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HONR 297 Environmental Models Chapter 3: Air Quality Modeling 3.7: The Basic Plume Model Puffs vs. Plumes Consider two distinct types of smoke stack releases shown to the right (Figure 3.15 from Hadlock). The top part of the figure shows a puff which is a momentary release of exhaust gases that forms a discrete cloud. The puff can move horizontally with any existing wind and is free to disperse or diffuse in three dimensions (x- and yhorizontally, z-vertically). Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment 2 Puffs vs. Plumes A plume, an example of which is shown in the bottom portion of Figure 3.15, is comprised of a steady continuous release of exhaust gases. The plume starts at the stack and continues for an indefinite distance in the direction of any prevailing wind (i.e. downwind). Concentration of the exhaust material gradually decreases the further one gets from the smoke stack source. Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment 3 Puffs vs. Plumes It turns out that the second type of smoke stack release is easier to model! Here are some reasons why: ◦ Significant dispersion takes place in only two directions – vertically, in the z-direction and horizontally, perpendicular to the axis of the plume (y-direction). For the figure at right, what would be the y-direction in the plume release? ◦ Along the axis of the plume, in the x-direction, change in concentration is so slight and gradual that there is only a small amount of dispersion in this direction. ◦ Thus, to model the plume release, we only need two input position variables, instead of three for the puff release! Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment 4 Gaussian Plume Model For plume releases, here is the model that has been developed for determining concentration of exhaust material: Equation (3) is known as the Gaussian plume model! Let’s look at each term and constant in the model … 5 Gaussian Plume Model C is the concentration at a point, measured in amount of pollutant per volume. ◦ [C] = mass/length^3 or weight/length^3. Q is the source term – it represents the amount of pollutants emanating from the stack per unit time. ◦ [Q] = mass/time or weight/time. 6 Gaussian Plume Model u is the wind velocity in the prevailing direction – it represents an average velocity over the time period we are interested in modeling a plume. ◦ [u] = length/time. ◦ Units of wind velocity are chosen to match the desired concentration units for a given situation. ◦ For example, if we are interested in average onehour concentrations at a given point, we’d choose an average wind velocity over one hour instead of an instantaneous value for u. 7 Gaussian Plume Model y is the horizontal coordinate, measured in the direction perpendicular to the axis of wind movement. ◦ [y] = length. z is the vertical coordinate, which measures elevation above the ground. ◦ [z] = length. H is the effective stack height - which is the original physical stack height h plus additional height which is added to account for the fact that exhaust gases exiting a smoke stack may move upwards a distance vertically due to heat and momentum before starting to move horizontally with the prevailing wind. ◦ [H] = length. 8 Gaussian Plume Model is a dispersion coefficient that accounts for dispersion in the y-direction. y ◦ This coefficient is analogous to the diffusion constants we saw in the one and two dimensional diffusion models we saw in Section 3.5 and 3.6! ◦ [ y] = length. ◦ y is determined by current meteorological conditions. ◦ y also depends on downwind axial distance x, so it is a function, not a constant. ◦ As we move downwind, one would expect the plume to be wider, which in turn would correspond to a larger value of y. is a dispersion coefficient that accounts for dispersion in the z-direction. z ◦ [ z] = length. ◦ z also is determined by meteorological conditions and axial distance x! 9 Gaussian Plume Model Let’s look at the underlying ideas behind equation (3)! First, using basic ideas from algebra (Laws of Exponents, etc.), we can rewrite the Gaussian plume model in the form: 10 Gaussian Plume Model Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment In this form, (4), we see that the concentration of pollutant C is a product of three quantities: ◦ Source strength ◦ Diffusion effect in the y-direction. ◦ Diffusion effect in the z-direction. 11 Source Strength The first term in (4) measures the strength of pollutant at the source, itself. It takes into account two factors: ◦ The rate Q at which pollutant is being injected into the atmosphere. ◦ The wind speed u – higher wind speed will lead to reduced concentration, due to larger amounts of air mixing with the pollutant being released. What about lower wind speed? 12 Diffusion in the y-direction The second term in (4) represents the diffusion effect in the y-direction. This term is very similar to terms we saw in the one-dimensional or two-dimensional diffusion formulas, from Sections 3.5 and 3.6 (equations (1) and (2)). Are there any differences? ◦ Yes – equations (3) and (4) are steady-state formulas – they assume the contaminant concentration doesn’t change over time, due to continuous output at the source. 13 Diffusion in the z-direction The third term in (4) represents the diffusion effect in the z-direction. ◦ Note that the Gaussian plume model, actually uses the vertical distance from the effective stack height H, z-H, this appears in the first exponent involving z. ◦ The second exponent involving z, z+H, is an error correction factor used to account for vertical diffusion being blocked in the downward direction once material from the plume reaches ground level. ◦ This can be significant in short smoke stacks, but is less significant in taller stacks (why?). For large H values, the second term will be small! 14 “Natural” Questions about the Gaussian Plume Equation 1. Why doesn’t x show up in equations (3) or (4)? Shouldn’t concentration C depend on axial distance downwind from the source? 2. How do we calculate dispersion coefficients y and z? 3. Where does the plume equation come from? How is it derived? 15 1. Why doesn’t x show up in (3) or (4)? Recall that dispersion coefficients y and z are functions of the axial distance x from the source, so concentration C does depend on x! Thus, we could write y(x) and z(x) in (3) and (4) if we wish, but this makes the equation more complicated. Also, note that y and z are functions of atmospheric stability class (see p. 71, Table 3.1). 16 2. How do we find y and z? Meteorologists have made extensive studies of the effects of atmospheric stability class and other factors such as wind speed. Using both experimental and theoretical mathematical models, they have come up with a set of well-accepted values of dispersion coefficients y and z as functions of axial downwind distance x and atmospheric stability class. Values can be read off of the graphs in Figures 3.16 and 3.17 (on the next two slides) for y and z, respectively! 17 Horizontal dispersion coefficient y as a function of x and stability class Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment 18 Vertical dispersion coefficient z as a function of x and stability class Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment 19 Vertical Dispersion Coefficients Note that the scales on the graphs are logarithmic, so each subdivision on an axis scale represents a multiplier for the base to the left. For example, the marks between 100 and 1000 correspond to 2x100, 3x100, 4x100, … , 9x100. A point halfway between 100 and 1000 would correspond to a number between 300 and 400. 20 3. Where does (3) or (4) come from? The mathematics leading to the Gaussian plume equation involves partial derivatives and statistics! For more details see Chapter 6. Section 6.2 shows how the Gaussian normal distribution from statistics is related to the diffusion equations we have been studying – it turns out that there is a direct relationship! This is why equation (3) or (4) is called the Gaussian plume equation. Recall that diffusion results from molecules taking “random walks”! 21 3-D Plume Coordinate System and Dispersion Effects Figure 3.18 (on the next slide) gives a threedimensional representation of both the plume spreading out and the contaminant concentration as a function of vertical and horizontal distance. The elliptical cross-sections represent the physical plume spreading out as x increases. The bell-shaped curves represent the fact that contaminant concentration decreases as one moves away from the plume’s center axis. Also note that the effective stack height H and stack height h are shown on this graph! 22 3-D Plume Coordinate System and Dispersion Effects Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment 23 An Applied Example! You live two miles due east of a coal-fired utility power plant that produces electricity for your city. The stack on the plant is 350 feet high, and the ground is level. On a given day, the sun is shining brightly and the wind is blowing from the southwest to the northeast at 10 mph. Measurements at the plant stack of the concentration of nitrogen oxides in the exhaust gases show that such pollutants are being released at the rate of 80 lb/min. What would you expect the concentration of nitrogen oxides (NOx)to be at your residence? Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment 24 An Applied Example! Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment 25 An Applied Example! In Figure 3.19, choose an xy-coordinate system with the x-axis along the plume centerline through the power plant. Thus, the x-axis makes a 45-degree angle with a line from the power plant to your house, since the wind is blowing from southwest to northeast. The y-axis is perpendicular to the x-axis – choose the origin of the xy-coordinate system to be located at the power plant, since this is the source point. 26 An Applied Example! Using the 45-45-90 right triangle formed, with the hypotenuse corresponding to the side between the power plant and your house, we can find the coordinates of the house! The hypotenuse has length 2 miles. Since the hypotenuse of a 45-45-90 right triangle is √2 times the length of the triangle legs, it follows that the triangle legs have length 2/√2 = √2 miles 1.414 miles! 27 An Applied Example! It follows that the xy-coordinates of your house are (in miles) (x, y) = (√2, -√2) (1.414, -1.414). To use the Gaussian plume equation (3) (or (4)), we need to know Q, u, y, z, and diffusion coefficients y and z. 28 An Applied Example! Here’s what we know: Q = 80 lb/min (source term) u = 10 mi/hr (average wind speed) x = √2 mi (downwind distance) y = -√2 mi (horizontal displacement) z = 0 mi (vertical elevation) Use Table 3.1 on p. 71 to find the atmospheric stability class! ◦ Since we have bright sun and a wind of 10 mi/hr, it follows from Table 3.1 that the atmospheric stability class is B. 29 An Applied Example! From Figure 3.16, we can find the dispersion coefficient y using the downwind distance x and atmospheric stability class. Note that the graph in Figure 3.16 has units of meters on each axis and our length units for Q and stack height are given in feet. Thus, we need to convert our x value to meters, read off y in meters, and convert y back into feet! 30 An Applied Example! x = √2 mi 1.414 mi = (1.414 mi)*(5280 ft/mi)*(0.3048 m/ft) = 2275.956 m. Atmospheric stability class is B. Thus, y 300 m = (300 m)*(3.28084 ft/m) = 984.252 ft. 31 An Applied Example! To find z, using the same x (in m) and stability class values with the graph in Figure 3.17, we find that 230 m = (230 m)*(3.28084 ft/m) = z 754.593 ft. 32 An Applied Example! Finally, we need an H – value for the effective stack height. Since we are only given physical stack height h = 350 ft, we will choose this for H. Note that this is a more conservative estimate, because using a smaller value for H will raise the concentration at ground level. 33 An Applied Example! Using Mathematica along with equation (3), we find that after converting miles to feet and hours to minutes, C = 1.12 x 10-20 lb/ft3. Air contaminant concentrations are usually measured in parts per million (ppm) or parts per billion (ppb) instead of lb/ft3, so given that 1 ppm of NOx corresponds to 1.1 x 10-7 lb/ft3, it follows that C = (1.12 x 10-20 lb/ft3)*(1 ppm NOx)/(1.1 x 10-7 lb/ft3 NOx) = 1 x 10-13 ppm. Since contaminants in air are undetectable at concentration levels less than 1 ppb = 10-3 ppm, it follows that the level of NOx concentration is 10-10 ppb, well below detection limits – hence we are safe! 34 PLUME Spreadsheet Our textbook author has included a spreadsheet program, PLUME, that will work in Excel! If you don’t have a copy that came with your book, a copy can be downloaded from our class web page – see the Hadlock Textbook Floppy Disk Files (ZIP) link. PLUME includes built-in formulas to compute the dispersion coefficients y and z! 35 PLUME Spreadsheet Using PLUME, reproduce the work done above in the Applied Example (Hadlock, p. 95 problem #1. Be sure to enter quantities with the correct units within PLUME. Compare your results within PLUME to those we found above – do they agree? If not, justify or resolve any discrepancies. Using PLUME, try Hadlock problems # 3 and 4 on p. 95. Hint: For problem #3, part (d), the longterm average concentration will be 0.25*(concentration when wind blows towards home) + 0.75*(concentration when wind does not blow towards home). 36 Resources Charles Hadlock, Mathematical Modeling in the Environment – Chapter 3, Section 7 Figures 3.15, 3.16, 3.17, 3.18, and 3.19 used with permission from the publisher (MAA). 37