwise sort of uniform distribution. For example, we can consider this sort of pulse, a pulse being sort of a cluster or a group of pollutants. And you'll notice that that pulse might be distributed where it moves a little faster in one region and a little slower in another. This is particularly true about things like streams, where if you had stream flow, fluid flow in a stream, that you would get your pulse if there was a pollutant right in the middle that was placed in the stream. It would be spread out a little bit differently in the middle of the stream where things flow faster or more slowly. Or also true if you compare things, if you're thinking about a vertical column where we might be going higher up, that near the ground, things would be a little more slowly than they might a little bit further up, that there would be different sort of wind speeds at different heights. So if you can combine that spreading due to advection as well as the spreading due to diffusion, you end up with sort of a spread out version. And we can often describe that if we plot space on a horizontal axis and concentration of a particular material on the vertical axis, you will often get this distribution where you have a peak in the middle, something that's called a bell curve where you have a distribution for a strong high distribution in the middle with lesser amounts on either side. This is also called a Gaussian distribution if it fits a particular mathematical formulation. And since we see that, we can also use sort of Gaussian formulas for the Gaussian distribution to represent mathematically what we expect to happen when we combine diffusive and advective processes. So here, for example, is an attempt to show our distributions in three dimensions. You'll see over here on the left, there is a picture of a smokestack of a particular height. And when that smokestack releases its pollutant, that pollutant is going to tend to rise somewhat, again, due to the usually heat of the smokestack itself. In fact, it needs to be heated in order to get it to rise up the smokestack in the first place. And that rise is generally going to mean that the majority or sort of the center of the plume is going to rise to a certain part. And this value H is called the effective height of the smokestack, the effective height. So usually, that's a value that's, well, almost always, that's a value that's higher than the actual physical height of the smokestack. And it has a number of parameters that sort of affect the effective height, that sometimes it will be affected on the wind that's passing by, some temperatures inside and outside temperatures and things along those lines. But if we can calculate a value for the effective height, that's sort of where we're assuming the majority of the pollution is going to end up or sort of the center of the pollution is going to rise to that particular height. As it rises, there's a particular wind direction. And effective processes are going to make the plume move in that direction. We define sort of a center place, sort of a mean location for a mean location of all the pollutants. We define that sort of center line as where the mean location is. And then you can see distributed about that center line. If we look at it along a horizontal line, we can see that it peaks on the center line. And we have less distribution as we move side to side. What we call a lateral motion. And we're going to go ahead and call that our y-axis. Whereas motion along the center line is the x-axis. There's the x-axis and our y-axis. And vertical motions are z-axis. So as we move out side to side, you'll see that the distribution, the further we get away from the center line, we reduce the amount of pollution. Similarly, the same thing happens as you get further above or further below that center line. Notice, however, you don't get exactly the same distribution. The distribution does sort of have that same sort of slope and peak as we go up into the atmosphere. But as we go down to the ground, we do have some interference with the ground that prevents it from continuing to reduce. Some of the pollutants will effectively bump against the ground or bump against an air layer that's flowing along the ground. And we get a slightly different look to our distribution once we get down toward the ground. And we need to sort of take that into account. Notice the shape of our Gaussian plume here. If we consider the shape of our distribution there, you'll notice that it sort of fades out. And this should look somewhat familiar to you as the shape associated with exponential decay. As we get toward that end there, this effectively looks like an exponential decay of some sort. And we will see that momentarily in the equations. So here's an example of our dispersion equation looking fairly complicated. However, all of these values can be accounted for. And we should be able to either gain some measurements for them so in order to create a relationship to find the concentration of a pollutant. What we're trying to do is find the concentration in grams per cubic meter or basically in some weight of pollutant per unit volume. And this is going to be a spatial distribution. We're interested in what the concentration is at some point in space relative to the smokestack, the location of the smokestack. So in order to find this, we need a few pieces of information. One of them, here's Q. Q is the pollution emission rate. How much pollution do we know is coming out at the top of the smokestack? Usually that's something that we can measure and we can require to be measured in order to sort of determine that. And the higher that rate, obviously the higher the concentration of everything around will be. There are a few constants here. We have the wind speed is important. Notice the faster the wind speed, the more advection we have, the more distribution we have, which is going to reduce the concentration at any one place. You notice it's in the denominator. It's going to reduce the concentration at any one location. You'll also see here that there are indeed exponential terms. In fact, multiple exponential terms multiplied by each other, and they are all associated with negative values, exponential decay. Notice that means that as we're moving away, here's the value y, and here's the value z. As we move away from sort of our center line, the value y is centered at 0. Somewhere the smokestack is along that line, whereas the value z is centered at our effective stack height. So if we go above the effective stack height or below the effective stack height, we are going to change the concentration in a fashion that decays. There's a negative value here, exponentially. There's the E value. Notice all of these values are modified by these values here, the sigma y and the sigma z values. Sigma y and sigma z, sigma y and sigma z. But these values are our standard deviations away from the plume. How do we determine those standard deviations? You might recognize the term standard deviation from statistics. Well, what we have to do is we have to sort of measure, we have to get a measurement of how far, basically, of the diffusion relationships. And to do so, we basically need to gather those using some empirical relationships. In other words, we need to measure what happens using actual stacks or some sort of model, physical model of those stacks to get those values, what our expectations are. Notice those values are going to depend on some other measurements, so we include something called atmospheric stability. We choose our atmospheric stability from a range from A very unstable to F very stable. The more stable the atmosphere is, the more likely we are going to stay in a nice straight line and not sort of swirl and have turbulence and move around. The more turbulence and swirling we have in the air, the more dispersion we're going to have. So here's an example. Notice one of the things that we didn't have in our original equation is we had relationships between y and z, but we didn't have a relationship to x. That x relationship wasn't actually included there. That's because the x, these standard deviation values all have a dependency, sigma y and sigma z both have a dependency on x, where x is the distance downwind, this distance here in the direction of the wind that the plume is following. And people have taken some time to do some measurements and create some values there. And you'll see, based on these resources in the literature, here's a couple of graphs where we can actually look and say, we can follow on the x-axis our distance downwind in kilometers. For example, 10 to the first, this would be 10 kilometers. And then we can move up to one of the values here. And depending on whether or not we're talking about our stable or unstable and our qualifications A through F, our stable or unstable conditions, we can then let's say that we're extremely unstable. We can read to that A line and then find a value in meters that we can use for the standard deviation. So this would say that our standard deviation, in other words, how much we would expect, a measure of how much we would expect the pollution to spread out, will by the time we reach 10 kilometers downwind, we would expect to see a spread of about 10 to the third meters or one kilometer spread, at least in terms of standard deviation. Notice that would change if we had a more stable environment. And if we're talking about distribution for the z direction, we would use this graph, which obviously, if we were very unstable, it would put us somewhere off of the graph if we're looking for values for the sigma z. So these standard deviations define the shape of the plume, whether or not it's going to be long and skinny, again with stable conditions, or fat and spread out, a wide plume. And those change as we go along the x-axis in the direction of the primary wind. Here's a close-up view of each of those. For reference purposes, you can again see how we get a different spread along the z component than we do along the y component. So what do we do with that information? Well, you notice with that graph, every place in space in three dimensions, you can do a calculation and consider what the concentration value is going to be there. Well, now we're talking about in three dimensions, how do we see what that looks like? Well, one of the things we can do is create contour plots. We pick a certain, in this case, the picture on the left is a vertical cross-section. We look at it from not from side to side, but consider the picture where we're slicing. We're going to slice, so we're sort of looking at it from this side. So our eye is looking at it from this side. That would be a vertical cross-section. We're looking at it from that side. And then the picture on the other side, we're going to look at it from above, looking at it down. So let's go back to that picture there. So if we're looking at it from the side and the wind is blowing across the face there, we can sort of see where the concentration is greatest. Obviously, we'd expect it to be greatest closest to the plume. And then it spreads out, and the further you are up and away, the less pollutant you can expect. But you can also see that a great amount of this pollutant is going to be concentrated immediately downwind of the plume, of the smokestack. Similarly, if we look at it from the top, looking down and the wind is blowing aside, the further you get to the side of the smokestack, if you live directly in the line, you're going to be experiencing the largest sort of cross-section or the largest amount of exposure. But if you move off to either side, you're going to have lower concentrations from a similarly elevated source. There's a couple of different variations that we can look at for this formula. Notice this formula up here includes R instead of X and doesn't include the standard deviations. So it'd be sort of a variation of the formula that you'd be likely to use, maybe if you didn't have quite as strong of a prevailing wind and you were interested in sort of a longer term effect when the wind might be changing direction at different points in time, in which case we're talking about a radius a distance away from the plume. And if you wanted to figure out how much of the pollution existed at ground level, we could solve these files by setting z equal to 0 and calculate a concentration for people who are actually standing on the ground, which is obviously of interest to us as long as the ground level is consistent. Obviously, if you have significant changes in your elevation, you would actually calculate this at different levels. There are other ways that you can add more detail, more accuracy, equals more work. You could actually figure out and do some research on how to account for the thermal inversion. Once you got to a certain height, whether or not the concentrations were kept within a particular boundary, you could also account for the gravity, which isn't included in this model but is one of the ones that we mentioned. Usually, again, depending on the size of the particulates, they may not settle out in the same sort of time frame. It might take them on the order of days for something to settle out, whereas in this particular case, the motion might be on the order of hours. If we're talking about how long it takes to sort of affect things, the rates might be significantly different. And if you had multiple sources, for example, the picture of our smokestack had one tall smokestack and many smaller smokestacks, you'd obviously have to add things together to figure out what your concentrations would be and determine the different types of pollution that might be existing there.