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19.9. Homework Exercises

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    19.9.1. Broaden Knowledge & Comprehension

    B1. Search the web for the government agency of your country that regulates air pollution. In the USA, it is the Environmental Protection Agency (EPA). Find the current air pollution standards for the chemicals listed in Table 19-1.

    B2. Search the web for an air quality report for your local region (such as town, city, state, or province). Determine how air quality has changed during the past decade or two.

    B3. Search the web for a site that gives current air pollution readings for your region. In some cities, this pollution reading is updated every several minutes, or every hour. If that is the case, see how the pollution reading varies hour by hour during a typical workday.

    B4. Search the web for information on health effects of different exposures to different pollutants.

    B5. Air-pollution models are computer codes that use equations similar to the ones in this chapter, to predict air-pollution concentration. Search the web for a list of names of a few of the popular air-pollution models endorsed by your country or region.

    B6. Search the web for inventories of emission rates for pollutants in your regions. What are the biggest polluters?

    B7. Search the web for an explanation of emissions trading. Discuss why such a policy is or is not good for industry, government, and people.

    B8. Search the web for information on acid rain. What is it? How does it form? What does it do?

    B9. Search the web for information on forest death (waldsterben) caused by pollution or acid rain.

    B10. Search the web for instruments that can measure concentration of the chemicals listed in Table 19-1.

    B11. Search the web for “web-cam” cameras that show a view of a major city, and discuss how the visibility during fair weather changes during the daily cycle on a workday.

    B12. Search the web for information of plume rise and/or concentration predictions for complex (mountainous) terrain.

    B13. Search the web for information to help you discuss the relationship between “good” ozone in the stratosphere and mesosphere, vs. “bad” ozone in the boundary layer.

    B14. For some of the major industry in your area, search the web for information on control technologies that can, or have, helped to reduce pollution emissions.

    B15. Search the web for satellite photos of emissions from major sources, such as a large industrial complex, smelter, volcano, or a power plant. Use the highest-resolution photographs to look at lateral plume dispersion, and compare with the dispersion equations in this chapter.

    B16. Search the web for information on forward or backward trajectories, as used in air pollution. One example is the Chernobyl nuclear accident, where radioactivity measurements in Scandinavia were used with a back trajectory to suggest that the source of the radioactivity was in the former Soviet Union.

    B17. Search the web for information on chemical reactions of air pollutants in the atmosphere.

    B18. Search the web for satellite photos and other information on an urban plume (the pollutant plume downwind of a whole city).

    B19. To simplify the presentation of air-quality data to the general public, many governments have created an air-quality index that summarizes with a simple number how clean or dirty the air is. For your national government (or for the USA if your own government doesn’t have one), search the web for info about the air-quality index. How is it defined in terms of concentrations of different pollutants? How do you interpret the index value in terms of visibility and/or health hazards?

    19.9.2. Apply

    A1. Given the following pollutant concentrations in µm m–3, convert to volume fraction units ppmv assuming standard sea-level conditions:

    a. SO2 1300 b. SO2 900 c. SO2 365
    d. SO2 300 e. SO2 80 f. SO2 60
    g. NO2 400 h. NO2 280 i. NO2 200
    j. NO2 150 k. NO2 40 m. CO 40,000
    n. CO 35,000 o. CO 20,000 p. CO 15,000
    q. O3 235 r. O3 160 s. O3 157
    t. O3 100 u. O3 50 v. O3 30

    A2. Same as previous exercise, but for a summer day in Denver, Colorado, USA, where T = 25°C and P = 82 kPa.

    A3. Create a table similar to Table 19-1, but where all the ppm values of volume fraction have been converted into concentration units of µg m–3.

    A4. Given wind measurements in the table below.

    1. Find the mean wind speed component in each direction
    2. Create a table showing the deviation from the mean at each time for each wind component.
    3. Find the velocity variance in each direction.
    4. Find the standard deviation of velocity for each wind direction.
    5. Determine if the turbulence is isotropic.
    6. Speculate on the cross-section shape of smoke plumes as they disperse in this atmosphere.
    t (min) U (m s–1) V (m s–1) W (m s–1)
    1 8 1 0
    2 11 2 –1
    3 12 0 1
    4 7 –3 1
    5 12 0 –1

    A5. Determine the Pasquill-Gifford turbulence type

    1. Strong sunshine, clear skies, winds 1 m s–1
    2. Thick overcast, winds 10 m s–1, night
    3. Clear skies, winds 2.5 m s–1, night
    4. Noon, thin overcast, winds 3 m s–1.
    5. Cold air advection 2 m s–1 over a warm lake.
    6. Sunset, heavy overcast, calm.
    7. Sunrise, calm, clear.
    8. Strong sunshine, clear skies, winds 10 m s–1.
    9. Thin overcast, nighttime, wind 2 m s–1.
    10. Thin overcast, nighttime, wind 5 m s–1.
    11. Thin overcast, 9 am, wind 3.5 m s–1.

    A6. Given turbulence kinetic energy (TKE) buoyant generation (B) and shear generation (S) rates in this table (both in units of m2·s–3), answer questions (i) - (vi) below

    B S
    a. 0.004 0.0
    b. 0.004 0.002
    c. 0.004 0.004
    d. 0.004 0.006
    e. 0.002 0.0
    f. 0.002 0.002
    g. 0.002 0.004
    h. 0.002 0.006
    i. 0.0 0.0
    j. 0.0 0.002
    k. 0.0 0.004
    m. 0.0 0.006
    n. –0.002 0.0
    o. –0.002 0.002
    p. –0.002 0.004
    q. –0.002 0.006
    r. –0.004 0.0
    s. –0.004 0.002
    t. –0.004 0.004
    u. –0.004 0.006
    1. Specify the nature of flow/convection
    2. Estimate the Pasquill-Gifford turbulence type.
    3. Classify the static stability (from strongly stable to strongly unstable)
    4. Estimate the flux Richardson number Rf = –B/S
    5. Determine the dispersion isotropy
    6. Is turbulence intensity (TKE) strong or weak?

    A7. Given the table below with pollutant concentrations c (µg m–3) measured at various heights z (km), answer these 5 questions.

    1. Find the height of center of mass.
    2. Find the vertical height variance.
    3. Find the vertical height standard deviation.
    4. Find the total amount of pollutant emitted.
    5. Find the nominal plume spread (depth)
    z (km) c (µg m–3)
    Question: a b c d e
    1.5 0 0 0 0 0
    1.4 0 10 0 86 0
    1.3 5 25 0 220 0
    1.2 25 50 0 430 0
    1.1 20 75 0 350 0.04
    1.0 45 85 0 195 0.06
    0.9 55 90 2 50 0.14
    0.8 40 93 8 5 0.18
    0.7 30 89 23 0 0.13
    0.6 10 73 23 0 0.07
    0.5 0 56 7 0 0.01
    0.4 0 30 3 0 0
    0.3 0 15 0 0 0
    0.2 0 5 0 0 0
    0.1 0 0 0 0 0

    A8.(§) For the previous problem, find the best-fit Gaussian curve through the data, and plot the data and curve on the same graph.

    A9. Using the data from question A7, find the nominal plume width from edge to edge.

    A10. Given lateral and vertical velocity variances of 1.0 and 0.5 m2 s–2, respectively. Find the variance of plume spread in the lateral and vertical, at distance 3 km downwind of a source in a wind of speed 5 m s–1. Use a Lagrangian time scale of:

    a. 15 s b. 30 s c. 1 min d. 2 min
    e. 5 min f. 10 min g. 15 min h. 20 min
    i. 5 s j. 45 s m. 12 min n. 30 min

    A11.(§) For a Lagrangian time scale of 2 minutes and wind speed of 10 m s–1, plot the standard deviation of vertical plume spread vs. downwind distance for a vertical velocity variance (m2 s–2) of:

    a. 0.1 b. 0.2 c. 0.3 d. 0.4 e. 0.5
    f. 0.6 g. 0.8 h. 1.0 i. 1.5 j. 2
    k. 2.5 m. 3 n. 4 o. 5 p. 8

    A12.(§) For the previous problem, plot σz if

    1. only the near-source equation
    2. only the far source equation

    A13. Given the following emission parameters:

    Wo (m s–1) Ro (m) ∆θ (K)
    a. 5 3 200
    b. 30 1 50
    c. 20 2 100
    d. 2 2 50
    e. 5 1 50
    f. 30 2 100
    g. 20 3 50
    h. 2 4 20

    Find the momentum and buoyant length scales for the plume-rise equations. Assume |g|/θa ≈ 0.0333 m·s–2·K–1 , and M = 5 m s–1 for all cases.

    A14.(§) For the previous problem, plot the plume centerline height vs. distance if the physical stack height is 100 m and the atmosphere is statically neutral.

    A15. For buoyant length scale of 5 m, physical stack height 10 m, environmental temperature 10°C, and wind speed 2 m s–1, find the equilibrium plume centerline height in a statically stable boundary layer, given ambient potential temperature gradients of ∆θ/∆z (K km–1):

    a. 1 b. 2 c. 3 d. 4 e. 5 f. 6 g. 7
    h. 8 i. 9 j. 10 k. 12 m. 15 n. 18 o. 20

    A16. Given σy = σz = 300 m, zCL = 500 m, z = 200 m, Q = 100 g s–1, M = 10 m s–1. For a neutral boundary layer, find the concentration at y (km) =

    a. 0 b. 0.1 c. 0.2 d. 0.3 e. 0.4 f. 0.5 g. 0.7
    h. 1 i. 1.5 j. 2 k. 3 m. 4 n. 5 o. 6

    A17(§). Plot the concentration footprint at the surface downwind of a stack, given: σv = 1 m s–1, σw = 0.5 m s–1, M = 2 m s–1, Lagrangian time scale = 1 minute, Q = 400 g s–1 of SO2, in a stable boundary layer. Use a plume equilibrium centerline height (m) of:

    a. 10 b. 20 c. 30 d. 40 e. 50 f. 60 g. 70
    h. 15 i. 25 j. 35 k. 45 m. 55 n. 65 o. 75

    A18. Calculate the dimensionless downwind distance, given a convective mixed layer depth of 2 km, wind speed 3 m s–1, and surface kinematic heat flux of 0.15 K·m s–1. Assume |g|/Tv ≈ 0.0333 m·s–2·K–1 . The actual distance x (km) is:

    a. 0.2 b. 0.5 c. 1 d. 2 e. 3 f. 4
    g. 5 h. 7 i. 10 j. 20 k. 30 m. 50

    A19. If w* = 1 m s–1, mixed layer depth is 1 km, wind speed is 5 m s–1, Q = 100 g s–1, find the

    1. dimensionless downwind distance at x = 2 km
    2. dimensionless concentration if c = 100 µg m–3
    3. dimensionless crosswind integrated concentration if cy = 1 mg m–2

    A20.(§) For a convective mixed layer, plot dimensionless plume centerline height with dimensionless downwind distance, for dimensionless source heights of:

    a. 0 b. 0.01 c. 0.02 d. 0.03 e. 0.04
    f. 0.05 g. 0.06 h. 0.07 i. 0.08 j. 0.09
    k. 0.1 m. 0.12 n. 0.15 o. 0.2 p. 0.22

    A21.(§) For the previous problem, plot isopleths of dimensionless crosswind integrated concentration, similar to Fig. 19.8, for convective mixed layers.

    A22. Source emissions of 300 g s–1 of SO2 occur at height 200 m. The environment is statically unstable, with a Deardorff convective velocity of 1 m s–1, and a mean wind speed of 5 m s–1.

    Find the concentration at the ground at distances 1, 2, 3, and 4 km downwind from the source, directly beneath the plume centerline. Assume the mixed layer depth (km) is:

    a. 0.5 b. 0.75 c. 1.0 d. 1.25 e. 1.5 f. 1.75
    g. 2.0 h. 2.5 i. 3.0 j. 3.5 k. 4.0 m. 5.0

    (Hint: Interpolate between figures if needed, or derive your own figures.)

    19.9.3. Evaluate & Analyze

    E1. Compare the two equations for variance: (19.5) and (19.9). Why is the one weighted by pollution concentration, and the other not?

    E2. To help understand complicated figures such as Fig. 19.3, it helps to separate out the various parts. Using the info from that figure, produce a separate sketch of the following on a background grid of B and S values:

    1. TKE (arbitrary relative intensity)
    2. Rf
    3. Flow type
    4. Static stability
    5. Pasquill-Gifford turbulence type
    6. Dispersion isotropy (plume cross section)
    7. Suggest why these different variables are related to each other.

    E3. Fig. 19.3 shows how dispersion isotropy can change as the relative magnitudes of the shear and buoyancy TKE production terms change. Also, the total amount of spread increases as the TKE intensity increases. Discuss how the shape and spread of smoke plumes vary in different parts of that figure, and sketch what the result would look like to a viewer on the ground.

    E4. Eq. (19.8) gives the center of mass (i.e., plume centerline height) in the vertical direction. Create a similar equation for plume center of mass in the horizontal, using a cylindrical coordinate system centered on the emission point.

    In eq. (19.10) use Q1 = 100 g m–1 and \(\ \bar{z}\) = 0. Plot on graph paper the Gaussian curve using σz (m) =

    a. 100 b. 200 c. 300 d. 400

    Compare the areas under each curve, and discuss the significance of the result.

    E6. Why does a “nominal” plume edge need to be defined? Why cannot the Gaussian distribution be used, with the definition that plume edge happens where the concentration becomes zero. Discuss, and support your arguments with results from the Gaussian distribution equation.

    E7. The Lagrangian time scale is different for different size eddies. In nature, there is a superposition of turbulent eddies acting simultaneously. Describe the dispersion of a smoke plume under the influence of such a spectrum of turbulent eddies.

    E8. While Taylor’s statistical theory equations give plume spread as a function of downwind distance, x, these equations are also complex functions of the Lagrangian time scale tL. For a fixed value of downwind distance, plot curves of the variation of plume spread (eq. 19.13) as a function of tL. Discuss the meaning of the result.

    E9. a. Derive eqs. (19.14) and (19.15) for near-source and far-source dispersion from Taylor’s statistical theory equations (19.13).

    b. Why do the near and far source dispersion equations appear as straight lines in a log-log graph (see the Sample Application near eq. (19.15)?

    E10. Plot the following sounding on the boundarylayer θ – z thermo diagram from the Atmospheric Stability chapter. Determine the static stability vs. height. Determine boundary-layer structure, including location and thickness of components of the boundary layer (surface layer, stable BL or convective mixed layer, capping inversion or entrainment zone, free atmosphere). Speculate whether it is daytime or nighttime, and whether it is winter or summer. For daytime situations, calculate the mixed-layer depth. This depth controls pollution concentration (shallow depths are associated with periods of high pollutant concentration called air-pollution episodes, and during calm winds to air stagnation events). [Hint: Review how to nonlocally determine the static stability, as given in the ABL and Stability chapters.]

    z (m) a. T (°C) b. T (°C) c. T (°C) d. T (°C)
    2500 –11 8 –5 5
    2000 –10 10 –5 0
    1700 –8 8 –5 3
    1500 –10 10 0 5
    1000 –5 15 0 10
    500 0 18 5 15
    100 4 18 9 20
    0 7 15 10 25

    E11. For the ambient sounding of the previous exercise, assume that a smoke stack of height 100 m emits effluent of temperature 6°C with water-vapor mixing ratio 3 g kg–1. (Hint, assume the smoke is an air parcel, and use a thermo diagram.)

    1. How high would the plume rise, assuming no dilution with the environment?
    2. Would steam condense in the plume?

    E12. For plume rise in statically neutral conditions, write a simplified version of the plume-rise equation (19.16) for the special case of:

    1. momentum only
    2. buoyancy only

    Also, what are the limitations and range of applicability of the full equation and the simplified equations?

    E13. For plume rise in statically stable conditions, the amount of rise depends on the Brunt-Väisälä frequency. As the static stability becomes weaker, the Brunt-Väisälä frequency changes, and so changes the plume centerline height. In the limit of extremely weak static stability, compare this plume rise equation with the plume rise equation for statically neutral conditions. Also, discuss the limitations of each of the equations.

    E14. In eq. (19.20), the “reflected” part of the Gaussian concentration equation was created by pretending that there is an imaginary source of emissions an equal distance underground as the true source is above ground. Otherwise, the real and imaginary sources are at the same horizontal location and have the same emission rate.

    In eq. (19.20), identify which term is the “reflection” term, and show why it works as if there were emissions from below ground.

    E15. In the Sample Application in the Gaussian Concentration Distribution subsection, the concentration footprints at ground level have a maximum value neither right at the stack, nor do concentrations monotonically increase with increasing distances from the stack. Why? Also, why are the two figures in that Sample Application so different?

    E16. Show that eq. (19.20) reduces to eq. (19.21) for receptors at the ground.

    E17. For Gaussian concentration eq. (19.21), how does concentration vary with:

    a. σy b. σz c. M

    E18. Give a physical interpretation of crosswind integrated concentration, using a different approach than was used in Fig. 19.6.

    E19. For plume rise and pollution concentration in a statically unstable boundary layer, what is the reason for, or advantage of, using dimensionless variables?

    E20. If the Deardorff velocity increases, how does the dispersion of pollutants in an unstable boundary layer change?

    E21. In Fig. 19.8, at large distances downwind from the source, all of the figures show the dimensionless concentration approaching a value of 1.0. Why does it approach that value, and what is the significance or justification for such behavior?

    19.9.4. Synthesize

    S1. Suppose that there was not a diurnal cycle, but that the atmospheric temperature profile was steady, and equal to the standard atmosphere. How would local and global dispersion of pollutants from tall smoke stacks be different, if at all?

    S2. In the present atmosphere, larger-size turbulent eddies often have more energy than smaller size one. What if the energy distribution were reversed, with the vigor of mixing increasing as eddy sizes decrease. How would that change local dispersion, if at all?

    S3. What if tracers were not passive, but had a special magnetic attraction only to each other. Describe how dispersion would change, if at all.

    S4. What if a plume that is rising in a statically neutral environment has buoyancy from both the initial temperature of the effluent out of the top of the stack, and also from additional heat gained while it was dispersing.

    A real example was the black smoke plumes from the oil well fires during the Gulf War. Sunlight was strongly absorbed by the black soot and unburned petroleum in the smoke, causing solar warming of the black smoke plume.

    Describe any resulting changes to plume rise.

    S5. Suppose that smoke stacks produced smoke rings, instead of smoke plumes. How would dispersion be different, if at all?

    S6. When pollutants are removed from exhaust gas before the gas is emitted from the top of a smoke stack, those pollutants don’t magically disappear. Instead, they are converted into water pollution (to be dumped into a stream or ocean), or solid waste (to be buried in a dump or landfill). Which is better? Why?

    S7. Propose methods whereby life on Earth could produce zero pollution. Defend your proposals.

    S8. What if the same emission rate of pollutions occurs on a fair-weather day with light winds, and an overcast rainy day with stronger winds. Compare the dispersion and pollution concentrations at the surface for those situations. Which leads to the least concentration at the surface, locally? Which is better globally?

    S9. Suppose that all atmospheric turbulence was extremely anisotropic, such that there was zero dispersion in the vertical , but normal dispersion in the horizontal.

    1. How would that affect pollution concentrations at the surface, for emissions from tall smoke stacks?
    2. How would it affect climate, if at all?

    S10. What if ambient wind speed was exactly zero. Discuss the behavior of emission plumes, and how the resulting plume rise and concentration equations would need to be modified.

    S11. What if pollutants that were emitted into the atmosphere were never lost or removed from the atmosphere. Discuss how the weather and climate would be different, if at all?

    S12. If there were no pollutants in the atmosphere (and hence no cloud and ice nuclei), discuss how the weather and climate would be different, if at all.

    S13. Divide the current global pollutant emissions by the global population, to get the net emissions per person. Given the present rate of population increase, discuss how pollution emissions will change over the next century, and how it will affect the quality of life on Earth, if at all.

    This page titled 19.9. Homework Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Roland Stull via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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