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HONR 297 Environmental Models

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 ).

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HONR 297 Environmental Models

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  1. HONR 297Environmental Models Chapter 3: Air Quality Modeling 3.7: The Basic Plume Model

  2. 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 y- horizontally, z-vertically). Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

  3. 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

  4. 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

  5. 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 …

  6. 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.

  7. 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 one-hour concentrations at a given point, we’d choose an average wind velocity over one hour instead of an instantaneous value for u.

  8. 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.

  9. Gaussian Plume Model • y is a dispersion coefficient that accounts for dispersion in the y-direction. • 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. • z is a dispersion coefficient that accounts for dispersion in the z-direction. • [z] = length. • z also is determined by meteorological conditions and axial distance x!

  10. 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:

  11. Gaussian Plume Model • 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. Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

  12. 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?

  13. 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.

  14. 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!

  15. “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?

  16. 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).

  17. 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!

  18. Horizontal dispersion coefficient y as a function of x and stability class Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

  19. Vertical dispersion coefficient z as a function of x and stability class Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

  20. 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.

  21. 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”!

  22. 3-D Plume Coordinate System and Dispersion Effects • Figure 3.18 (on the next slide) gives a three-dimensional 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!

  23. 3-D Plume Coordinate System and Dispersion Effects Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

  24. 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

  25. An Applied Example! Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

  26. 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.

  27. 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!

  28. 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.

  29. 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.

  30. 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 yin meters, and convert y back into feet!

  31. 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.

  32. 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 z  230 m = (230 m)*(3.28084 ft/m) = 754.593 ft.

  33. 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.

  34. 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 10-13ppm. • Since contaminants in air are undetectable at concentration levels less than 1 ppb = 10-3ppm, it follows that the level of NOx concentration is 10-10 ppb, well below detection limits – hence we are safe!

  35. 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!

  36. 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 long-term average concentration will be 0.25*(concentration when wind blows towards home) + 0.75*(concentration when wind does not blow towards home).

  37. 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).

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