WORKBOOK OF ATMOSPHERIC DISPERSION ESTIMATES

D. BRUCE TURNER sources Field Research Office, al Science Services Administration

Env

ENVIRONMENTAL PROTECTION AGENCY Office of Air Programs Research Triangle Park, North Carolina Revised 1970

The A P s e r i e s of reports is issued by the Office of Air Programs Environmental Protection Agency, to report the results of scientific and engineering studies, and information of general interest in the field of a i r pollution. Information reported in this s e r i e s includes coverage of Air P r o g r a m intramural activities and of cooperative studies conducted in conjunction with state and local agencies, r e s e a r c h institutes, and industrial organizations. Copies of AP reports a r e available f r e e of charge to Federal employees, current contractors and grantees, andnonprofit organizations - a s supplies permit from the office of Technical Informationand Publications, Office of Air P r o g r a m s , Environmental Protection Agency, P. 0. Box 12055, Research Triangle P a r k , North Carolina 27709. Other requesters may purchase copies f r o m the Superintendent of Documents, Washington, D. C. 20402.

-

4th printing July 197 1

Office of Air P r o g r a m s Publication No. AP-26

For sale by the Superintendent of Documents, U.S. Oovemment Prlnting Offioe. Washington, D.C. 20402- Pdoe $1.00 Stock Number 5503-W16

PREFACE
This workbook presents some computational techniques currently used by scientists working with atmospheric dispersion problems. Because the basic working equations are general, their application to specific problems usually requires special care and judgment; such considerations are illustrated by 26 example problems. This workbook is intended as an aid to meteorologists and air pollution scientists who are required to estimate atmospheric concentrations of contaminants from various types of sources. It is not intended as a complete do-it-yourself manual for atmospheric dispersion estimates; all of the numerous complications that arise in making best estimates of dispersion cannot be so easily resolved. Awareness of the possible complexities can enable the user to appreciate the validity of his "first approximations" and to realize when the services of a professional air pollution meteorologist are required.

ACKNOWLEDGMENTS
The author wishes to express his appreciation to Robert A. McCormick, Paul A. Humphrey, and other members of the Field Research Office for their helpful discussions and review; to Jean J. Schueneman, Chief, Criteria and Standards Development, National Center for Air Pollution Control, who suggested this workbook; to Phyllis Polland and Frank Schiermeier, who checked the problem solutions; to Ruth Umfleet and Edna Beasley for their aid; and to the National Center for Air Pollution Control, Public Health Service, and Air Resources Laboratory, Environmental Science Services Administ

CONTENTS
ABSTRACT ........................................................................................................................ vii
........................................... Chapter 1. INTRODUCTION ...........................................

1 3

Chapter 2 BACKGROUND

.

Chapter 3. ESTIMATES OF ATMOSPHERIC DISPERSION 5 Coordinate System ............................................................................. 5 . . Diffusion Equations 5 Effects of Stability ......................................... 6 ......................................... Estimation of Vertical and Horizontal Dispersion .............................. 7 Evaluation of Wind Speed ................................................................ 7 Plots of Concentrations against Distance ........................................ 7 Accuracy of Estimates ........................................................................... 7 Graphs for Estimates of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 ........................... Plotting Ground-Level Concentration Isopleths ................ 10 ................ . . Areas Within Isopleths .......................................................................... 17 Calculation of Maximum Ground-Level Concentrations .................... 17 Review of Assumptions ......................................................................... 17 Chapter 4. EFFECTIVE HEIGHT OF EMISSION .......................................... 31 General Considerations 31 Effective Height of Emission and Maximum Concentration ....... 31 ....... Estimates of Required Stack Heights . . 31 Effect of Evaporative Cooling .............................................................. 32 Effect of Aerodynamic Downwash ........... 32 Chapter 5. SPECIAL TOPICS ...................................................................................... 35 Concentrations in an Inversion Break-up Fumigation ........................ 35 Plume Trapping .................................................................................... 36 Concentrations at Ground Level Compared to Concentrations a t the Level of Effective Stack Height from Elevated Con- . tinuous Sources 36 Total Dosage from a Finite Release ...................................................... 37 Crosswind-Integrated Conbentration . 37 Estimation of Concentrations for Sampling Times Longer than a Few Minutes ...................................................................... . 37 Estimation of Seasonal or Annual Average Concentrations a t a Receptor from a Single Pollutant Source .................................... 38 Meteorological Conditions Associated with Maximum Ground-Level Concentrations ...................:............................................38 Concentrations a t a Receptor Point from Several Sources ................ 39 Area Sources ............................................................................................ 39 Topography 40 Line Sources ............................................................................................40 Instantaneous Sources ............................................................................ 41 Chapter 6. RELATION TO OTHER DIFFUSION EQUATIONS ........................ 43 Chapter 7. EXAMPLE PROBLEMS ........................................................................ 45 Appendices: .......................................................................................................................... 57
1 - Abbreviations and Symbols ............................................................... 59 2 - Characteristics of the Gaussian Distribution ....................................61 3 - Solutions to Exponentials .................................................................. 65 4 - Constants, Conversion Eguations, Conversion Tables ......... 69 .........

. .

ABSTRACT
This workbook presents methods of practical application of the binormal continuous plume dispersion model to estimate concentrations of air pollutants. Estimates of dispersion are those of Pasquill as restated by Gifford. Emphasis is on the estimation of concentrations from contiiuous sources for sampling times up to 1hour. Some of the topics discussed are determination of effective height of emission, extension of concentration estimates to longer sampling intervals, inversion break-up fumigation concentrations, and concentrations from area, line, and multiple sources. Twenty-six example problems and their solutions are given. Some graphical aids to computation are included.

vii

Chapter 1

-INTRODUCTION temperature structure. When temperaturedecreases with height at a rate higher than 5.4"F per 1000 ft (1°C per 100 meters), the atmosphere is in unstable equilibrium and vertical motions are enhanced. When temperature decreases a t a lower ,te inC~ases with height (inversion), vertical motions are damped or reduced. Examples of typvariations in temperature and And speed with height for daytime and nighttime conditions are illustrated in Figure 1-1.

During recent years methods of estimating atmospheric dispersion have undergone considerable revision, primarily due to results of experimental measurements. In most dispersion problems the relevant atmospheric layer is that nearest the ground, varying in thickness from several hundred in both to a few thousand meters' thermal and mechanical turbulence and in wind velocity are greatestin the layer in contact with the surface. Turbulence induced by buoyancy forces in the atmosphere is closely related to the vertical
600

-

500 -

.. .

. A

400 -

a

E

< I
E

300 -

I

I

I

J

-

1

0

1

2

3

4

5

6

7

WIND SPEED, m/soc

speed with height (after Smith, 1963).
The transfer of momentum upward or downward in the atmosphere is also related to stability; when the atmosphere is unstable, usually in the daytime, upward motions transfer the momentum "deficiency" due to eddy friction losses near the earthys surface through a relatively deep layer, causing the wind speed to increase more slowly with height than at night (except in the lowest few meters). In addition to thermal turbulence, roughness elements on the ground engender mechanical turbulence, which affects both the dispersion of material in the atmosphere and the wind prose (variation of wind with height). Examples of these effects on the resulting wind profile are shown in Figure 1-2. As wind speed increases, the effluent from a continuous source is introduced into a greater volume of air per unit time interval. In addition to this dilution by wind speed, the spreading of the material (normal to the mean direction of transPO^) by turbulence is a major factor in the dispersion process. The procedures presented here to estimate atmospheric dispersion are applicable when mean wind speed and direction can be determined, but measurements of turbulence, such as the standard deviation of wind direction fluctuations, are not available. If such measurements are at hand, techniques such as those outlined by Pasquill (1961) are likely to give more accurate results. The diffusion param-

-

eters presented here are most applicable to groundlevel or low-level releases (from the surface to about 20 meters), although they are commonly applied at higher elevations without fl experimental validaul tion. It is assumed that stability is the same throughout the diilusing layer, and no turbulent transfer occurs through layers of dissimilar stability characteristics. Because mean values for wind directions and speeds are required, neither the variation of wind speed nor the variation of wind direction with height in the mixing layer are taken into account. This usually is not a problem in neutral or unstable (e.g., daytime) situations, but can cause over-estimations of downwind concentrations in stable conditions.
600

REFERENCES Davenport, A. G., 1963: The relationship of wind structure to wind loading. Presented a t Int. Conf. on The Wind Effects on Buildings and Structures, 26-28 June 63, Natl. Physical Laboratow, Teddington, Middlesex, Em. Pasquill, F., 1961: The estimat&n of the dispersion of wind borne material. Meteorol. Mag. 90, 1063, 33-49. Smith, M. E., 1963: The use and misuse of the atmosphere, 15 pp., Brookhaven Lecture Series, No. 24, 13 Feb 63, BNL 784 (T-298) Brookhaven National Laboratory.
SUBURBS
1

1

URBAN A R E A

L E V E L COUNTRY

500

,400 rn +

, .

-

rn

E

300

2
"A

I

i

200

100

0

Figure 1-2. Examples of variation of wind with height over different size roughness elements (ngures are percentages of gradient wind); (from Davenport, 1963).

. A

d .

2

ATMOSPHWIC DISPERSION ESTIMATES

Chapter 2

-BACKGROUND than those suggested in this workbook, the parameter values can be used with the equations given here. REFERENCES Bosanquet, C. H., and J. L. Pearson, 1936: The spread of smoke and gases from chimneys. Trans. Faraday Soc., 32, 1249-1263. Cramer, H. E., 1957: A practical method for estimating the dispersion of atmospheric contaminants. Proc. 1st Natl. Conf. on Appl. Meteorol. Amer. Meterol. Soc. Cramer, H. E., F. A. Record, and H. C. Vaughan, 1958: The study of the diffusion of gases or aerosols in the lower atmosphere. Final Report Contract AF 19(604)-1058 Mass. Inst. of Tech., Dept. of Meteorol. Cramer, H. E., 1959a: A brief survey of the meteorological aspects of atmospheric pollution. Bull. Amer. Meteorol. Soc., 40, 4, 165-171. Cramer, H. E., 1959b: Engineering estimates of atmospheric dispersal capacity. Amer. Ind. Hyg. Assoc. J., 20, 3, 183-189. Gifford, F. A., 1961: Uses of routine meteorological observations for estimating atmospheric dispkrsion. Nuclear Safety, 2, 4, 47-51. Hay, J. S., and F. Pasquill, 1957: Diffusion from a fixed source a t a height of a few hundred feet in the atmosphere. J. Fluid Mech., 2, 299-310. Hay, J. S., and F. Pasquill, 1959: Diffusion from a continuous source in relation to the spectrum and scale of turbulence. pp 345-365 in Atmospheric Diffusion and Air Pollution, edited by F. N. Frenkiel and P. A. Sheppard, Advances in Geophysics, 6, New York, Academic Press, 471 pp. Pasquill, F., 1961: The estimation of the dispersion of windborne material. Meteorol. Mag., 90,1063, 33-49. G., Sutton, 0. 1932: A theory of eddy difIusion in the atmosphere. Roc. Roy. Soc., A, 135, 143165.

For a number of years estimates of concentrations were calculated either from the equations of Sutton (1932) with the atmospheric dispersion parameters C,, C,, and n, or from the equations of Bosanquet (1936) with the dispersion parameters D and - ~ ~ - a.

-~ -

Hay and Pasquill (1957) have presented experimental evidence that the vertical distribution of spreading particles from an elevated point is related to the standard deviation of the wind eleva, tion angle, u, a t the point of release. Cramer (1957) derived a diffusion equation incorporating standard deviations of Gaussian distributions: U, for the distribution of material in the plume across wind , in the horizontal, and 0 for the vertical distribution of material in the plume. (See Appendix 2 for properties of Gaussian distributions.) These statistics were related to the standard deviations of azimuth angle. uA. and elevation angle, uR. calculated from , -- . . wind measurements made with a bi-directional wind vane (bivane). Values for diffusion parameters based on field diffusion tests were suggested by Cramer, et al. (1958) (and also in Cramer 1959a and 1959b). Hay and Pasquill (1959) also presented a method for deriving the spread of pollutants from records of wind fluctuation. Pasquill (1961) has further proposed a method for estimating diffusion when such detailed wind data are not available. This method expresses the height and angular spread of a diffusing plume in terms of more commonly observed weather parameters. Suggested curves of height and angular spread as a function of distance downwind were given for several "stability" classes. Gifford (1961) converted Pasquill's values of angular spread and height into standard deviations of plume concentration distribution, u, and us. Pasquill's method, with Gifford's conversion incorporated, is used in this workbook (see Chapter 3) for diffusion estimates.

-

Advantages of this system are that (1) only two dispersion parameters are required and (2) results of most diffusion experiments are now being reported in terms of the standard deviations of plume spread. More field dispersion experiments are being conducted and wiU be conducted under conditions of varying surface roughness and atmospheric stability. If the dispersion parameters from a specific experiment are considered to be more representative

Background

Chapter 3 -ESTIMATES

OF ATMOSPHERIC DISPERSION becomes essentially level, and is the sum of the physical stack height, h, and the plume rise, AH. The following assumptions are made: the plume spread has a Gaussian distribution (see Appendix 2) in both the horizontal and vertical planes, with standard deviations of plume concentration distribution in the horizontal and vertical of ar and an, respectively; the mean wind speed affecting the plume is u; the uniform emission rate of pollutants is Q; and total reflection of the plume takes place a t the earth's surface, i.e., there is no deposition or reaction a t the surface (see problem 9).

This chapter outlines the basic procedures to be used in making dispersion estimates as suggested by Pasquill (1961) and modified by Gifford (1961).

COORDINATE SYSTEM i In the system considered here the origin is a t ground level a t or beneath the point of emission, with the x-axis extending horizontally in the &rection of the mean wind. The y-axis is in the horizontal plane perpendicular to the x-axis, and the z-axis extends vertically. The plume travels along or parallel to the x-axis. Figure 3-1 illustrates the coordinate system.
DIFFUSION EQUATIONS

x

The concentration, 2, of gas or aerosols (particles less than about 20 microns diameter) a t x,y,z from a continuous source with an effective emission height, H, is given by equation 3.1. The notation used to depict this concentration is x (x,y,z;H). H is the height of the plume centerline when i t

I [-+(+)'I exp Q (x,y,z;H) = 2r a,. az u e . x*

[- +(5) ]
1 -T

+exp

*Note: exp a l b = e-alb where e is the base of natural logarithms and is approximately equal to 2.7183.

tz

Estimates

5

Any consistent set of units may be used. The most common is:

x (g m-9

or, for radioactivity (curies m-9 Q (g sec-') or (curies sec-') u (m sec-l) UY, HAY, and z (m) US, This equation is the same as equation (8.35) p. 293 of Sutton (1953) when U'S are substituted for Sutton's parameters through equations like (8.27) p. 286. For evaluations of the exponentids found in Eq. (3.1) and those that follow, see Appendix 3. x is a mean over the same time interval as the time interval for which the U'S and u are representative. The values of both U~ and U, are evaluated in terms of the downwind distance, x. Eq. (3.1) is valid where diffusion in the direction of the plume travel can be neglected, that is, no diffusion in the x direction. This may be assumed if the release is continuous or if the duration of release is equal to or greater than the travel time (x/u) from the source to the location of interest. For concentrations calculated a t ground level, i.e., z = 0, (see problem 3) the equation simplifies to:

sented, and the effect of distance from the source is considered in the graphs determining the parameter values. Values for and uz are estimated from the stability of the atmosphere, which is in turn estimated from the wind speed a t a height of about 10 meters and, during the day, the incoming solar radiation or, during the night, the cloud cover (Pasquill, 1961). Stability categories (in six classes) are given in Table 3-1. Class A is the most unstable, class F the most stable class considered here. Night refers to the period from 1hour before sunset to 1 hour after sunrise. Note that the neutral class, D, can be assumed for overcast conditions during day or night, regardless of wind speed.
Table 3-1
Surface Wind Speed (at m sec-1

Y

b

KEY TO STABILITY CATEGORIES
Day Night Thinly Overcast or '318 Sliaht ' 4 1 8 Low C~oud Cloud

Incoming Solar Radiation Strona Moderate .

<

2 2-3

3-5
5-6 > 6

A A-B B C C

A-B B B-C C-D D

B C C D D

E

F

D
D D

E
D D

The neutral class, D, should be assumed for overcast conditions during day or night.

Where the concentration is to be calculated along the centerline of the plume (y = O), (see problem 2) further simplification results:

For a ground-level source with no effective plume rise (H = O), (see problem 1):

EFFECTS OF STABILITY
The values of U, and UB vary with the turbulent structure of the atmosphere, height above the surface, surface roughness, sampling time over which the concentration is to be estimated. wind sveed, and distance from the source. For the parameter values given here, the sampling time is assumed to be about 10 minutes, the height to be the lowest several hundred meters of the atmosphere, and the surface to be relatively open country. The turbulent structure of the atmosphere and wind speed are considered in the stability classes pre-

"Strong" incoming solar radiation corresponds to a solar altitude greater than 60" with clear skies; "slicht" insolation corres~onds a solar altitude to fro; 15" to 35" with clear skies. Table 170, Solar Altitude and Azimuth, in the Smithsonian Meteorological Tables (List, 1951) can be used in determining the solar altitude. Cloudiness will decrease incoming solar radiation and should be considered along with solar altitude in determining solar radiation. Incoming radiation that would be strong with clear skies can he expected to be reduced to moderate with broken (5/s to 7/8 cloud cover) middle clouds and to slight with broken low clouds. An objective system of classifying stability from hourlv meteoroloeical observations based on the above method has been suggested (Turner, 1961). These methods will give representative indications of stability over open country or rural areas, but are less reliable for urban areas. This difference is due primarily to the influence of the city's lareer surface rouehness and heat island effects upon the stabilityVregime over urban areas. The greatest difference occurs on calm clear nights; on such nights conditions over rural areas are very stable, but over urban areas they are slightly unstable or near neutral to a height several times the average building height, with a stable layer above (Duckworth and Sandberg, 1954; DeMarrais, 1961).
ATMOSPHERIC DISPERSION ESTIMATES

i

: :

.

1
,
,

i
I

1

*

Some preliminary results of a dispersion experiment in St. Louis (Pooler, 1965) showed that the dispersion over the city during the daytime behaved somewhat like types B and C; for one night experiment uyvaried with distance between types D and E. ESTIMATION OF VERTICAL AND HORIZONTAL DISPERSION Having determined the stability class from Table 3.1, one can evaluate the estimates of uy and u as a function of downwind distance from the , source, x, using Figures 3-2 and 3-3. These values of q and 0 are representative for a sampling time , of about 10 minutes. For estimation of concentrations for longer time periods see Chapter 5. Figures 3-2 and 3-3 apply strictly only to open level country and probably underestimate the plume dispersion potential from low-level sources in built-up areas. Although the vertical spread may be less than the values for class F with very light winds on a clear night, quantitative estimates of concentrations are nearly impossible for this condition. With very light windson a clear night for ground-level sources free of topographic influences, frequent shifts in wind Arection usually occur which serve to spread the plume horizontally. For elevated sources under these extremely stable situations, significant concentrations usually do not reach ground level until the stability changes. A stable layer existing above an unstable layer will have the effect of restricting the vertical diffusion. The dispersion computation can be modified for this situation by considering the height of the base of the stable layer, L. At a height 2.15 U~ above the plume centerline the concentration is onetenth the plume centerline concentration a t the same distance. When one-tenth the plume centerline concentration extends to the stable layer, a t height L, it is reasonable to assume that the distribution starts being affected by the "lid." The following method is suggested to take care of this situation. Allow u to increase with distance to a value of , L/2.15 or 0.47 L. At this distance x,, the plume is assumed to have a Gaussian distribution in the vertical. Assume that by the time the plume travels twice this far, 2 XL, plume has become uniformly the distrihuted between the earth's surface and the height L, i.e., concentration does not vary with height (see Figure 3-4). For the distances greater than 2 x,, the concentration for any height between the ground and L can be calculated from:

(see problem 6). Note that Eq. (3.5) assumes normal or Gaussian distribution of the plume only in the horizontal plane. The same result can he obtained from the following equation where uzL is an effective dispersion parameter because v'Z L = 2.5066 L and 0.8 WL= 2.51 L.

x (x,Y,z;H) = " UY uzL u

exp

-+(+)'I

for any z from 0 to L for x >2,1,; XL is where uz = 0.47 L The value of = 0.8 L EVALUATION OF WIND SPEED For the wind speed, u, a mean through the vertical extent of the plume should be used. This would be from the height H - 2 through H 2 . Of course, if 2 u is greater than H then the , wind can be averaged from the ground to H 2 US. However, the "surface wind" value may be all that is available. The surface wind is most applicable to surface or low-level emissions, especially under stable conditions.

+

+

PLOTS OF CONCENTRATIONS AGAINST DISTANCE i To gain maximum insight into a diffusion problem i t is often desirable to plot centerline concentrations against distance downwind. A convenient procedure is to determine the ground-level centerline concentrations for a number of downwind distances and plot these values on log-log graph paper. By connecting the points, one may estimate concentrations for intermediate downwind distances (see problem 6). ACCURACY OF ESTIMATES Because of a multitude of scientific and technical limitations the diffusion computation method presented in this manualmay provide best estimates but not infallible predictions. In the unstable and stable cases, severalfold errors in estimate of US can occur for the longer travel distances. In some cases the uz may be expected to be correct within a factor of 2, however. These are: (1) all stabilities for distance of travel out to a few hundred meters; (2) neutral to moderately unstable conditions for distances out to a few kilometers; and (3) unstable conditions in the lower 1000 meters of the atmosphere with a marked inversion above for distances out to 10 km or more. Uncertainties in the estimates of U are in general less than those of us. , The ground-level centerline concentrations for these

for any z from 0 to L for x 2 XL;XL is where uz = 0.47 L '
Estimates

1
DISTANCE D O W N W I N D , k m

10

Figure 3-2. Horizontal dispersion coefficient as a function of downwind distance from the source.
I

ATMOSPHERIC DISPERSION ESTIMATES

DISTANCE DOWNWIND, km

Figure 3-3. Vertical dispersion coefficient as a function of downwind distance from the source.
Estimates
399-901 0

- 89 - 2

\j

i

, '. .
:

Figure 3-4. Variations in concentration in the vertical beneath a more stable layer. three cases (where wZ can be expected to be within a factor of 2) should be correct within a factor of 3, including errors in G and u. The relative confidence , . in the w's (in decreasing order) is indicated by the heavy lines and dashed lines in Figures 3-2 and 3-3. Estimates of H, the effective height of theplume, may be in error because of uncertainties in the estimation of AH, the plume rise. Also, for problems that require estimates of concentration a t a specific point, the difficulty of determining the mean wind over a given time interval and consequently the location of the x-axis can cause considerable uncertainty. GRAPHS FOR ESTIMATES OF DIFFUSION PLOTTING GROUND-LEVEL CONCENTRATION ISOPLETHS Often one wishes to determine the locations where concentrations equal or exceed a given magnitude. First, the axial position of the plume must be determined by the mean wind direction. For plotting isopleths of ground-level concentrations, the relationship between ground-level centerline concentrations and ground-level off-axis concentrations can be used: (3.7) The y coordinate of a particular isopleth from the x-axis can be determined a t each downwind distance, x. Suppose that one wishes to know the g m-8 isopleth a t an X off-axis distance to the of 600 m, under stability type B, where the groundlevel centerline concentration a t this distance is 2.9 X g m3.

. ... .i .

*

*

T~ avoid repetitious computations, Figure 3-5 ( A through F) gives relative ground-level concentrations times wind speed (x u / ~ ) against downwind distances for various effectiveheights of emission and limits to the vertical mixing for each stability class (1 figure for each stability). Computations were made from Eq. (3.3), (3.4), and (3.5). Estimates of actual concentrations may be determined by multiplying ordinate values by Q/u.
10

2.9 x

lo-" lo-"

[ - %(%)'1
= 0.345

x (x,y,O;H)
=

x

(x,O,O;H)

-

ATMOSPHERIC DISPERSION ESTIMATES

01 .

I
DISTANCE.km

10

100

Figure 3-5A. xu/Q with distance for various heights of emission (H) and limits to vertical dispersion (L), A stability.
Estimates
11

:

la.

DISTANCE, k m

Figure 3-5B. xu/Q with distance for various heights of emission (H) and limits to vertical dispersion (L), B stability.
12
ATMOSPHERIC DISPERSION ESTIMATES

0.1

I
DISTANCE, k m

I0

100

!

Figure 3-5C. xu/Q with distance for various heights of emission (HI and limits to vertical dispersion (L), C stability.
Estimates
13

i

10-7
0.1
1
10

100

DISTANCE, kn

Figure 3-50, xu/Q with distance for various heights of emiss~on(I+ and limits to vertical dispersion
14

(L), D stability.

ATMOSPHERIC DISPERSION ESTIMATES

. .

DISTANCE, L m

Figure 3-5E. xu/Q with distance for various heights of emission (H) and limits to vertical dispersion (L), E stability.
Estimates

15

0.1

I DISTANCE. k m

10

100
I

Figure 3-5F. xu/Q with distance for various heights of emission (H) and limits to vertical dispersion (L), F stability.
16 ATMOSPHERIC DISPERSION ESTIMATES

I

!

From Table A-1 (Appendix 3) when exp

From Figure 3-2, for stability B and x = 600 m, a, 92. Therefore y = (1.46) (92) = 134 meters. This is the distance of the isopleth from the x-axis a t a downwind distance of 600 meteis.
=

height, the product u u, UZ will not change appreciably. The greater the effective height, the more likely i t is that the stability may not be the same from the ground to this height. With the longer travel distances such as the points of maximum concentrations for stable conditions (Types E or F), the stability may change before the plume travels the entire distance. REVIEW OF ASSUMPTIONS The preceding has been based on these assumptions, which should be clearly understood:

This can also be determined from:

The position corresponding to the downwind distance and off-axis distance can then be plotted. After a number of points have been plotted, the concentration isopleth may be drawn (see problems 8 and 26). Figures 3-6 and 3-7 give ground-level isopleths of Xu/Q for various stabilities for sources a t H = 0 and H = 100 meters. For example, to g m-3 isopleth resulting from a locate the ground-level source of 20 g sec-' under B stability conditions with wind speed 2 m sec-I, one must first determine the corresponding value of xu/Q since this is the quantity graphed in Figure 3-6. xu/& = x 2/20 = lo-% Therefore the xu/& isopleth m-2 correin Figure 3-6B having a value of g m-3. sponds to a x isopleth with a value of AREAS WITHIN ISOPLETHS Figure 3-8 gives areas within isopleths of groundlevel concentration in terms of x u/Q for a groundlevel source for various stability categories (Gifford, 1962; Hilsmeier and Giff ord, 1962). For the example just given, the area of the g m-S isopleth m-2 x u/Q isopleth) is about 5 x lo4 meter2. CALCULATION OF MAXIMUM GROUND-LEVEL CONCENTRATIONS Figure 3-9 gives the distance to the point of maximum concentration, x,,,, and the relative maximum concentration, x u/Q,,,, as a function of effective height of emission and stability class (Martin, 1965). This figure was prepared from graphs of concentration versus distance, as in Figure 3-5. The maximum concentration can he determined by iinding x u/Q as a function of effective emission height and stability and multiplying by Q U . In using Figure 3-9, the user must keep in mind that the dispersion a t higher levels may differ considerably from that determined by the ~ / s and us used here. As noted, however, since U, gener; ally decreases with height and u increases with
*"Inu denotes natural logarithms, i.e., to the base e,

(i) Continuous emission from the source or emission times equal to or greater than travel times to the downwind position under consideration, so that diffusion in the direction of transport may be neglected. (ii) The material diffused is a stable gas or aerosol (less than 20 microns diameter) which remains suspended in the air over long periods of time. (iii) The equation of continuity:

is fulfilled, i.e., none of the material emitted is removed from the plume as it moves downwind and there is complete reflection a t the ground. (iv) The mean wind direction specifies the x-axis, and a mean wind speed representative of the diffusing layer is chosen. (v) Except where specifically mentioned, the plume constituents are distributed normally in both the cross-wind and vertical directions. (vi) The u's given in Figures 3-2 and 3-3 represent time periods of about 10 minutes. REFERENCES DeMarrais, G. A,, 1961: Vertical temperature difference observed over an urban area. Bull. Amer. Meteorol. Soc., 42, 8, 548-554. Duckworth, F. S., and J. S. Sandberg, 1954: The effect of cities upon horizontal and vertical temperature gradients. Bull. Amer. Meteorol. Soc., 35, 5, 198-207. Gifford, F. A., 1961: Use of routine meteorological observations for estimating atmospheric dispersion. Nuclear Safety, 2, 4, 47-51. Gifford, F. A., 1962: The area within ground-level dosage isopleths. Nuclear Safety, 4, 2, 91-92.

Estimates

1

' I

3 l N V I S I O ONIMSSOUI
,

,
!

ATMOSPHERIC DISPERSION ESTIMATES

Estimates

ATMOSPHERIC DISPERSION ESTIMATES

Estimates

ATMOSPHERIC DISPERSION ESTIMATES

I
I

Estimates

'1 '(11 33NVISIII aNIMSSOU3

ATMOSPHERIC DISPERSION ESTIMATES

Estimates
339-$01 0 - 69

-3

Estimates

Xu -,

m-2

Figure 3-8. Area within isopleths for a ground-level source (from Hilsmeier and Gifford).
Hilsmeier, W. F., and F. A. Gifford, 1962: Graphs for estimating atmospheric diffusion. 0R0-545, Oak Ridge, Term. Atomic Energy Commission, 10 ww.
A A

of windborne material. 1063, 33-49.

Meteorol. Mag., 90,

.,

!

Pooler, F,, 1965: personal communication^ Sutton, 0. G., 1953: Micrometeorology, New York, McGraw-Hill. 333 pp. Turner, D. B., 1961: Relationships between 24hour mean air quality measurements and meteoroloeical factors in Nashville. Tennessee. J. Air 6011. Cont. Assoc., 11, 483-489. e List, R. J., 1951: Smithsonian Meteorological Tables, Sixth Revised Edition, 497-505, Washington, D. C., Smithsonian Institution, 527 pp. Martin, D. O., 1965: Personal communication. Pasquill, F., 1961: The estimation of the dispersion

,

ATMOSPHERIC DISPERSION ESTIMATES

I
I

-0

Estimates

Chapter 4GENERAL CONSIDERATIONS

EFFECTIVE HEIGHT OF EMISSION v. = stack gas exit velocity, m sec-' d = the inside stack diameter, m u = wind speed, m sec-' p = atmospheric pressure, mb T. = stack gas temperature, "K T, = air temperature, O K and 2.68 x is a constant having units of mb-I m -I. Holland (1953) suggests that a value between 1.1 and 1.2 times the AH from the equation should be used for unstable conditions; a value between 0.8 and 0.9 times the AH from the equation should be used for stable conditions. Since the plume rise from a stack occurs over some distance downwind, Eq. (4.1) should not be applied within the first few hundred meters of the stack.
EFFE~TIVEH~~~~~ OF EMISSION mD MAXImM CONCENTRATION

In most problems one must estimate the effective stack height, H, a t which the plume becomes essentially level. Rarely will this height correspond f to the physical height of the stack, h. I the plume is caught in the turbulent wake of the stack or of buildings in the vicinity of the stack, the effluent will be mixed rapidly downward toward the ground (aerodynamic downwash). If the plume is emitted free of these turbulent zones, a number of emission factors and meteorological factors influence the rise of the plume. The emission factors are: velocity of the effluent a t the top of the stack, v,; temperature of the effluent a t the top of the stack, T,; and diameter of the stack opening, d. The meteorological factors influencing plume rise are wind speed, u; temperature of the air, T,; shear of the wind speed with height, du/dz; and atmospheric stahilitv. No theorv on nlume rise takes into account all of these var:ables; even if such a theory were available, measurements of all of the parameters would seldom be available. Most of the equations that have been formulated for computing the effective height of emission are semi-empirical. For a recent review of equations for effective height of emission see Moses, Strom, and Carson (1964). Moses and Strom (1961), having compared actual and calculated plume heights by means of six plume rise equations, report "There is no one formula which is outstanding in all respects." The formulas of Davidson-Bryant (1949), Holland (1953), Bosanquet-Carey-Halton (1950), and Bosanquet (1957) all give generally satisfactory results in the test situations. The experiments conducted by Moses and Strom involved plume rise from a stack of less than 0.5 meter diameter, stack gas exit velocities less than 15 m sec-', and effluent temperature not more than 35°C higher than that of the ambient air. The equation of Holland was developed with experimental data from larger sources than those of Moses and Strom (stack diameters from 1.7 to 4.3 meters and stack temperatures from 82 to 204°C); Holland's equation is used in the solution of the problems given in this workbook. This equation frequently underestimates the effective height of emission; therefore its use often provides a slight "safety" factor. Holland's equation is:

If the effective heights of emission were the same under all atmospheric conditions, the highest ground-level concentrations from a given source would occur with the lightest winds. Generally, however, emission conditions are such that the effective stack height is an inverse function of w 4 d speed as indicated in Eq. (4.1). The maximum ground-level concentration occurs a t some intermediate wind speed, a t which a balance is reached between the dilution due to wind speed and the effect of height of emission. This critical wind speed will vary with stability. In order to determine the critical wind speed, the effective stack height as a function of wind speed should first be determined. The maximum concentration for each wind speed and stability can then be calculated from Figure 3-9 as a function of effective height of emission and stability. When the maximum concentration as a function of wind speed is plotted on log-log graph paper, curves can be drawn for each stability class; the critical wind speed corresponds to the point of highest maximum concentration on the curve (see problem 14).
ESTIMATES OF REQUIRED STACK HEIGHTS Estimates of the stack height required to produce concentrations below a given value may be made through the use of Figure 3-9 by obtaining solutions for various wind speeds. Use of this figure considers maximum concentrations a t any distance from the source. In some situations high concentrations upon the property of the emitter are of little concern, but

_I

where: AH

= the

rise of the plume above the stack, m

Effective Height

maximum concentrations beyond the property line are of the utmost importance. For first approximations it can he assumed that the maximum concentration occurs where \/2oz = H and that a t this distance the u's are related to the maximum concentration by:
(Ti

us=
7

-

Q u e X,. Z,,

=

-

0.117 Q
1.7

xmn,

(4.2)

Knowing the source strength, Q, and the concentration not to be exceeded XI.,., one can determine the necessary O, vz for a given wind speed. Figure 4-1 shows O, o, as a function of distance for the various stability classes. The value of vy on and a design distance, x, (the distance beyond which x is less than some pre-determined value), will determine a point on this graph yielding a stability class or point between classes. The o for this stability , (or point between stabilities) can then be determined from Figure 3-3. The required effective stack height for this wind speed can then be approximated by H = \/Z (see problem 15). Since Eq. (4.2) is an approximation, the resulting height should be used with Eq. (3.3) to ensure that the maximum concentration is sufficiently low. If enough is known about the proposed source to allow use of an equation for effective height of emission, the relation between AH and u can be determined. The physical stack height required a t the wind speed for which H was determined is H AH. The same procedure, starting with the determination of -, o must be used with other wind , speeds to determine the maximum required physical stack height (see problem 16).
US

EFFECT OF EVAPORATIVE COOLING

When effluent gases are washed to absorb certain constituents prior to emission, the gases are cooled and become saturated with water vapor. Upon release of the gases from the absorption tower, further cooling due to contact with cold surfaces of ductwork or stack is likely. This cooling causes condensation of water droplets in the gas stream. Upon release of the gases from the stack, the water droplets evaporate, withdrawing the latent heat of vaporization from the air and cooling the plume. The resulting negative buoyancy reduces the effective stack height (Scorer, 1959).
EFFECT OF AERODYNAMIC DOWNWASH

least twice its height and extends downwind 5 to 10 times its height. Building the stack 2.5 times the height of the highest building adjacent to the stack usually overcomes the effects of building turbulence (Hawkins and Nonhebel, 1955). Ensuring that the exit velocity of the stack gas is more than 1.5 times the wind speed will usually prevent downwash in the wake of the stack. Most of the knowledge about the turbulent wakes around stacks and buildings has been gained through wind tunnel studies ( ~ h & lock and Lesher, 1954; Strom, 1955-1956; Strom, et al, 1957; and Halitsky, 1962). By use of models of building shapes and stacks, one may determine the wind speeds required to cause downwash for various wind directions. With a wind tunnel the meteorological variables most easily accounted for are wind speed and wind direction (by rotation of the model within the tunnel). The emission factors that may be considered are the size and shape of the plant building; the shape, height, and diameter of the stack; the amount of emission; and the stackgas velocity. Through wind tunnel studies, the critical wind speeds that will cause downwash from various directions can he determined for a given set of plant factors. The average number of hours of downwash per year can then be calculated by determining the frequency of wind speeds greater than the critical speeds for each direction (Sherlock and Lesher, 1954) if climatological data representative of the site are available. Maximum downwash about a rectangular structure occurs when the direction of the wind is a t an angle of 45 degrees from the major axis of the structure; minimum downwash occurs with wind flow parallel to the major axis of the structure (Sherlock and Lesher, 1954). Halitsky (1961, 1963) has shown that the effluent from flush openings on flat roofs frequently flows in a direction opposite to that of the free atmospheric wind, owing to counter-flow along the roof in the turbulent wake above the building. In addition to the effect of aerodynamic downwqsh upon the release of air pollutants from stacks and buildings, one must also consider the effects of aerodynamic downwash when exposing meteorological instruments near or upon buildings. Where the pollution is emitted from a vent or opening on a building and is immediately influenced by the turbulent wake of the building, the pollution is rapidly distributed within this turbulent wake. To account for mixing in the turbulent wake, one may assume binormal distributions of concentrations a t the source, with horizontal and vertical standard deviations of us, and o . , The standard deviations are related to the width and height of the building, for example, letting 4.3 uY, equal the width of the building and 2.15 o., equal
ATMOSPHERIC DISPERSION ESTIMATES

4

.,
>

The influence of mechanical turbulence around a building or stack can significantly alter the effective stack height. This is especially true with high winds, when the beneficial effect of high stackgas velocity is a t a minimum and the plume is emitted nearly horizontally. The region of disturbed flow surrounds an isolated building, generally to a t

Dislonre Downwind. km

Figure 4-1. The product of ws
Effective Height

as a function of downwind distance frorti the source.

the height. Values other than 4.3 and 2.15 can be used. When these values are used 97 % of the distribution is included within these limits. Virtual distances x, and x, can be found such that a t x,, u7=v,,,andatx,, o,, = . These x's will differ with stability. Equations applicable to point sources can then be used, determining U, as a function of x, and as a function of x x,. x

Hawkins, J. E., and G. Nonhebel, 1955: Chimneys and the dispersal of smoke. J. Inst. Fuel, 28, 530-546. Holland, J. Z., 1953: A meteorological survey of the Oak Ridge area. 554-559 Atomic Energy Comm., Report ORO-99, Washington, D.C., 584 pp. Moses, H., and G. H. Strom, 1961: A comparison of observed plume rises with values obtained from well-known formulas. J. Air Poll. Cont. Assoc., 11, 10, 455-466. Moses, H., G. H. Strom, and J. E. Carson, 1964: Effects of meteorological and engineering factors on stack plume rise. Nuclear Safety, 6, 1, 1-19. Scorer, R. S., 1959: The behavior of plumes. Int. J. Air Poll., 1 , 198-220. Sherlock, R. H., and E. J. Lesher, 1954: Role of chimney design in dispersion of waste gases. Air Repair, 4, 2, 1-10. Strom, G. H., 1955-1956: Wind tunnel scale model studies of air pollution from industrial plants. Ind. Wastes, Sept. - Oct. 1955, Nov. - Dec. 1955, and Jan. - Feb. 1956. Strom, G. H., M. Hackman, and E. J. Kaplin, 1957: Atmospheric dispersal of industrial stack gases determined by concentration measurements in scale model wind tunnel experiments. J. Air Poll. Cont. Assoc., 7, 3, 198-203.

+

UB

+

REFERENCES Bosanquet, C. H., W. F. Carey, and E. M. Halton, 1950: Dust from chimney stacks. Proc. Inst. Mech. Eng., 162, 355-367. Bosanquet, C. H., 1957: The rise of a hot waste gas plume. J. Inst. Fuel, 30, 197, 322-328. Davidson, W. F., 1949: The dispersion and spreading of gases and dust from chimneys. Trans. Conf. on Ind. Wastes, 14th Ann. Meeting, Ind. Hygiene Found. Amer., 38-55. Halitsky, J., 1961: Wind tunnel model test of exhaust gas recirculation a t the NIH Clinical Center. Tech. Rep. No. 785.1, New York Univ. Halitsky, J., 1962: Diffusion of vented gas around buildings. J. Air Poll. Cont. Assoc., 12, 2,74-80. Halitsky, J., 1963: Gas diffusion near buildings, theoretical concepts and wind tunnel model experiments with prismatic building shapes. Geophysical Sciences Lab. Rep. No. 63-3. New York Univ.

I

ATMOSPHERIC DISPERSION ESTIMATES

I

Chapter 5
CONCENTRATIONS IN AN INVERSION BREAK-UP FUMIGATION

-SPECIAL TOPICS
A difficulty is encountered in estimatimg a reasonable value for the horizontal dispersion since in mixing the stable plume through a vertical depth some additional horizontal spreading occurs (see problem 12). ~f this spreading is ignored and the for stable conditions used, the probable result would be estimated concentrations higher than actual concentrations. or, using an appro-ation suggested by ~ i and ~ Hewson (1962) that the ~ l ~ edge of the plume spreads outward with an angle of IS0, the for the inversion break-up fumigation equals the for stable conditions plus one-eighth the effective height of emission. ~h~ origin of this concept be seen in Figure 5-1 and the following equation, where the edge o the plume is the point f a t which the concentration falls to 1/10 that a t the centerline (at a distance of 2.15 from the plume center). fly A surface-based inversion may be eliminated by the upward transfer of sensible heat from the ground surface when that surface is warmer than the overlying air. This situation occurs when the ground is being warmed by solar radiation or when air flows from a cold to a relatively warm surface. In either situation pollutants previously emitted above the surface into the stable layer will be mixed vertically when they are reached by the thermal eddies, and ground-level concentrations can increase. This process, called "fumigation" was described by Hewson and Gill (1944) and Hewson (1945). Equations for estimating concentrations with these conditions have been given by Holland (1953), Hewson (1955), Gifford (1960a), Bierly and Hewson (1962), and Pooler (1965). To estimate ground-level concentrations under inversion break-up fumigations, one assumes that the plume was initiallv emitted into a stable laver. ~heiefore, and uz iharacteristic of stable conditions must be selected for the particular distance of concern. An equation for the ground-level concentration when the inversion has been eliminated to a height hi is: xr (X,Y,O;H) =

WTF

=

-

2.15 ar (stable) H tan 15" 2.15 (stable) H/8

+

+

(5.4)

A Gaussian distribution in the horizontal is assumed.
,2u BOUNDARY O F

r

IJ where p =

013

1
1

exp (-0.5
~ Y E

p2) dp

1

u hr
2 ]

[ -+(*) h, -H flz (5.1)
C

1
I

4

2.15 u y t H t o n 15' 1
I

-

and m y E is discussed below. Values for the integral in brackets can be found in. most statistical tables. For example, see pages 2%276, Burington (1953). This factor accounts for the portion of the plume that is mixed dawnward.. If the inversion is eliminated up to the effective stack height, half of the plume is presumed, to. be: mixed downward, the other half remaining. in thestable air above. Eq. (5.1) can be approximated, when the fumigation concentration is near its; maximum by:

Figure 5-1. Diagram showing assumed height, hi and uy during fumigation, for use in equation (5.2).
Eq. (5.4) should not be applied near the stack, for if the inversion has been eliminated to"-aheight sufficient to include the e n t i i plume, the emission is taking place under unstable not stable conditions. Therefore, the nearest downwind distance to be considered for an estimate of fumigation concentrations must be great enough, based on the time reqvired to eliminate the inversion, that this portion of the plume was initially emitted into stable air. This distance is x = ut,, where u is the mean
35

Special Topics

wind in the stable layer and t, is the time required to eliminate the inversion from h, the physical height of the stack to hi (Eq. 5.3).

t, is dependent upon both the strength of the inversion and the rate of heating a t the surface. Pooler (1965) has derived an expression for estimating this time:

layer aloft. Bierly and Hewson (1962) have suggested the use of an equation that accounts for the multiple eddy reflections from both the ground and the stable layer:

x

(x,O,z;H)

=

2~ u

vy vz

where t,

required for the mixing layer to develop from the top of the stack to the top of the plume, sec p, =ambient air density, g m-8 c, = specific heat of air a t constant pressure, cal g-' O K' R = net rate of sensible heating of an air column by solar radiation, cal m-%sec-I
6,
=

= time

vertical potential temperature gradient,

"K m-' ST r (the adiabatic lapse rate) hi = height of base o the inversion sufficient f to be above the plume, m h = physical height of the stack, m Note that hi -h heated and is the average height o the f layer. Although R depends on season, and cloud cover and varies continuously with time, Pooler has used a value o 67 cal m-%sec-l as an average for f fumigation. Hewson (1945) also suggested a method of estimating the time required to eliminate an inversion to a height z by use o an equation of Taylor's f (1915, p. 8):

-=+

+ exp - 2
+ exp2

zVH+2NL
4

(w)

is the thickness of the layer to be

(

NL )'])(5.8)

where L is the height of the stable layer and J = 3 or 4 is sufficient to include the important reflections. A good approximation of this lengthy equation can be made by assuming no effect of the stable layer until sz = 0.47 L (see Chapter 3). It is assumed that a t this distance, XL,the stable layer begins to affect the vertical distribution so that a t the downwind distance, 2 xL, uniform vertical mixing has taken place and the following equation can be used:

where:

t = time required to eliminate the inversion to height z, sec z =height to which the inversion has been eliminated, m K =eddy difhsivity for heat, m2 sec-I For distances between XL and 2 XL the best approximation to the ground-level centerline concentration is that read from a straight l i e drawn between the concentrations for points XL and 2 XL on a log-log plot of ground-level centerline concentration as a function of distance. CONCENTRATIONS AT GROUND LEVEL COMPARED TO CONCENTRATIONS AT THE LEVEL OF EFFECTIVE STACK HEIGHT FROM ELEVATED CONTINUOUS SOURCES There are several interesting relationships between ground-level concentrations and concentrations a t the level of the plume centerline. One of ATMOSPHERIC DISPERSION ESTIMATES

Rewriting to compare with Eq. (5.5),

Hewson (1945) has suggested a value of 3 mz set-I for K. PLUME TRAPPING Plume trapping occurs when the plume is trapped between the ground surface and a stable
36

these is a t the distance of maximum concentration a t the ground. As a rough approximation the maximum ground-level concentration occurs a t the dis1 tance where u - -H. This approximation is

accurately. The estimate of this path is usually increasingly difficult with shorter release times. D, can also be given in curie sec m-3 if QT is in curies. CROSSWIND-INTEGRATED CONCENTRATION The ground-level crosswind-integrated concentration is often of interest. For a continuous elevated source this concentration is determined from Eq. (3.2) integrated with respect to y fram -" to +" (Gifford 1960a) giving: exp -- (5.11) u In diffusion experiments the ground-level crosswind-integrated concentration is often determined a t particular downwind distances from a crosswind line or arc of sampling measurements made a t this distance. When the source strength, Q, and average wind speed, u, are known, uz can be estimated indirectly even though no measurements were made in the vertical. If any of the tracer is lost through reaction or deposition, the resulting U, from such estimates will not represent the vertical dispersion (see problem 18).
Xcwr
=
&U%

"

- I / ' %

much better for unsiable conditions than for stable conditions. With this approximation, the ratio of concentration a t plume centerline to that a t the ground is:

[ (E)' ]

- [1.0 + exp- 0.5(2
-

2

\/B2]

exp - 0.5 ( \/2) '

This calculation indicates that a t the distance of maximum ground-level concentration the concentration a t plume centerline is greater by about one-third.

It is also of interest to determine the relationship between o; and H such that the concentration a t ground-level a t a given distance from the source is the same as the concentration a t plume level. This condition should occur where:

ESTIMATION OF CONCENTRATIONS FOR SAMPLING TIMES LONGER THAN A FEW MINUTES Concentrations directly downwind from a sourte decrease with sampling time mainly because of a larger uY due to increased meander of wind direction. Stewart, Gale, and Crooks (1958) reported that this decrease in concentration follows a one-fifth power law with the sampling time for sampling periods from about 3 minutes to about half an hour. Cramer (1959) indicates that this same power law applies for sampling times from 3 seconds to 10 minutes. Both of these studies were based on observations taken near the height of release. Gifford (1960b) indicates that ratios of peak to mean concentrations are much higher than those given by the above power law where observations of concentrations are made a t heights considerably different from the height of release or considerably removed from the plume axis. He also indicates that for increasing distances from an elevated source, the ratios of peak to average concentrations observed a t ground level approach unity. Singer (1961) and Singer, e t al. (1963) show that ratios of peak to mean concentrations depend also. on the stability of the atmosphere and the type of terrain that the plume is passing over. Nonhebel (1960) reports that Meade deduced a relation between calculated concentrations a t ground level and the sampling time from "a study of published data on lateral and vertical diffusion coefficients in steady winds." These relations are shown in Table 5-1.
37

The value H / u ~= 1.10 satisfies this expression, which can be written as U = 0.91 H (see problem . 10). TOTAL DOSAGE FROM A FINITE RELEASE The total dosage, which is the integration of concentration over the time of passage of a plume or puff, can be obtained from:

where DT= total dosage, g sec m-3 and Q = total release, g T The a's should be representative of the time period over which the release takes place, and care should be taken to consider the x-axis along the trajectory or path of the plume or puff travel. Large errors can easily occur if the path is not known
Special Topics

Table 5-1 VARIATION OF CALCULATED CONCENTRATION WITH SAMPLING TIME
Sampling Time Ratio of Calculated Concentration to 3-minute Concentration

3 minutes 15 minutes
1 hour

1.00 0.82 0.61 0.51 0.36

3 hours 24 hours

depending upon whether a stable layer aloft is affecting the distribution. The estimation of x for a particular direction and downwind distance can he accomplished by choosing a representative wind speed for each speed class and solving the appropriate equation (5.13 or 5.14) for all wind speed classes and stabilities. Note that a SSW wind affects a receptor to the NNE of a source. One obtains the average concentration for a given direction and distance by summing all the concentrations and weighting each one according to its frequency for the particular stability and wind speed class. I desired, a different effective f height of emission can be used for various wind speeds. The average concentration can be expressed by:

This table indicates a power relation with time: x a t - O 1 7 . Note that these estimates were based upon published dispersion coefficients rather than upon sampling results. Information in the references cited indicates that effects of sampling time are exceedingly complex. If i t is necessary to estimate concentrations from a single source for the time intervals greater than a few minutes, the best estimate apparently can he obtained from:

where x, is the desired concentration estimate for the sampling time, t.; xk is the concentration estimate for the shorter sampling time, tk, (probably about 10 minutes); and p should be between 0.17 and 0.2. Eq. (5.12) probably would be applied most appropriately to sampling times less than 2 hours (see problem 19).
ESTIMATION OF SEASONAL OR ANNUAL AVERAGE CONCENTRATIONS AT A RECEPTOR FROM A SINGLE POLLUTANT SOURCE

For a source that emits a t a constant rate from hour to hour and day to day, estimates of seasonal or annual average concentrations can be made for any distance in any direction if stability wind"rose" data are available for the period under study. A wind rose gives the frequency of occurrence for each wind direction (usually to 16 points) and wind speed class (9 classes in standard Weather Bureau use) for the period under consideration (from 1 month to 10 years). A stability wind rose gives the same type of information for each stability class.

where f (e, S, N) is the frequency during the period of interest that the wind is from the direction e , for the stability condition, S, and wind speed class N. nZSis the vertical dispersion parameter evaluated a t the distance x for thestability condition S. UN is the representative wind speed for class N. H, is the effective height o release for the wind f speed u,. Where stability wind rose information cannot be obtained, a first-order approximation may be made of seasonal or annual average concentrations by using the appropriate wind rose in the same manner, and assuming the neutral s t a b i t y class, D, only.

L ,

If the wind directions are taken to 16 points and it is assumed that the wind directions within each sector are distributed randomly over a period of a month or a season, i t can further be assumed that the effluent is uniformly distributed in the horizontal within the sector (Holland, 1953, p. 540). The appropriate equation for average concentration is then either:

METEOROLOGICAL CONDITIONS ASSOCIATI.:D WITH 3lAXllfllM GHOUND-LEVEL CONCENTRATIONS
1. For ground-level sources maximum concentrations occur with stable conditions.

ATMOSPHERIC DISPERSION ESTIMATES

2. For elevated sources maximum "instantaneo~s" concentrations occur with unstable conditions when portions of the plume that have undergone little dispersion are brought to the ground. These occur close to the point of emission (on the order of 1 to 3 stack heights). These concentrations are usually of little general interest because of their very short duration; they cannot be estimated from the material presented in this workbook. 3. For elevated sources maximum concentrations for time periods of a few minutes occur with unstable conditions; although the concentrations fluctuate considerably under these conditions, the concentrations averaged over a few minutes are still high compared to those found under other conditions. The distance of this maximum concentration occurs near the stack (from 1 to 5 stack heights downwind) and the concentration drops off rapidly downwind with increasing distance. 4. For elevated sources maximum concentrations for time periods of about half an hour can occur with fumigation conditions when an unstable layer increases vertically to mix downward a plume previously discharged within a stable layer. With small AH, the fumigation can occur close to the source but will be of relatively short duration. For large AH, the fumigation will occur some distance from the stack (perhaps 30 to 40 km), but can persist for a longer time interval. Concentrations considerably lower than those associated with fumigations, but of significance can occur with neutral or unstable conditions when the dispersion upward is severely limited by the existence of a more stable layer above the plume, for example,an inversion. 5. Under stable conditions the maximum concentrations a t ground-level from elevated sources are less than those occurring under unstable conditions and occur a t greater distances from the source. However, the difference between maximum ground-level concentrations for stable and unstable conditions is only a factor of 2 for effective heights of 25 meters and a factor of 5 for H of 75 m. Because the maximum occurs a t greater distances, concentrations that are below the maximum but still significant can occur over large areas. This becomes increasingly significant if emissions are corning from more than one source.
CONCENTRATIONS AT A RECEPTOR POINT FROM SEVERAL SOURCES

source-receptor geometry can then be worked out merely by drawing or visualizing an x-axis oriented upwind from the receptor and determining the crosswind distances of each source in relation to this x-axis. As pointed out by Gifford (1959), the concentration a t (0, 0, 0) from a source a t (x, y, H) on a coordinate system with the x-axis oriented upwind is the same as the concentration a t (x, y, 0) from a source a t (0, 0, H) on a coordniate system with the x-axis downwind (Figure 5-2). The total concentration is then given by summing the individual contributions from each source (see problem 20).

Figure 52. Comparison of source-oriented and receptororiented coordinate systems.
It is often difficult to determine the atmospheric conditions of wind direction, wind speed, and stability that will result in the maximum combined concentrations from two or more sources; drawing isopleths of concentration for various wind speeds and stabilities and orienting these according to wind direction is one approach.
AREA SOURCES

Sometimes, especially for multiple sources, it is convenient to consider the receptor as being a t the origin of the diffusion coordinate system. The
Special Topics

-

In dealing with diffusion of air pollutants in areas having large numbers of sources, e.g., as in urban areas, there may be too many sources of most atmospheric contaminants to consider each source
39

individually. Often an approximation can be made by combining all of the emissions in a given area and treating this area as a source having an initial horizontal standard deviation, my,. A virtual distance, x,, can then be found that will give this standard deviation. This is just the distance that will yield the appropriate value for a, from Figure 3-2. Values of x, will vary with stability. Then equations for point sources may be used, determining o, as a function of x x,, a slight variation of the suggestion by Holland (1953). This procedure treats the area source as a cross-wind line source with a normal distribution, a fairly good approximation for the distribution across an area source. The initial standard deviation for a square area source can be approximated by u,, s/4.3, where s is the length of a side of the area (see problem 22). I the emissions within an area are from varying f effective stack heights, the variation may be approximated by using a u,,. Thus H would be the mean effective height of release and us, the standard deviation of the initial vertical distribution of sources. A virtual distance, x,, can be found, and point source equations used for estimating concentrations, determining as as a function of x + x,.

+ exp [-- (2:-y

)']I

1

IexP

[-T

+

B is the distance from the x-axis to the restricting bluff, and the positive y axis is defined to be in the direction of the bluff. The restriction of horizontal dispersion by valley sides is somewhat analogous to restriction of the vertical dispersion by a stable layer aloft. When the o, becomes great enough, the concentrations can be assumed to be uniform across the width of the valley and the concentration calculated according to the following equation, where in this case Y is the width of the valley. x= 2Q exp \/2;;uZYu

[--( ) '1 :
-

TOPOGRAPHY

LENE SOURCES

Under conditions of irregular topography the direct application of a standard dispersion equation is often invalid. In some situations the best one may be able to do without the benefit of in situ experiments is to estimate the upper limit of the concentrations likely to occur. For example, to calculate concentrations on a hillside downwind from and facing the source and a t about the effective source height, the equation for concentrations a t ground-level from a groundlevel source (Eq. 3.4) will yield the highest expected concentrations. This would closely approximate the situation under stable conditions, when the pollutant plume would be most likely to encounter the hillside. Under unstable conditions the flow is more likely to rise over the hill (see problem 21). With downslope flow when the receptor is a t a lower elevation than the source, a likely assumption is that the flow parallels the slope; i.e., no allowance is made for the difference between groundlevel elevations at the source and a t the receptor. Where a steep ridge or bluff restricts the horizontal dispersion, the flow is likely to be parallel to such a bluff. An assumption of complete reflection a t the bluff, similar to eddy reflection a t the ground from an elevated source, is in order. This may be accomplished by using:

Concentrations downwind of a continuously emitting infinite line source, when the wind direction is normal to the lime, can be expressed by rewriting equation (12) p. 154 of Sutton (1932):

.

(5.17)

4

x (x,y,O;H) =

42 q u % uz

exp

[-- (H)'] :-

Here q is the source strength per unit distance, for example, g sec-I m --'. Note that the horizontal dispersion parameter, uY, does not appear in this equation, since it is assumed that lateral dispersion from one segment of the line is compensated by dispersion in the opposite direction from adjacent segments. Also y does not appear, since concentration a t a given x is the same for any value of y (see problem 23). Concentrations from infinite line sources when the wind is not perpendicular to the line can be approximated. If the angle between the wind direction and line source is @, the equation for concentration downwind of the line source is:

This equation should not be used where 0 is less than 45".
ATMOSPHERIC DISPERSION ESTIMATES

When estimating concentrations from finite line sources, one must account for "edge effects" caused by the end of the line source. These effects will of course extend to greater cross-wind distances as the distance from the source increases. For concentrations from a finite line source oriented crosswind, define the x-axis in the direction of the mean wind and passing through the receptor of interest. The limits of the line source can be defined as extending from y, to y, where y, is less than y,. The equation for concentration (from Sutton's (1932) equation ( l l ) , p. 154), is:

The symbols have the usual meaning, with the important exceptions that Q, represents the total mass of the release and the u's are not those evaluated with respect to the dispersion of a continuous source a t a Gxed point in space. In Eq. (5.21) the u's refer to dispersion statistics following the motion of the expanding puff. The us is the standard deviation of the concentration distribution in the puff in the downwind direction, and t is the time after release. Note that there is no dilution in the downwind direction by wind speed. The speed of the wind mainly serves to give the downwind position of the center of the puff, as shown by examination of the exponential involving uX. Wind speed may influence the dispersion indirectly because the dispersion parameters u,, uy, and uz may be functions of wind speed. The u,.'s and u;s for an instantaneous source are less than those for a few minutes given in Figure 3-2 and 3-3. Slade (1965) has suggested values for a uY and uz for quasi-instantaneous sources. These are given in Table 5-2. The problem remains to make best estimates of u,. Much less is known of diffusion in the downwind direction than is known of lateral and vertical dispersion. In general one should expect the u, value to be about the same as u7. Initial dimensions of the puff, i.e., from an explosion, may be approximated by iinding a virtual distance to give the appropriate initial standard deviation for each direction. Then ur will be d e t e ~ mined as a function of x x,, uz as a function of x x,, and a, as a function of x x,.

exp (-0.5 where p,
=

pZ)dp
=

(5.20)
Yz %

Yl - ,p,
"r

The value of the integral can be determined from tabulations given in most statistical tables (for example, see Burrington (1953), pp. 273-276; also see problem 24).

INSTANTANEOUS SOURCES
Thus far we have considered only sources that were emitting continuously or for time periods equal to or greater than the travel times from the source to the point of interest. Cases of instantaneous release, as from an explosion, or short-term releases on the order of seconds, are often of practical concern. To determine concentrations a t any position downwind, one must consider the time interval after the time of release and diffusion in the downwind direction as well as lateral and vertical diffusion. Of considerable importance, but very difficult, is the determination of the path or trajectory of the "puff." This is most important if concentrations are to be determined a t specific points. Determining the trajectory is of less importance if knowledge of the magnitude of the concentrations for particular downwind distances or travel times is required without the need to know exactly a t what points these concentrations occur. Rewriting Sutton's (1932) equation (13), p. 155, results in an equation that may be used for estimates of concentration downwind from a release from height, H :

+

+

+

Table 5-2 ESTIMATION O DISPERSION PARAMETERS FOR F QUASI-INSTANTANEOUS S U C S (FROM SLADE, 1965) ORE x= 0 ,

100m
0 ,

x=4km
' Y T

uz

Unstable Neutral
Very Stable

10

15

300 120 35

220 50 7

4

3.8
0.75

1.3

REFERENCES Bierly, E. W., and E. W. Hewson, 1962: Some restrictive meteorological conditions to be considered in the design of stacks. J. Appl. Meteorol., 1, 3, 383-390. Burington, R. S., 1953: Handbook of Mathematical Tables and Formulas. Sandusky, Ohio, Handbook Publishers, 296 pp. Cramer, H. E., 1959: Engineering estimates of atmospheric dispersal capacity. Amer. Ind. Hyg. Assoc. J., 20, 3, 183-189.

(The numerical value of ( 2 ~ ) is / ~ ~ 15.75.)
Special Topics
339-901 0

- 69 - 4

Gifford, F. A., 1959: Computation of pollution from several sources. Int. J. Air Poll., 2, 109110. Gifford, F. A., 1960a: Atmospheric dispersion calculations using the generalized Gaussian plume model. Nuclear Safety, 2, 2,56-59,67-68. Gifford, F. A., 1960b: Peak to average concentration ratios according to a fluctuating plume dispersion model. Int. J. Air Poll., 3, 4, 253-260. Hewson, E. W., and G. C. Gi, 1944: Meteorological investigations in Columbia River Valley near Trail, B. C., pp 23-228 in Report submitted to the Trail Smelter Arbitral Tribunal by R. S. Dean and R. E. Swain, Bur. of Mines Bull 453, Washington, Govt. Print. Off., 304 pp. Hewson, E. W., 1945: The meteorological control of atmospheric pollution by heavy industry. Quart. J. R. Meteorol. Soc., 71, 266-282. Hewson, E. W., 1955: Stack heights required to minimize ground concentrations. Trans. ASME 77, 1163-1172. Holland, J. Z., 1953: A meteorological survey of the Oak Ridge area, p. 540. Atomic Energy Comm., Report ORO-99, Washington, D. C., 584 pp.

Nonhebel, G., 1960: Recommendations on heights for new industrial c h i i e y s . J. Inst. Fuel, 33, 479-513. Pooler, F., 1965: Potential dispersion of plumes from large power plants. P H s Publ. No. 999AP-16, 1965. 13 pp. Singer, I. A., 1961: The relation between peak and mean concentrations. J. Air Poll. Cont. Assoc., 11, 336-341. Singer, I. A., K. Imai, and R. G. Del Campos, 1963: Peak to mean pollutant concentration ratios for various terrain and vegetation cover. J. Air Poll. Cont. Assoc., 13, 40-42. Slade, D. H., 1965: Dispersion estimates from pollutant releases of a few seconds to 8 hours in duration. Unpublished Weather Bureau Report. Aug. 1965. Stewart, N. G., H. J. Gale, and R. N. Crooks, 1958: The atmospheric difision of gases discharged from the c h i i e y of the Harwell Reactor BEPO. Int. J. Air Poll., 1, 87-102. Sutton, 0. G., 1932: A theory of eddy diffusion in the atmosphere. Proc. Roy. Soc. London, A, 135, 143-165. Taylor, G. I., 1915: Eddy motion in the atmosphere. Phil. Trans. Roy. Soc., A, 215, 1-26.

ATMOSPHERIC DISPERSION ESTIMATES

Chapter 6

-RELATION TO OTB[ER

DIFFUSION EQUATIONS

Most other widely used diffusion equations are variant foms of the ones presented here. With respect to ground-level concentrations from an elevated source (Eq. 3.2):

NOTE: Calder wrote the equation for the concentration a t (x, y, z) from a ground-level source. For Eq. (6.3) it is assumed that the concentration at ground level from an elevated source is the same as the concentraton at an elevated point from a ground-level source. Table 6-1 lists the expressions used in these equations that are equivalent to UT and US (continuous source) in this paper.

Other well-known equations can be compared: Bosanquet and Pearson (1936):

Table 6-1 EXPRESSIONS EQUIVALENT TO U, AND VARIOUS DIFFUSION EQUATIONS. Equation Bosanquet and Pearson q
X

o ,

IN

where p and q are dimensionless diffusion coefficients. Sutton (1947):

Sutton Calder
\/Takv,x
U

flkv,x
U

where n is a dimensionless constant and C, and C. are diffusioncoefficients in mnJ2. Calder (1952):

REFERENCES i Bosanquet, C. H., and J. L. Pearson, 1936: The spread of smoke and gases from chimneys. Trans. Faraday Soc., 32, 1249-1263. Calder, K. L., 1952: Some recent British work on the problem of diffusion in the lower atmosphere, 787-792 in Air Pollution, Proc. U. S. Tech. Conf. Air Poll., New York, McGraw-Hill, 847 pp. Sutton, 0.G., 1947: The problem of difision in the lower atmosphere. Quart. J. Roy. Met Soc., 73,257-281.

v' where a =--, the ratio of horizontal eddy velocity w' to vertical eddy velocity, k is von K m a n ' s constant approximately equal to 0.4, and v, = k u H In (z) where z, is a roughness parameter, m.

Other Equations

Chapter 7

-EXAMPLE PROBLEMS
All but the exponential involving y has been found in the preceding problem. Therefore:

The following 26 example problems and their solutions illustrate the application of most of the techniques and equations presented in this workbook. PROBLEM 1: It is estimated that a burning dump emits 3 g sec-I of oxides of nitrogen. What is the concentration of oxides of nitrogen, averaged over approximately 10 minutes, from this source directly downwind a t a distance of 3 km on an overcast night with wind speed of 7 m sec-I? Assume this dump to be a point ground-level source with no effective rise. SOLUTION: Overcast conditions with a wind speed of 7 m sec-I indicate that stability class D is most applicable (Statement, bottom of Table 3-1). For x = 3 km and stability D, a?,= 190 m from Figure 3-2 and UB = 65 m from Flgure 3-3. Eq. (3.4) for estimation of concentrations directly downwind (y = 0) from a ground-level source is applicable: 3 Q x (x,O,O;O) = ?i ay uz u - ?i 190 (65) 7 = 1.1x g m-3 of oxides of nitrogen. PROBLEM 2: I t is estimated that 80 g sec-' of sulfur dioxide is being emitted from a petroleum refinery from an average effective height of 60 meters. At 0800 on an overcast winter morning with the surface wind 6 m secl, what is the ground-level concentration directly downwind from the refinery a t a distance of 500 meters? SOLUTION: For overcast conditions, D class stability applies. With D stability a t x = 500 m, ay = 36 m, a = 18.5 m. Using Eq. (3.3): .

x

(500,50,0; 60) = 3.3 x exp [-0.5 (50/36)2] = 3.3 x 10-"0.381) = 1.3 x g m-3 of SO, PROBLEM 4: A power plant burns 10 tons per hour of coal containing 3 percent sulfur; the effluent is released from a single stack. On a sunny summer afternoon the wind a t 10 meters above ground is 4 m sec' from the northeast. The morning radiosonde taken a t a nearby Weather Bureau station has indicated that a frontal inversion aloft will limit the vertical mixing to 1500 meters. The 1200-meter wind is from 30" a t 5 m sec-'. The effective height of emission is 150 meters. From Figure 3-9, what is the distance to the maximum ground-level concentration and what is the concentration a t this point? SOLUTION: To determine the source strength, the amount of sulfur burned is: 10 tons hr-I x 2000 lb t o r 1 x 0.03 sulfur = 600 lb sulfur hr-l. Sulfur has a molecular weight of 32 and combines with 0, with a molecular weight of 3?; therefore for every mass unit of sulfur burned, there result two mass units of SO,.

Q=
X

64 (molecular weight of SO,) 32 (molecular weight of sulfur) 600 lb hr-I (453.6 g lb-')

x (x,o,o;H) = 77 ay a u exp Q.

[

$ (+)

80 36 (18.5) 6 exp [-0.5 (60/18.5)2] = 6.37 x exp [-0.5 (3.24)2] The exponential is solved using Table A-1 (Appendix 3). = 6.37 x (5.25 x lo-') x = 3.3 x g m-= of SO, rr -

'1

3600 sec hr-' = 151 g sec-' of SO, On a sunny summer afternoon the insolation should be strong. From Table 3-1, strong insolation and 4m sec-l winds yield class-B stability. From Figure 3-9, the distance to the point of maximum concentration is 1 km for class-B stability and effective height of 150 meters. From Figure 3-3 a t this distance as = 110 m. This is much less than 0.47 L. Therefore, a t this distance, the limit of mixing of 1500 meters will not affect the ground-level concentration. From Figure 3-9, the maximum for B stability and this effective height of 150 m is 7.5 x 10P.

PROBLEM 3: Under the conditions of problem 2, what is the concentration a t the same distance downwind but a t a distance 50 meters from the x-axis? That is: x (500, 50, 0; 60) = 7 SOLUTION: Using Eq. (3.2):

=

2.8 x

g m-S of SO,

PROBLEM 5: For the power plant in problem 4, a t what distance does the maximum groundExample Problems

level concentration occur and what is this concentration on an overcast day with wind speed 4 m sec-I? SOLUTION: On an overcast day the stability class would be D. From Figure 3-9 for D stability and H of 150 m, the distance to the point of maximum ground-level concentration is 5.6 km, and the maximum xu/Q is 3.0 x
Xmsr =

Table 7-1 CALCULATION O CONCENTRATIONS F R F O VARIOUS DISTANCES (PROBLEM 6)

3.0 x

lo-= x 151

4 = 1.1x 10-4gm-3 x, u, a , , km m s e c l m

PROBLEM 6: For the conditions given in problem 4, draw a graph of ground-level centerline sulfur dioxide concentration with distance from 100 meters to 100 km. Use log-log graph paper. SOLUTION: The frontal inversion limits the mixing to L = 1500 meters. The distance a t which uz = 0.47 L = 705 m is XI. = 5.5 km. At distances less than this, Eq. (3.3) is used to calculate concentrations:

L, m

X' g m-8

11.0 30 100

4.5 1300 1500 4.5 3000 1500 4.5 8200 1500

OB 6.9 x 1 3.0 x 1O-B 1.1x 1O-B

At distance equal to or greater than 2 XL, which 1 is 1 km, Eq. (3.5) is used:

PROBLEM 7: For the conditions given in problem 4, draw a graph of ground-level concentration versus crosswind distance a t a downwind distance of 1km. SOLUTION: From problem 4 the ground-level centerline concentration a t 1 km is 2.8 x lo-" g m-8. To determine the concentrations a t distances y from the x-axis, the ground-level centerline concentration must be multiplied by the factor exp

Solutions for the equations are given in Table 7-1. The values of concentration are plotted against distance in Figure 7-1.

[ - % (+) '1

a? = 157 meters a t x = 1 km. Values for this computation are given in Table 7-2.

Table 7-2 DETERMINATION OF CROSSWIND CONCENTRATIONS (PROBLEM 7)

a 100 i- 200

a 300 1.400 a 500

0.64 1.27 1.91 2.55 3.18

0.815 0.446 0.161 3.87 x 106.37 x 1 r 3

2.3 x 1~ 1.3 x lw4 4.5 x 101.1x 1W6 1.8 x lwe

These concentrations are plotted in Figure 7-2.

PROBLEM 8: For the conditions given in problem 4, determine the position of the 10" g m-" ground level isopleth, and determine its area. SOLUTION: From the solution to problem 6, the g m-" graph (Figure 7-1) shows that the isopleth intersects the x-axis a t approximately x -- 350 meters and x = 8.6 kilometers.
ATMOSPHERIC DISPERSION ESTIMATES

Figure 7-2. Concentration as a function of crosswind distance (Problem 7).
The values necessary to determine the isopleth half widths, y, are given in Table 7-3.

Table 7-3

DETERMINATION OF ISOPLETH WD H ITS (PROBLEM 8) lcenterlinel, X,

x (isoPleth)
(centerline)

km

m

g

m-3

y/aY

Y , m

Figure 7-3. Location of the 10" g m-"round-level pleth (Problem 8).

isoa

PROBLEM 9: For the conditions given in problem 4, determine the profile of concentration with height from ground level to z = 450 meters a t x = 1 km, y = 0 meters, and draw a graph of concentration against height above ground. SOLUTION: Eq. (3.1) is used to solve this problem. The exponential involving y is equal to 1. At x = 1 km, a, = 157 m, as = 110 m. (From problem 4). The orientation of the x-axis will be toward 225" close to the 'source, curving more toward 210" to 215" azimuth a t greater distances because of the change of wind direction with height. The isopleth is shown in Figure 7-3. Since the isopleth approximates an ellipse, the area may be estimated by 7 ab where a is the semimaior axis and b is the semiminor axis.

Values for the estimation of x(z) are given in Table 7-4. PROBLEM 10: For the conditions given in problem 4, determine the distance a t which the ground-level centerline concentration equals the centerline concentration a t 150 meters above ground. Verify by computation of x (x,O,O) and x (x,0,150). SOLUTION: The distance a t which concentrations a t the ground and a t plume height are equal should occur where az = 0.91 H (See Chapter 5). For B stability and H = 150 m, US = 0.91 (150) = 136 m occurs a t x = 1.2 km. At this distance ay = 181 m.
47

Example Problems

Table 7-4

DETERMINATION OF CONCENTRATIONS FOR VARIOUS HEIGHTS (PROBLEM 9)

x lo-. exp [- 1/2 (1.10)2] = 4.88 x (0.546) =2 . 7 ~ gm-3
= 4.88

These values are plotted in Figure 7-4.
500

I

I

I

I

PROBLEM 11: For the power plant in problem 4, what will the maximum ground-level concentration be beneath the plume centerline and a t what distance will i t occur on a clear night with wind speed 4 m sec-I? SOLUTION: A clear night with wind speed 4 m sec-I indicates E stability conditions. From Figure 3-9, the maximum concentration should occur a t a distance of 13 km, and the maximum xu/Q is 1.7 x
Xmnx =

1.7 x lo-* x 151 4 = 6.4 x 10-Q m-%f SO,
XU -

Q

X - =u

&

CONCENTRATION,

@

m-a

Figure 7-4. Concentration as a function of height (Problem 9).
Verifying:

PROBLEM 12: For the situation in problem 11, what would the fumigation concentration be the next morning a t this point (x = 13 km) when superadiabatic lapse rates extend to include most of the plume and i t is assumed that wind speed and direction remain unchanged? SOLUTION: The concentration during fumigation conditions is given by Eq. (5.2) with the exponential involving y equal to 1. in this problem.

d

b

x (x,O,O) =

Q n uy US U

exp

[- -+ (+)'I

For the stable conditions, which were assumed , to be class E, a t x = 13 km, o = 520 m., and US = 90 rn. Using Eq. (5.3) to solve for hl: h, = H 2 US = 150 2 (90) = 330 m. From the horizontal spreading suggested by Eq. (5,4):

+

+

ATMOSPHERIC DISPERSION ESTIMATES

UYF = v

(stable)

+ H/8

= 520

+ 19

= 539

=

8.5 x

g m-3 of SO,

Note that the fumigation concentrations under these conditions are about 1.3 times the maximum ground-level concentrations that occurred during the night (problem 11). PROBLEM 13: An air sampling station is located a t an azimuth of 203" from a cement plant a t a distance of 1500 meters. The cement plant releases fine particulates (less than 15 microns diameter) a t the rate of 750 pounds per hour from a 30-meter stack. What is the contribution from the cement plant to the total suspended particulate concentration a t the sampling station when the wind is from 30" a t 3 m sec-I on a clear day in the late fall a t 1600? SOLUTION: For this season and time of day the C class stability should apply. Since the sampling station is off the plume axis, the x and y distances can be calculated:

concentration as a function of wind speed for stability classes B and D. Determine the critical wind speed for these stabilities, i.e., the wind speed that results in the highest concentrations. Assume that the design atmospheric pressure is 970 mb and the design ambient air temperature is 20°C (293'K). SOLUTION: Using height equation: Holland's effective stack

x = 1500 cos 7" = 1489 y = 1500 sin 7' The source strength is: g sec-I = 94.5 g sec-' lb hrl At this distance, 1489 m, for stability C, ey = 150 m, = 87. The contribution to the concentration can be calculated from Eq. (3.2):
=

183

19.5 (2.5) u 48.8 =u
=

i

Q = 750 lb hr-' x 0.126

The effective stack heights for various wind speeds and stabilities are summarized in Table 7-5.

Table 7-5 EFFECTIVE STACK HElGHlS (PROBLEM 14)
Glass
U,

D h Class B

AH, m + AH, m 1.15 AH, m h

m sec-1

+ 1.15 AH, m =

94.5 1.23 x loJ

exp 1-0.5

(1.22)=]

exp 1 4 . 5 (0.345)2]

PROBLEM 14: A proposed source is to emit 72 g sec- of SO, from a stack 30 meters high with a diameter of 1.5 meters. The effluent gases are emitted a t a temperature of 250°F (394°K) with an exit velocity of 13 m sec-'. Plot on loglog paper a graph of maximum ground-level
Example Problems

By use of the appropriate height, H, the maximum concentration for each wind speed and stability can be determined by obtainmg the
49

maximum xu/& as a function of H and stability from Figure 3-9 and multiplying by the appropriate Q/u. The computations are summarized in Table 7-6, and plotted in Figure 7-5.

is a small town of 500 inhabitants 1700 meters northeast of the plant. Plant managers have decided that it is desirable to maintain concentrations below 2 0 ppb (parts per billion by volume), or approximately 2.9 x g m-3, for any period greater than 3 0 minutes. Wmd direction frequencies indicate that winds blow from the proposed location toward this town between 10 and 15 per cent of the time. What height stack should be erected? It is assumed that a design wind speed of 2 m secl wl be il sufficient, since the effective stack rise will be quite great with winds less than 2 m sec-l. Other than this stipulation, assume that the physical stack height and effective stack height are the same, to incorporate a slight safety factor. SOLUTION: The source strength is:

WlND SPEED, rn set-1

Figure 7-5. Maximum concentration as a function of wind speed (Problem 14).
Tabla 7-6
Stability

MAXIMUM CONCENTRATION AS A FUNCTION OF WIND SPEED (PROBLEM 14) u, rn sec-1

H, rn Xu/Q rn-2 Q u g m-1 g

, rn-3 Class

B

D

0.5 1.0 1.5 2 3 5 7 0.5 1.0 1.5 2 3 5 7 10 20

142.2 86.1 67.5 58.1 48.7 41.3 38.0 127.6 78.8 62.6 54.4 46.3 39.8 37.0 34.9 32.4

8.0 x lWB 2.0 x lW5 3.1 x lWJ 4.1 x 1W6 5.7 x 1W6 7.8 x lW5 8.7 x 1F6 4.4 x lWe 1.42x1W5 2.47x1W5 3.5 x 1W6 5.1 x 1 W4 7.3 x lWJ 8.2 x 1W6 9.4 x lW5 1.1 x lW4

144 72 48 36 24 14.4 10.3 144 72 48 36 14.4 10.3 7.2 3.6

1.15 x 1W3 1.44 x lW3 1.49 x 1WS+ 1.48 x lWS 1.37 x lW3 1.12 x lW3 8.96 x lW4 6.34 x 1W4 1.02 x lW8 1.19 x lW5 1.26 x 10-+ 1.22 x lW3 1.05 x lWs 8.45 x lW4 6.77 x lW4 3.96 x 1(r

1000 lb day-' x 453.6 g lb -I = 5,25 set* 86,400 sec day-I From Eq. (4.2): 0.117 Q - 0.117 (5.25) uy US = (2.9 x 1W5) 2 xa u = 1.06 x lo4 mZ At a design distance of 1500 meters (the limit of company property), o U, = 1.06 x lo4 gives , a point from Figure 4-1 about 0.2 from Class C to Class D along the line x = 1500 m. From Figure 3-3, vs = 80 for this stability. H = f l u s = 113 meters

Q=

3

The wind speeds that give the highest maximum concentrations for each stability are, from Figure 7-5: B 1.5, D 2.0. PROBLEM 15: A proposed pulp processing plant ton per day of hydrogen is expected to emit sulfide from a single stack. The company property extends a minimum of 1500 meters from the proposed location. The nearest receptor

PROBLEM 16: In problem 15 assume that the stack diameter is to be 8 ft, the temperature of the effluent 250" F, and the stack gas velocity 4 5 ft secw1. From Holland's equation for effective stack height and the method used in problem 15, determine the physical stack height required to satisfy the conditions in problem 15. In estimating AH, use T. = 6 8 ° F and p = 920 mb. SOLUTION: First determine the relation between AH and u from Holland's equation. v. = 4 5 ft sec-' = 13.7 m sec* d = 8 f t = 2.44 m T. = 250°F = 121°C = 394°K T. = 6 8 ° F = 20°C = 293°K p = 920 mb

,

v

ATMOSPHERIC DISPERSION ESTIMATES

60 sec min1 9x m9min-I

DT= 7.41 x lo-"
The relation between or on and u is:

sec m-S The total dosage is given in g sec m-8 from

The required computations using Figure 4-1 are summarized in Table 7-7:
Table 7-7 REQUIRED PHYSICAL STACK HEIGHT AS A FUNCTION O WIND SPEED (PROBLEM 16) F

where QTis the total release in grams. ?i u or o Dr , Therefore Qr = -P

[- +(+)

-

AH, m sec-1 m

u,

,, , , rnZ Stability to Give uy UB at 1500 m

H, =
US

h=

For slightly unstable conditions (Class C) a t x=8km,07--690m,o;=310m;y=2000m, u = 5 m see-I

vFes, HI-AH, m m

rn

15.0

4.24 x lo4 2.12 x lo4 1.41 x lo4 1.06 x lo4 8.48 x lo3 7.06~ lo3 4.24 x lo3 3.03 x lo3 2.12 x lo3 7 1.41 x lo8

0.9 from A to B 190 0.6 from B to C 120 0.9 from B to C 96 0.2 from C to D 76 0.4 from C to D 64 0.6 from C to D 56 D 42 0.5 from D to E 34 E 28 0.5 from E to F 23

=

24.9 exp [-0.5 (2.90)2]

No correction has been made for the facts t h i t the release is for 1hour and the standard deviations represent time periods of 3 to 15 minutes.
33 26

The required physical height is 68 meten.

PROBLEM 17: A dispersion study is being made over relatively open terrain with fluorescent particles whose size yields 1.8 x loToparticles per gram of tracer. Sampling is by membrane filters through which 9 x mSof air is drawn each minute. A study involving a 1-hour release, which can be considered from ground-level, is to take place during conditions forecast to be slightly unstable with winds 5 m set?. It is desirable to obtain a particle count of a t least 20 particles upon membrane filters located a t ground-level 2.0 lun from the plume centerline on the sampling arc 8 km from the source. What should the total release be, in grams, for this run? SOLUTION: The total dosage a t the sampler is determined by the total sample in grams divided by the sampling rate: 20 particles DT (g sec m-9 = 1.8 x 1010 particles g1
Example Problems

PROBLEM 18: A release of 2 kg of fluorescent particles is made based on the results of the computation in problem 17. The conditions are class C stability and wind speed 5 m sec-l. The crosswind-integrated ground-level dosage along the 8-km arc is determined from the samplers along this arc to be 8.2 x 10-I g sec mP. What is the effective on for this run? SOLUTION: given by: The crosswind-intagrated dosage is

Since the source is a t ground-level, the exponential has a value of 1. Solving for -=:

51

PROBLEM 19: At a point directly downwind from a ground-level source the 3- to 15-minute g concentration is estimated to be 3.4 x m-3. What would you estimate the 2-hour concentration to be a t this point, assuming no change in stability or wind velocity? SOLUTION: Using Eq. (5.12) and letting k min, s = 2 hours, and p = 0.2:
=3

\

SOURCE A x=24.6 k m y= 8.4 k m

= -

I 40 O 2

(3.4 x 10-9
= 1.6
=

3'4 2.09 Letting k 15 min, s
=

10-3

m-d

SCALE, k m

2 hours, and p

=

0.17

0

2 1

(&)
3.4 x 1.42

3.4 x 10-~ 0~) 10-3 m-d Figure 7-6. Locations of sources and receptor (Problem

20).

-

-- 1 I 8 O L 7 (3.4 x

= 2.4

The 2-hour concentration is estimated to be between 1.6 x and 2.4 x g m-a. PROBLEM 20: Two sources of SO, are shown as points A and B in Figure 7-6. On a sunny summer afternoon the surface wind is from 60" a t 6 m sec-l. Source A is a power plant emitting 1450 g sec-I SO, from two stacks whose physical height is 120 meters and whose AH, from Holland's equation, is AH (m) = 538 (m2 ~ e c - ~ ) / u (m sec3). Source B is a refinery emitting 126 g sec-I SO, from an effective height of 60 meters. The wind measured a t 160 meters on a nearby T V tower is from 70" a t 8.5 m sec-'. Assuming that the mean direction of travel of both plumes is 245", and there are no other sources of SO,, what is the concentration of SO, a t the receptor shown in the figure? SOLUTION: Calculate the effective height of Source A using the observed wind speed a t 160 meters.

For Source A, x = 24.6 km, y v ,
=

= 8.4

km m sm-I

1810 m,
7

uz =

1120 m, u

= 8.5

XA =

1450 exp [-0.5 1810 (1120) 8.5

8400 1450 exp [-0.5 (4.64)2] 5.42 x lo7 exp L 4 . 5 (0.164)21 = 2.67 x (2.11 x lo-') (0.987) xa = 5.6 x g m-S
-

For Source B, x = 13.0 km,y = 4.0 km. vr = 1050 m,
XB
=
T

uz =

640 m, u

=

7.0 m s e e 4000

126 1050 (640) 7

exp [-0.5
-

($1

'1
+

For a sunny summer afternoon with wind speed 6 m sec-I, the stability class to be expected is C. The equation to be used is Eq. (3.2):

126 exp [--0.5 (3.81)21 1.48 x lo7 exp 1-0.5 (0.0938)*] = 8.5 x lo-# (7.04 x lo-') (0.996) xB = 6.0 x 10- g m-3 x = Xa XB = 0.56 x 10-' 6.0 x lo-' - = 6.6 x g m -3

+

ATMOSPHERIC DISPERSION ESTIMATES

PROBLEM 21: A stack 15 meters high emits 3 g sec-' of a particular air pollutant. The suris rounding t e ~ a i n relatively flat except for a rounded hill about 3 km to the northeast whose crest extends 15 meters above the stack top. What is the highest 3- to 15-minute concentration of this pollutant that can be expected on the facing slope of the hill on a clear night when the wind is blowing directly from the stack toward the hill a t 4 m set-l? Assume that AH is less than 15 m. How much does the wind have to shift so that concentrations a t this point drop below lo-" g m-j? SOLUTION: A clear night with 4 m sec-' indicates class E stability. Eq. (3.4) for groundlevel concentrations from a ground-level source is most applicable (See Chapter 5). At 3 km for class E, a7 = 140 m, as = 43 m.

( I t may actually be more unstable, since this is in a built-up area.) To allow for the area source, let aYO= 1524/4.3 = 354. For class E the virtual distance, x, = 8.5 km. For x = 1524 m, az = 28.5. For x x =10,024 m, ay = 410 m. ,

+

To determine the crosswind distance from the plume centerline to produce a concentration of lo-' g m-3 Eq. (3.8) is used:

PROBLEM 23: An estimate is required of the total hydrocarbon concentration 300 meters downwind of an expressway a t 1730 on an overcast day with wind speed 4 m sec->. The expressway runs north-south and the wind is from the west. The measured traffic flow is 8000 vehicles per hour during this rush hour, and the average speed of the vehicles is 40 miles per hour. At this speed the average vehicle is exssecl of total hydropected to emit 2 x lo-" carbons. SOLUTION: The expressway may be considered as a continuous infinite line source. To obtain a source strength q in grams sec-I m-I, the number of vehicles per meter of highway must be calculated and multiplied by the emission per i vehicle. Vehicles/meter = Flow (vehicles hour-') Average speed (miles hour-') 1600 (m mile-') 8000 = 1.25 x 10- (vehicles ml) 40 x 1600 q = 1.25 x lo-' (vehicles m-l) x 2 z (g sec-I vehicle->) q = 2.5 x lW3(gsec-' m-I) Under overcast conditions with wind speed 4 m sec-' stability class D applies. Under D, a t x = 300 meters, a% = 12 m. From Eq. (5.18) :

484 tan e =-----3000

= 0.1614

A wind shift of 9.2" is required to reduce the concentration to lo-' g m-a. PROBLEM 22: An inventory of SO, emissions has been conducted in an urban area by square areas, 5000 f t (1524 meters) on a side. The emissions from one such area are estimated to be 6 g sec-I for the entire area. This square is composed of residences and a few small commercial establishments. What is the concentration resulting from this area a t the center of the adjacent square to the north when the wind is blowing from the south on a thinly overcast night with the wind a t 2.5 m sec-'? The average effective stack height of these sources is assumed to be 20 meters. SOLUTION: A thinly overcast night with wind speed 2.5 m sec-' indicates stability of class E.
Example Problems

.
= 4.2

x

g m-' of total hydrocarbons.

PROBLEM 24: A line of burning agricultural waste can be considered a finite line source 150 m long. It is estimated that the total emission of organics is a t a rate of 90 g sec-'. What is the 3- to 15-minute concentration of organics a t a distance of 400 m directly downwind from the center of the line when the wind is blowing a t . 3 m sec-' perpendicular to the line? Assume .

that i t is 1600 on a sunny fall afternoon. What is the condentration directly downwind from one end of the source? SOLUTION: Late afternoon a t this time of year implies slight insolation, which with 3 m sec-I winds yields stability class C . For C stability a t x = 400 m, w7 = 45 m, en = 26 m.

The accident has occurred on a relatively clear night with wind speed 2.5 m secl. What is the concentration in the air 3 kilometers directly downwind from the source a t 0400 due to all radioactive material? due to iodine-131? SOLUTION: Source strength tivity (corrected for decay) Leak rate = 0.001 day-' 86400 sec day1
=

leak rate x ac-

Eq. (5.20) is appropriate. Source strength of all products QA (curies sec-I) = 1.157 x -0.2
= 1.74 x

(1.5 x lo6)

1 1 : n
.

lo-'

(+) 4'2

To determine decay of materials with the halflife given, multiply by exp is time and L is half-life. Source strength of exp (-0.5 pZ)dp QI (curies sec-l)
= 1.157 x

("y

where t

(5.3 x lo4) exp
4

"Ft)
For I,,, L
= 6.95

x

lo6 sec

For a point downwind of one of the ends of the line:

QI = 6.13

x

exp

For a clear night with wind speed 2.5 m see*, class F applies. Approximate the spreading a t the reactor shell by 2.15 cr0 = 2.15 WSO = the = radius of the shell = 20 m W ~ O wzo = 9.3 m. The virtual distances to account for this are: x, = 250 m, x. = 560 m. exp (-0.5 pZ)dp

( c: 9 ;

)

PROBLEM 25: A core melt-down of a power reactor that has been operating for over a year occurs a t 0200, releasing 1.5 x loB curies of activity (1 second after the accident) into the atmosphere of the containment vessel. This total activity can be expected to decay according
, I ,

For concentration a t 0400, 3000 m downwind due to all radioactivity, t = 7200 seconds.
XA =

4.4 x 1,3 x

= 7.66
XA =

(1.74 x 10P) (7200)-"' x lo-' (0.17) curies m-8

to

cunes of this activity is due to iodine-131, which has a half-life of 8.04 days. The reactor building is hemispherically shaped with a radius of 20 meters. Assume the leak rate of the building is 0.1% day1.

1=o /

-o.z.

It is estimated that about 5.3 x

lo4

The concentration a t 0400, 3000 m downwind due to 1"' is:
XI = 4.4

-

x (7200) I

(6.13 x lo4) exp [-4.997 x 1 P

ATMOSPHERIC DISPERSION ESTIMATES

= 2.7 x lo-* (1.0) The decay of nificant for 2 hours
XI = 2.7

is insig-

Table 7-9 DETERMINATION OF CONCENTRATION A A S FUNCTION OF DS A C (PROBLEM 26) IT N E x, km a, x lo-$ curies m-3

PROBLEM 26: A spill estimated a t 2.9 x loe grams of unsymmetrical dimethyl hydrazine occurs a t 0300 on a clear night while a rocket is being fueled. A circular area 60 meters in diameter built around the launch pad is revetted into squares 20 feet on a side to c o n h e to as small an area as possible any spilled toxic liquids. I n this spill only one such 20- by 20-foot area is involved. At the current wind speed of 2 m sec-', i t is estimated that the evaporation rate will be 1100 g sec-l. The wind direction is predicted to be from 310" 2 15" for the next hour. Table 7-8 gives the emergency tolerance limits for UDMH vapor. Table 7-8 EMERGENCY TOLERANCE LMT FOR UDMH I IS VAPOR V R U EXPOSURE TIME ES S
Time, minutes Emergency Tolerance Limits, g m-3

m

x

+ x,, krn

a?' rn

g m-3

x,

0.1 0.3 0.6 1 3 6 10

2.3 5.6 9.7 14 27 37 47

0.14 0.34 0.64 1.04 3.04 6.04 10.04

5.5 12.5 22 35 93 175 275

13.9 2.5 8.2x 10-1 3.6x 1 0' 7.0 x 1W2 27 x . 1.4 x 1 F 2

These values of x are graphed as a function of x in Figure 7-7. The downwind concentration drops below the critical value of 2.5 x at a distance of 6.5 km.

5 15 30

1.2 x 1w* 8.6 x 1W2 4.9 x 1 w 2

What area should be evacuated? SOLUTION: From Table 3-1, the stability class is determined to be Class F. This is not a point . source but a small area source. Mowing 4 3 wy0 to equal the width of the wetted area, 6.1 meters (20 feet), wy0 = 1.4 meters. In attempting to determine the virtual distance, x i t is found to , , be less than 100 meters, and will be approximated by 40 meters. The release will take: 2.9 x lo6 g 1.1x lo3 g WC-'
= 2.64

x 10Ssec 44 min. =

DISTANCE, k m

Therefore the concentration for an exposure time of 1 hour (2.5 x g m-3) is of main concern. The equation for calculation of downwind concentrations is Eq. (3.4) :

Figure 7-7. Concentration of UDMH as a function of downwind distance (Problem 26).
Calculated widths within a given isopleth are summarized in Table 7-10. The maximum width of the area encompassed by an isopleth is about 140 meters from the downwind position. Since the wind direction is expected to be from 31Oo-t15",the sector a t an azimuth of 115" to 145"plus a 140-meter rectangle on either side should be evacuated. See Figure 7-8.

x (x,O,O;O) = of x

+ x,.

P

wr

WZ

u

where aY is a function

Values of the parameters and of TaMe 7-9.
Example Problems

x

are given in

Table 7-10

DETERMINATION O WIDTHS WITHIN F ISOPLETHS (PROBLEM 26) v,, x, kin x

+ x,, km m

(centerline), g m-'

x Lisop'eth)
(centerline)

Y o ,

Y, m

0.1 0.5 1.0 2.0 3.0 4.0 5.0 6.0

0.14 0.54 1.04 2.04 3.04 4.04 5.04 6.04

5.5 19 35 66 93 120 149 175

13.9

1.8 x l W 4 2.27 x 1Cr2 6 . 9 4 ~1F" 1.92 x 10-I 3.57 x lFi 5.20 x lo-'

3.55 2.75 2.31 1.82 1.44 1.14 0.82

20 52 80 120 134 137 122 68

1.1
3.6~ 10-1 1.3 x 10-1 7.0 x lo-' 4.8 x l W A 2.7 x 1Cr2

lo-' 3.5~ lF2 7 . 1 4 ~

9.26 x lF1 0.39

SCALE, k m

145'

Figure 7-8. Possible positions of the 2.5 x lo-' g m-" isopleth and the evacuation area (Problem 26).

ATMOSPHERIC DISPERSION ESTIMATES

APPENDICES

Appendix 1: ABBREVIATIONS AND SYMBOLS Abbreviations cal g calorie gram degrees Kelvin meter millibar second

t,, time required for the mixing layer to develop t, from the top of the stack to the top of the plume a time period ambient air temperature stack gas temperature a t stack top wind speed a mean wind speed for the wind speed class N. horizontal eddy velocity stack gas velocity a t the stack top a velocity used by Calder vertical eddy velocity distance downwind in the direction of the mean wind design distance, a particular downwind distance used for design purposes the distance a t which as = 0.47L a virtual distance so that ax (x,) equals the ini, tial standard deviation, a, a virtual distance so that a? (x,) equals the initial standard deviation, up, a virtual distance so that an (x,) equals the initial standard deviation, a . s crosswind distance height above ground level roughness parameter

T ,
T,
u
UN

"K m mb sec

Symbols ratio of horizontal eddy velocity to vertical eddy velocity c specific heat a t constant pressure , C, Sutton horizontal dispersion parameter C, Sutton vertical dispersion parameter d inside stack diameter a t stack top DT(x,y,O;H) Total dosage e 2.7183, the base of natural logarithms f (0,S,N) frequency of wind direction for a given stability and wind speed class h physical stack height h, height of the base of an inversion H effective height of emission H,, effective height of emission for a particular wind speed k von Karman's constant, approximately equal to 0.4 K eddy diffusivity L two uses: 1. the height of an air layer that is relatively stable compared to the layer beneath it; a lid 2. the half-life of a radioactive material n Sutton's exponent N an index for wind speed class p three uses: 1. Bosanquet's horizontal dispersion parameter 2. atmospheric pressure 3. a dummy variable in the equation for a Gaussian distribution. q two uses: 1. Bosanquet's vertical dispersion parameter 2. emission rate per length of a line source Q emission rate of a source Q total emission during an entire release R net rate of sensible heating of an air column by solar radiation s the length of the edge of a square area source S an index for stability t, a short time period
Appendix 1

a

v' v, v, w' x

x, x,. x, x, x, y z z,

the rate of change of potential temperature sz with height AH the rise of the plume centerline above the stack top 0 two uses: 1. wind direction azimuth or sector 2. potential temperature = 3.1416 pa ambient air density WA the standard deviation of azimuth (wind direction) as determined from a wind vane or bidirectional vane a the standard deviation of wind elevation angle as determined from a bi-directional vane a, the standard deviation in the downwind direction of a puff concentration distribution an initial downwind standard deviation a the standard deviation in the crosswind direca tion of the plume concentration distribution as,) an initial crosswind standard deviation a the standard deviation in the vertical of the plume concentration distribution as, an effective equal to 0.8 L a , an initial vertical standard deviation , the vertical standard deviation of the plume a concentration a t a particular downwind distance for the stability, S.

60

I

the angle between the wind direction and a line source x concentration xawr crosswind-integrated concentration x, a ground-level concentration for design purposes xr. inversion break-up fumigation concentration xk concentration measured over a sampling time,
@
tk

2.

concentration measured over a sampling time,

t

- relative concentration

Q

x,,,,,. maximum ground-level centerline concentration with respect to downwind distance

relative concentration normalized for wind speed x (x,y,z;H) concentration a t the point (x, y, z) from an elevated source with effective height, H. x (x,e) the long-term average concentration a t distance x, for a direction e from a source.

Q

*

60

ATMOSPHERIC DISPERSION ESTIMATES

Appendix 2: CHARACTERISTICS OF THE GAUSSIAN DISTRIBUTION The Gaussian or normal distribution can be depicted by the bellshaped curve shown in Figure A-1. The equation for t
Y=

This area is found from Eq. (A.2) : Area (to p)
=

2 .

7z
(-4.2)

1

1

exp (-4.5 pZ)dp

GS

Figure A-2 gives the ordinate value a t an from the center of the distribution (which occu a t x). This information is also given in Figure A-3 gives the area under the Gau from - to a parti

Figure A-4 gives the area under the Gaussian -p to +p. This can be found from Eq.

Area (-p

to +p)

=

(A.3)

0 PO 80 70 6-

Y
0 54

0 40 302-

-3

-2

-I

0 x-i -

I

2

3

Figure A-1. The Gaussian distribution curve.

Appendix 2

Figure A-2. Ordinate values of the Gaussian distribution.

- ATMOSPHERIC DISPERSION ESTIMATES

Figure A-3. Area under the Gaussian distribution curve from

-m

to p.

Appendix 2

Figure A-4. Area under the Gaussian distribution curve between -p and Sp.

- ATMOSPHERIC DISPERSION

ESTIMATES

Appendix 3:

SOLUTIONS TO EXPONENTIALS

Expressions of the form exp 1-0.5

AZl where

A is H/uZ or y/ur frequently must be evaluated.
Table A-1 gives B as a function of A where B = exp 1-0.5 AZl The sign and digits to the right of the E are to be considered as an exponent of 10. For example, if A is 3.51, B is given as 2.11E - 03 which means 2.11 x

.

Appendix 3

65

Table A-1

SOLUTIONS TO EXPONENTIALS B = exp I-0.5AZl The notation 2.16 E-1 means 2.16 x 10-I

Table A - l (continued)

SOLUTIONS

Appendix 4:

CONSTANTS, CONVERSION

EQUATIONS, CONVERSION TABLES

Constants

Conversion Equations and Tables T("C) = 5 (T("F) - 32) 9 1 ' T("K) = T("C) 273.16 T("F) = (9/5 T('C) ) 32

+

+

Appendi 4

69

CONVERSION FACTORS

-

VELOCITY M I (STAT) PER HR 2.2369 E 00 KNOTS MI(STAT1 PER DAY 5.3686 E 01

DESIRED UNITS METERS FT FT KM PER SEC PER SEC PER MIN PER H R GIVEN UNITS METERS PER SEC 1.0000 E 00 3.2808 E 00 1.9683 E 02 3.6000 E 00

1.9425 E 00

FT
PER SEC FT PER MIN
KM

5.0800 E-03

1.6667 E-02

1.0000 E 00

1.8288 E-02

1.1364 E-02

9.8681 €103

2.7273 E-01

PER HR
M I ISTATI

5 g PER H R KNOTS

4.4704 E-01

1.4667 E 00

8.8000 E 01

1.6093 E 00

1.0000 E 00

8.6839 E-01

2.4000 E 01

8 'a
3
n

M I ISTAT)

PER DAY

1,8627 Em02

6.1111 E-02

3.6667 E 00

6.7056 E-02

4.1667 Em02

3.6183 €102

1.0000 E 00

i3
'a

8: 2

! i

TO CONVERT A VALUE FROM A GIVEN UNIT TO A DESIRED UNIT, MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS AN0 BENEATH THE DESIRED UNIT* NOTE THAT E-XX MEANS 10 TO THE -XX POWER.

IP

CONVERSION FACTORS

EMISSION RATES

DESIRED UNITS GRAYS GRAMS KG KG LBS LBS LBS TONS TONS PER HOUR PER DAY PER MIN PER HOUR PER DAY PER HOUR PER DAY PER SEC PER MIN GIVEN UNITS GRAMS PER SEC GRAMS PER MIN

KG PER DAY LBS PER M1N LBS PER HOUR LBS PER DAY TONS PER HOUR TON3 PER OPY

1.1574 Em02

6.9444 E-01

4.1667 E-02

1.0000 E 00

1.5310 E-03

9.1859 E-02

2.2046 E 00

4.5930 E-05

1.1023 E-03

2.5200 E 02

1.5120 E 04

9.0718 E 02

2.1772 E 04

3.3333 E 01

2.0000 E 03

4r8000 E 04

1.0000 E 00

2.4000 E 01

T CONVERT A VALUE FROM A GIVEN UNIT TO A DESIRED UNIT. MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSlTE THE GIVEN UNITS O AN0 BENEATH THE DESIRED UNIT. NOTE THAT E-XX MEANS 10 TO THE -XX POWER*

CONVERSION FACTORS

-

LENGTH MICRON KILOMETER FOOT

OESIREO UNITS METER GlVEN UNITS METER

1.0000 E 06 1.0000 E 04

1.0000 E-03

3.2808 E 00 3,2808 E0 .2 3.2808 Em06 3.2808 E 03

1.oooo
El05

MICRON

1.0000 E-06

1.0000 E 00 1.0000 E 09

1.0000 E-09 1.0000 E 00

KILOMETER

1.0oo0
E 03

FOOT

Q

'
M

TO CONVERTA VALUE FROM A GIVEN UNIT TO A OESlREO UNIT. MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS AN0 BENEATH THE OESIREO UNIT. NOTE THAT E-XX MEANS 10 TO THE -XX POWER.

P

CONVERSION FACTORS

-

AREA SO KM SQ CM SQ I N C H SQ FOOT SO YARD ACRE SQ STAT MlLE SQ NAUT MILE

D E S I R E 0 U N I T S SQ METER GIVEN UNITS
SO METER

SO I N C H

I

SQ FOOT

SO YARD

ACRE

SO STAT MILE SO NAUT MILE TO CONVERT A VALUE FROM A G I V E N U N I T TO A D E S I R E D U N I T , M U L T I P L Y THE G I V E N VALUE BY THE FACTOR OPPOSITE THE G I V E N U N I T S AND BENEATH THE DESIREO UNIT. NOTF TUAT E-XX MEANS 1 0 TO THE -XX POWER.

CONVERSIOY F A C T O D S

-

VnLU4E
C!l
INCd

D ~ S T R F I ) U . I ~ T S cu ~ ' E T ~ R CITE*

CU FOOT

GIVEN l l N l T 5

CU STAT CU NAUT U 5 FLUID U S QUART MILE MILE 3UNCE

U S GALLON

CU INCH

1.63117 r-nj

1.6387 E-02

1.0000 F 00

5.7H70 E-04

3.9315 E-15

2,5746

E-1s

5.5412 E-01

1.7316 E-08

4.3290 E-03

CU STAT MlLE CU NAUT M~LF

4.168L E 09 6.3650 F n9

4.1681

E
6.7649 E 1~

2.5436 E 14 3.8n42 E 14

1.4720 E 11 2;2478 E 11

1.0000 E 00 1.5270 E 00

6.5486 E-01 1,0000 E 00

1.4094 E 14 2.1523 E 14

4.4045 E 06 6.7259 E 06

1.1011 E 12 1.6815

z i; E 12

1 m U 5 QUART

9.*535 E 02 3.7854 E-03

9.6693 E 03 3.7833 E 00

5.7750 E 07 2.3100 E 02

3.1420 E 04 1.3368 E-'11

2.2704 E-07 Y.0817 E-13

1.4868 E-07 5.9472 E-13

3.2000 E 07 1.2800 E 02

1.0000 E 00 4.0000 Em06

2.5000 E 05 1,0000 E 00

U 5 GALLON

8
M V1

TO CONVFRT A V A L U F FROM A G I V t N UNIT T D A DFSIRED UNIT. MULTIPLY THE GIVEN VALUE UY THE FACTOR OPPOSITE THE GIVEN U N I T S AND BENFATH THE OFSIRED UYIT. NOTt r d 4 T E-XX YE4NS 1 0 TO TdE -XX POWER.

CONVERSION FACTORS

-

MASS MICROGRAM KILOGRAY METRIC TON SHORT TON LONG TON GRAIN OUNCE LB (&VDP) 9.5274 E-02 5.5274 E-08 (AVDPI

DESIRED UNITS GRAY GlVEN U N I T S GRAM 1.0000 E 00 1.0000 E-06

1.0000 E 06 1.0000 E 00

1.0000 E-03 1.0000 E-09

1.0000 E-06 1.0000 Em12

1.1023 E-06 1.1023 E-12

9.8421 E-07 9.8421 E-13

1.5432 E 01 1.5432 E-05

2.2046 E-03 2.2046 E-09

MICROGRAM

METRIC TON

1.0000 E 06 9.0718 E 05 1.0160 E 06 6.4799 E-02 2.8349 E 01 2.8349 E 07 2.8349 E-02 2.8349 El05 3.1250 E-05 2.7902 E-05 4.3750 E 02 1.0000 E 00 o.4500 E-02

SHORT TON

LONG T ~ N

GRAIN

OUNCE (AVDP I

!

TO CONVERT A VALUE FROM A GIVEN AND BENEATH THE DESIRED UNIT.

UNIT T O A DESIRED UNIT, MULTIPLY THE GIVEN NOTE THAT E-XX MEANS 1 0 TO THE -XX POWER.

VALUE BY THE FACTOR OPPOSITE

THE GIVEN

UNITS

$

6

E 6

CONVERSION FACTORS

-

FLOW LITER LITER LITER PER SEC PER MIN PER HR 9.9997 E 02 5.9998 E 04 3.5999 E 06 CU FT CU FT CU FT PER SEE PER MIN PER HR 3.5314 E 01 2.1189 E 03 1.2713 E 05 CU CY PER SEC 1.0000 E 06

DESIRED UNITS CU YETER CU METER PER SEC PER H R GIVEN UNITS CU METER PER SEC CU METER PER HR LITER PER SEC LITER PER'MIN LITER PER H R 1,6667 E-05 2.7779 E-07 2.8317 E-02 6.0002 E-02 1.0000 E-03 1.0194 E 02 1.0000 E 00 3.6000 E 03

1.6667 E-02 2.7778 E-04 2.8316 E 01

1.0000 E 00 1.6667 E-02 1.6990 E 03

6.0000 E 01

5.8859 E-04 9.8098 E-06 1,0000 E 00

3.5315 E-02 5.8859 E-04 6.0000 E 01

L.1189 E 00 3.5315 E-02 3.6000 E 03

1,6667 E 01 2.7779 Em01 2.8317 E 04

1,0000
E 00 1.0194 E 05

,

% z 8

CU PT PER SEC

z

CU PT PER MIN CU FT PER H R CU C M PER SEC 7.8658 E-06 2.8317 E-02 3.6000 E-03 7.8656 E-03 9.9997 E-04 4.7194 E-01 5.9998 E-02 2.8316 E 01 3.5999 E 00 2.7778 E-04 3.5314 E-05 1.6667 E-02 2.1189 E-03 1,0000 E 00 1.2713 E-01 7.8658 E 00 1.0000 E 00

ij
W m

1,0000
E-06

! I

3 2
M

B *

TO CONVERT A VALUE FROM A GIVEN UNIT T O A DESIRED U N I T S MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS A Y O BENEATH THE DESIRED UNIT. NOTE THAT E-XX MEANS 10 TO THE -XX POWER.

c .
CONVERSION FACTORS

..

-

CONCENTRATIONv DENSITY

M PER G MlCROGRAM MICROGRAM GRAIN PER OUNCE PER L B PER DESIRED U N I T S GRAM PER GRAM PER L B PER CU METER CU METER PER CU M PER L I T E R CU FT CU FT CU FT CU FT CU METER GIVEN U N I T S GRAM PER CU METER M PER G CU METER MICROGRAM PER CU M M l CROGRAM PER L I T E R 9.9997 Em04 9.9997 E-01 9.9997 E 02 1.0000 E 00 4.3699 E-04 9.9883 E-07 6.2427 E-08 L.8316 E-05 2.2046 Em06 1.0000 E-03 1.0000 E 00 1.0000 E 03 1.0000 E 00 4.3700 E-04 9.9885 Em07

6.2428 Em08

L.8317 Em05

2.2016 E-06

OUNCE PER CU FT
L B PER CU FT

1.6018 E 04

1.6018 E 07

1.6018 E 10

1.6019 E 07

7.0000 E 03

1.6000 E 01

1.0000 E 00

4.5359 E 02

3.5314 E 01

LB PER CU METER

4.5359 E 02

4.5359 E 05

4.5359 E 08

4.5360 E 05

1.9822 E 02

4.5307 E-01

2.8317 E-02

1.2844 E 01

1,0000 E 00

TO CONVERT A VALUE FROM A GIVEN U N I T TO A OESIRED U N I T * MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN U N I T S A Y D BENFATH THE DFSIRED UNIT. NOTE THAT E-XX MEANS 1 0 TO THE -XX POWER.

.. .. ..
- 0 N I

r-m

m

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W

N . .W

r-0 r-0

m

C O N B .w

I-N

r-m U O d l d

m

0

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0

m o no m
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0

m

n m no
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0

0

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40

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mm

00

m

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no o .W m

dd

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2 w.15 w w a a=

ATMOSPHERIC DISPERSION ESTIMATES

CONVERSION FACTORS

-

PRESSURE BAR ATMOSPHERE DYNES KG LBS M MERCURY I N MERCURY M PER SQ CM PER SQ CM PER SQ I N 9.8692 €104 1.0000 E 03 1.0197 €103 1.4504 ErO2 7.5006 Em01 2.9530 E-02

DESIRED UNITS MILLIBAR GIVEN UNITS MILLIBAR 1.0000 E 00

1.0000 E-03

BAR

ATMOSPHERE

3.0133 E 03

1.0133 E 00

1.0000 E 00

1.0133 E 06

1.0332 E 00

1.4696 E 01

7.6000 E 02

1.9921 E 01

OYNES , PER 90 C M KG M PER S0 C LBS PER S I N O M MERCURY M 6.8947 E 01 6.8947 E-02 6.8046 E-02 6.8947 E 04 7.0307 E-02 1.0000 E 00 5.1715 E 01 2.0360 E 00

TO CONVERT A VALUE FROM A GlVEN UNIT TO A DESIRED UNIT, MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS AN0 BENEATH THE OESIRED UNIT. NOTE THAT E-XX MEANS 1 0 TO THE -XX POWER.

CONVERSION FACTORS

-

TlME HOUR WEEK MONTH ( 2 8 ) MONTH (301 MONTH (311 YEAR ( 3 6 5 ) YEAR 13661

DESIRED UNlTS SECOND GlVEN UNlTS SECOND

WEEK

MONTH 128)

4.1336 E-07

,

MONTH ( 3 0 )

%

z
E
rn

MONTH ( 3 1 )

3.7336 E-07 3.1710 E-08 3.1623 E-08

2.2401 E-05 1.9026 Em06 1.8974 E-06

1.3441 E-03 1.1416 E-04 1.1384 Em04

2.2981 E-01 1.9178 E-02 1.9126 E-02

9.0323 E-01 7.6732 E-02 7.6503 E-02

9.6774 Em01 8.2192 Em02 8.1967 Em02

1.0000 E 00 8.4932 E-02 8.4699 Em02

!el774 E 01 1.0000 E 00 9.9727 E.01

1.1806 E 01 1.0027 E 00 1.0000 E 00

YEAR 1365)

5

g
M V1

! i

YEAR ( 3 6 6 )

T CONVERT A VALUE FROM A GlVEN UNIT TO A DESIRED UNIT* MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GlVEN UNlTS O AN0 BENEATH THE DESlRED UNIT. NOTE THAT E-XX MEANS 10 TO THE -XX POWER.

Appendix 4

2
7

0

3
0 00 O D d or-

m
4

m

00 O I d oc

0

0

0

00 00

..

ao
O D 0 d m

Y)

W

4

. . d no (DO
4 W

W

ATMOSPHERIC DISPERSION ESTIMATES

CONVERSION FACTORS

-

ENERGY PER U N I T AREA BTU I N T KW-HR CAL ( 1 5 ) M PER SO FT PER SQ M PER SQ C ASS JOULES PER SO C M

DESIRED UNITS LANGLEY GlVEN UNITS

CPL 1151 PER SO C)r

'BTU
PER SO F T I N T KW-HR PER SO M ABS JOULES PER SO C M

2.7133 E-01 8.6029 E 01

2.7133 E-01 8.6029 E 01

1.0000 E 00 3.1706 E 02

3.1540 E-03

1.1357 E 00 3.6007 E 02

1 .OOOO E 00

TO CONVERT A VALUE FROM A GIVEN UNIT 7 0 A DESIRED UNIT. MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS AND BENEATH THE DESIRED UNIT. NOTE THAT E-xx MEANS 10 TO THE -XX POWER.

CONVERSION FACTORS

-

POWER PER UNIT AREA

(CAL ARE 15 OEGI

DESIRED UNITS CAL PER SO CAL PER 5Q LANGLEY CAL PER 5 0 BTU PER SP BTU PER S RBS WATT O M PER MIN C PER DAY FT PER MIN FT PER DAY PER S CY M O M PER SEC C PER MIN GtVEN UNlTS CAL PER 5 0 M PER SEC CAL PER S O C PER MIN M LANGLEY PER MIN CAL PER SO C PER DAY M BTU PER S O FT PER MIN BTU PER SO FT PER DAY
A B S WATT M PER SO C

1.6667 E 02

1.0000 E 00

1.0000 E 00

1.4400 E 03

3.6855 E 00

5.3071 E 03

6.9758 E-02

1.1574 E-01 4.5222 E 01

6.9444 E-04 2.7133 Em01

6.9444 E-04 2.7133 E-01

1.0060 E 00 3.9072 E 02

2.5594 E-03 1.0000 E 00

3.6855 E 00 1.4400 E 03

4.8443 E-05 1.8928 E-02

2.3892 E 03

1.4335 E 01

1.4335 E 01

2.0643 E 04

5.2833 E 01

7.6079 E 04

1.0000 E 00

T CONVERT A VALUE FROM A GIVEN UNIT 1 0 A DESIRED O UNIT* MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS AN0 BENEATH THE DESIRE0 UNIT. NOTE THAT E-XX MEANS 10 TO THE - X X POWER.