Large particle diffusion from an elevated line source - a comparative evaluation of a theoretical model with field diffusion experiments

Agricultural Meteorology, 12(1973) 4 2 5 - - 4 3 9 © Elsevier Scientific Publishing C o m p a n y , A m s t e r d a m -- Printed in The Netherlands

L A R G E P A R T I C L E D I F F U S I O N F R O M A N E L E V A T E D L I N E S O U R C E -A C O M P A R A T I V E E V A L U A T I O N O F A T H E O R E T I C A L M O D E L WITH FIELD DIFFUSION EXPERIMENTS

WILLIAM A. S P R I G G

Office of Environmental Modification, National Oceanic and Atmospheric Administration, Rockville, Md. (U.S.A.) (Accepted for publication October 1, 1973)

ABSTRACT Sprigg, W. A., 1973. Large particle diffusion f r o m an elevated line source -- a comparative evaluation o f a theoretical model with field diffusion experiments. Agric. Meteorol., 12: 425--439. A theoretical model of atmospheric diffusion of a polydispersed material from an elevated line source is used to predict d o w n w i n d deposition of large particles (nominally 100 p diameter) released during six separate field diffusion experiments. T w o equations are used. One, where diffusion is d e p e n d e n t on the distribution of particles as advected in a steady-state condition. The second includes factors to account for atmospheric turbulence and diffusion. When the correct e q u a t i o n is chosen for a given turbulence condition, in all but two of the diffusion trials the m o d e l is within 5 m of predicting the p o i n t of m a x i m u m deposition; in all six trials the greatest discrepancy is 15 m. The m o d e l is reasonably capable o f predicting values of d o w n w i n d deposition. Wind profile fitting terms are shown to be m o s t accurate under thermally stable atmospheric conditions. INTRODUCTION

The problem of predicting downwind concentrations of airborne released material is c o m p o u n d e d by the great variety of possible sources. For example, gases and various size particulates will diffuse quite differently under identical atmospheric conditions because of differences in fall velocities. Also, it is obvious that a line source and a point source will contribute differently to effluent deposition at a given point downwind from the source, even though identical quantities may be released from each source. This report presents the results of a test of a theoretical diffusion model's ability to predict the downwind deposition of a polydispersed material from an elevated line source under varied atmospheric stability conditions. A solution to the line source t y p e of problem finds practical applicability in emissions from aircraft operations and surface transportation, such as highways. Accurate prediction of downwind deposition of material released in aerial crop spraying operations is necessary, and the elevated line source

426 diffusion problem is ideally formulated. The size particle treated in this study, approximately 100 p, is especially pertinent in aerial crop spraying operations ( Boving et al., 1971). DISCUSSION One a t t e m p t at describing atmospheric diffusion from the line source is based upon the K-theory model, where diffusion takes place under the influence of a gradient of concentration of the diffusing material: dx/dt = V . K m V x

(1)

The proportionality factor, Kin, is defined as eddy diffusivity. Development of the line source diffusion model by Calder (1961), Rounds (1955), and Smith (1957), among others, consists of solutions to eq.1 which attempt to eliminate constricting assumptions for specific practical applications. If we assume an infinite line source of diffusing material oriented along the y-axis of a cartesian coordinate system at x -- 0 and at specified height, z, eq.1 becomes: d x / d t = ~x

-~1

+ -~

If we further stipulate a steady-state source strength, air movement restricted to the x-direction, and assume the contribution to diffusion in the x-direction by velocity of air flow is much greater than the contribution due to eddy motion, the diffusion equation becomes: ~x

~z

Mean wind velocity in the x-direction is designated by ~. One method toward improvement of the particulate diffusion solution has been to describe the release characteristics more adequately. Davidson and Herbach (1964) have extended a solution of eq.2 by Rounds (1955) to include the effects of a polydispersed cloud, with material having significant terminal velocities. The Davidson and Herbach solution may be written: D/Q = P

exp(--F/X)F(1 + P , ) (F*/x)a +p*

(3)

Where F denotes the gamma function and P and F are defined as: p = (1 + Po ) exp(¢ + u2/2) + Po exp[2(¢ + u2)] [1 + Po + exp(¢ + u2)] 2 F

F(1 +Po) 1 + Po + exp(¢ + u 2)

The mean value of the logarithm of terminal velocities of the dispersed material

427 is p; the logarithmic standard deviation of particle terminal velocities is v. F, Po, and ¢ may be determined from: F = h~(h)/[(1 + a ) 2 k u . ] Po/(1 + Po) 1 F(1 +Po) [(1 +Po)/e] l÷P° - v(27r)l/2 exp(--v2/2) ¢=p--ln~7 Where ~ = (1 + cOku. The effects of surface roughness are approximated by inclusion of the friction velocity, u., and the roughness parameter, Zo. The wind profile fitting term, a, may be approximated from eq.4: ~(z) .~ (u. / k ) ( z / h ) ~ l n ( h / z o )

(4)

Estimates of u. and Zo may be taken from other empirical studies; for example, see Hage (1961a), Priestley (1959), and Sutton (1953). Eq.4 may be derived from Calder's (1949) wind profile eq.5, and the familiar logarithmic wind law eq.6. The approximation in eq.4 enters because q, a fitting term in eq.5, is assumed independel~t of height. ~(z) = u, q(z/z o )c,

(5)

~(z) = ( u , / k ) In (Z/Zo)

(6)

The unique assumption made in defining the solution, eq.3, is that particle terminal velocities are log-normally distributed with respect to mass. If this assumption is true we may consider the case where diffusion is dependent, not upon atmospheric diffusion, but rather on the spread or distribution of particles as they are advected in a steady-state condition downstream from the source. Davidson and Herbach consider that the distribution of mass is a function of terminal velocity and the following relation holds: 0j

Q(Vs)dV~= Q

If there exists a log-normal distribution of terminal velocity with respect to mass, the probability curve has the form: Q(V~) = {Q/[vVs(2zr) ~ ]} e x p [ - ( l n V~ -- p)2/2v2] For a continuous line source and the assumption of a log-normal distribution of terminal velocity, Davidson and Herbach (1964) derive the following deposition function: D / Q = [1/vx(27r)~] exp {--[ln(F/x)--c~] 2/2v2} Where F, ¢, p and v are defined as in eq.3.

(7)

428 OBJECTIVE A series of experiments conducted at Suffield Experimental Station, Ralston, Alberta, Canada, have been selected to test the applicability of eq.3 and 7 of Davidson and Herbach (1964) in predicting downwind deposition of large particulates from a continuous elevated source. These experiments have been well documented by Hage et al. (1960a, b), Hage (1961a, b) and Walker

(1965). Discussion of experiment During 1959 and 1960 several field trials were conducted over a gently rolling prairie, devoid of obstacles which might unduly influence the natural wind flow. Solid glass spheres were released from a continuous point source at 15 m above ground. The glass sphere deposition density and crosswind integrated deposits were determined by particle counts on horizontal flatplate sampling surfaces which were placed according to a grid arrayed in concentric arcs at specified distances from the source, all at ground level. Emission data is provided in Table I. TABLEI Emission data for diffusion trials 1--6 of Suffield Experimental Station; release height 15 m Trial

1

2

3

4

5

6

Date (mo-day-yr) Starting time (MST) Duration (min) Amount emitted (gm) Source strength (gin sec-1)

2/26/59

3/15/60

1/9/59

1/9/59

9/1/59

11/27/59

19h50 30

18h51 60

llh20 60

15h30 44

11h27 49

12h25 60

152.7

221.8

427.5

497.0

261.2

138.5

.08483

.06161

.011875

.18826

.08884

.03847

The glass spheres were predominantly spherical in shape. However, some irregularities were noted; most of the irregular spheres were pear-shaped or elongated. A fluorescent dye was used in coating the glass spheres to facilitate easier visual counting under ultraviolet light. Meteorological measurements were carried out by using a portable tower located 30 to 50 yards from the emission source in a direction normal to the expected wind. Sheppard-Casella cup anemometers measured wind speeds, while shielded copper-constantan thermocouples measured vertical temperature gradients between 8 m and 0.5, 1.0, 2.0 and 4.0 m each minute. A bivane m o u n t e d at the height of emission measured vertical wind fluctuations. Wind data is given in Table II. Additional meteorological data is provided in Table III.

429 T A B L E II Wind speeds ( m sec -1) at selected heights (m) d u r i n g diffusion trials 1--6 Height (m)

0.5 1.0 2.0 4.0 8.0 16.0

Trials: 1

2

3

4

5

6

2.39 2.92 3.65 4.69 6.02 7.65

3.55 4.20 4.91 5.69 6.62 7.64

3.30 4.05 4.81 5.64 6.50 7.39

2.86 3.51 4.20 5.02 5.94 6.96

5.14 6.33 7.30 8.09 8.72 9.14

2.85 3.48 4.08 4.65 5.19 5.65

T A B L E III Selected m e t e o r o l o g i c a l data f r o m m e a s u r e m e n t s d u r i n g diffusion trials 1--6 o f Suffield Experimental Station* Trial

(T4m -- To.5m ) ATe°C)

u, ( m sec -1)

zo (cm)

a

St. dev. vert. vane angle fluct.

oz (**) 1 2 3 4 5 6

1.1 0.3 0.1 0.4 --0.8 --1.4

0.52 0.45 0.45 0.45 0.45 0.37

4.3 1.9 2.3 3.3 0.5 3.3

.37 .20 .18 .24 .11 .14

.009 .012 .028 .036 .057 .047

* F r i c t i o n velocity, u . , r o u g h n e s s p a r a m e t e r , Zo, a n d t h e wind profile fitting t e r m , a, are d e f i n e d u n d e r t h e Discussion-section. ** In radians ( 2 0 sec t i m e average).

Procedure

The basic feature of eq.3 as derived by Davidson and Herbach (1964) is the lognormal distribution in particle terminal velocity. The assumption of a log-normal distribution in size of nonuniform particles has been documented as valid by Hatch and Choate (1929) and again by Smith and Jordan (1964). However, extending the assumption to a log-normal distribution in terminal velocity is questionable. Therefore, it is desirable to test the actual distribution of terminal velocities of the particles used in the experiment. If the terminal velocity distribution is not log-normal, use of eq.3 or 7 would not be condoned. From the given distribution of particle size with mass, terminal velocities were computed for selected mass percentiles (Table IV). Fig.1 shows a log-normal probability plot of the results. It is evident from the graph that

43O T A B L E IV

Particle cumulative mass-diameter-terminal velocity distribution Mass (%):

Diameter (p) Terminal velocity (m see -l )

0.1

0.2

0.4

0.9

4.3

13,4

22.2

33.5

45.3

75

80

85

90

95

100

102

104

106

.342

.368

.406

.446

.486

.530

.549

.564

.581

Mass (%): Diameter (p) Terminal velocity (m sec -l)

1.0

[

l

I

57.0

67.9

77.8

86.9

95.5

98.3

99.3

99.7

99.9

108

110

112

115

120

125

130

135

140

.598

.624

.632

.655

.700

.740

.786

.833

.881

i

1

] l a f l l r

i

I

i

I

i

O

0.8

O

i

O

(Y

O

0.6

0.4

0

0

0

0 ._1 i..i > /

0.2

Z

w o,

o12 o15 t

A

I

]

I

2

5

~0

20

I

I

50 40

MASS

I

I

50 60

I

i

1

I

70

80

90

95

i

t,,,,I,

98 99

i

L J

99.9

(PERCENT)

Fig.1. Mass--terminal velocity distribution of glass spheres emitted during diffusion trials 1--6 of Suffield Experimental Station.

the assumption of a log-normal distribution of terminal velocity is valid for this experiment. Due to the large size of particles being released during each trial, computation of terminal velocities could not be accomplished by Stokes' Law. When dealing with particles of diameter greater than approximately 80 p substantial errors evolve if Stokes' Law is invoked. This has been documented by Langmuir (1961), Drinker and Hatch {1954), and Johnson (1954). At large particle sizes sufficient account is not made by Stokes' Law of drag forces, which for large particles increases as (rV~) 2 instead of (rV~), as is the case for smaller particles. The equation for determining particle terminal velocity, as given by Johnson (1954) is: V~ = [2r2g(pD - - PA)/9~] ( 2 4 / C D R e )

(8)

431 Where the terms PD, PA, r, and g are, respectively, droplet density, air density, droplet radius, and acceleration of gravity. Stokes' Law is defined for the case where the term 24/CDRe approaches unity. Since Stokes' Law cannot be invoked, eq.8 must be evaluated. This presents some difficulty in that the Reynolds number, Re, is a function of the terminal velocity, and the drag coefficient, CD, is a function of the Reynolds number. A trial-and-error method in evaluation of eq.8 is the usual means of solution. A less cumbersome technique has been developed by McDonald (1960) in which he plots a curve relating CDRe 2 as a function of Re. By computing the value of CDRe 2 and consulting the plot, the corresponding Reynolds number may be determined. Then, from the following equation: Vs = pRe/2pAr

the terminal velocity may be computed. Fig.2 is a plot of CDRe 2 against R e for the range of values encountered in the present problem. Table IV also lists the computed terminal velocities for selected cumulative mass distributions. 10

I

8 6

d

Re

2

,

, I0

20

o/,

, 40

,

,

L ,,I

60

80 I00

I 200

~

I

I

400

CDRe z

Fig.2. CDRe 2 versus Re; w h e r e C D a n d Re are respectively drag c o e f f i c i e n t a n d R e y n o l d s number.

The mass mean diameter of the glass spheres used in the experiment was given by Hage (1961a) as 106.7 p. The logarithm of the corresponding terminal velocity gives another of the values necessary to evaluate eq.3 and 7. Thus, p, the mean value of the logarithm of the terminal velocities was determined to be approximately --0.53. A value of 0.19 was used in the calculations for the logarithmic standard deviation of particle terminal velocities. Evaluation of eq.3 also requires knowledge of the values attained by Zo, u. and ~ in each of the six trials investigated. The values of Zo corresponding to each experiment were calculated by Hage (1961a) for the Suffield terrain

432

and are reported in Table III. These values are in agreement with those published by Sutton (1953) and Priestley (1959) for similar terrain. Values of u, were calculated from the logarithmic wind law, eq.6, using the previously established values of Zo and z = 16 m. The value assigned Von Karman's constant, k, ranges from 0.38 to 0.45 according to various authors. See, for example, Hess {1959} and Rounds (1955). Table V illustrates TABLE V Calculated values of friction velocity, u, (m sec-]), assuming two extreme values of Von Karman's constant, k, for diffusion trials 1--6

Ifk=.38 Ifk=.45

Trial: 1

2

3

4

5

6

.49 .58

.43 .51

.43 .51

.43 .51

.43 .51

.35 .41

...... /,,

16.0

/

8.0

I-"1" (.9

..... /,,, ..... /,

/

/

4.0

~ 2.o

J

"T

/

.

~.0

i

0.5

i

i

i

2

i

i

4

i

6

i

i

i

8

TRIAL

~

4.0

z

2.0

,

4

I

I

6

I

i

8

2

ill

i

i

4

i

i

6

8

SPEED ( M / S E C ) TRIAL

I

/ 8.0

i

2

WIND A

'

J

2

TRIAL

....... / i

]

3

/

•

o

/

,o

]

0,5

J 1 2

J L h ~ J ~ , 4 6 8

' L I J I / i/ 2 4 6

WIND B

TRIAL

4

t I

I 8

I

' 2

4

= ~ , 6 8

F

SPEED ( M / S E C ) TRIAL

5

TRIAL

6

Fig.3. Vertical wind profiles, observed (dashed line) and calculated (solid line) for diffusion trials, A. 1--3; and B. 4--6.

433 the range of values taken by u, which would be calculated from eq.6 using the given extreme values of k. For the purpose of this report a value of k = 0.40 has been assumed for all computations. The values of u, calculated by the author, and given in Table III, are in satisfactory agreement with those calculated by Hage (1961a}. Calculations of a were made from eq.4. Observed and theoretical wind profiles were matched at height z = 8 m for all trials except trial 5, which was matched at z = 15 m, for computation of a, and are shown in Fig.3.

I

I

I

2O

TRIAL

I

18 ~16

/"/\'.i\'I,

I-

~ ~o

~a -

6

..I w I1:

4

//

2 O 0

/

Z' I00

200 DOWNWIND

DISTANCE

300

400

(M)

Fig.4. Trial 1 relative deposition density curves: observed (solid line); predicted by eq.3 (dashed line); predicted by eq.7 (broken dashed line).

A

TRIAL

/

,'7" /

~s o

2

\,

\,

\

6

~2 0

IO0

200

300

40(

DISTANCE F)OWNWlND (M)

Fig.5. Trial 2 relative deposition density curves: observed (solid line); predicted by eq.3 (dashed line); predicted by eq.7 (broken dashed line).

434 I

I

1

I

A ~E

TRIAL

/ /'

12 I-

~8

/

~6

~,.

~4 ,.",2 0

\ \ L

\ \,

.//

,' /

0

/

3

"

/I

~

I 200

I00 DISTANCE

-

"~-,

DOWNWIND

-

~

~

J 300

400

(M)

Fig.6. Trial 3 relative deposition density curves: observed (solid line); predicted by eq.3 (dashed line); predicted by eq.7 (broken dashed line). I

I

I

P~

TRIAL

/\

i

~,o ~8 a

/

'\\

4

\

J

6

..-1.2 e: 0

.'/./'1/,

~-,,

,oo

. . . . . . .

~oo

o,s'r..~

oow~w,.o

~oo

.oo

(,,)

Fig.7. Trial 4 relative deposition density curves: observed (solid line); predicted by eq.3 (dashed line); predicted by eq.7 (broken dashed line). I

I

I

I

TRIAL

~D2

5

~E z o

I0

~-

8

0 o.

6

w >

4

•

\

/

\

/

\

~2 _J

~o 0

oo

2oo D ST,NCE

3oo

,oo

D0WNW,ND

Fig.8. Trial 5 relative deposition density curves: observed (solid line); predicted by eq.3 (dashed line); predicted by eq.7 (broken dashed line).

435 I

I

{

~18

f',,

TRIAL

6

/ \

i

zl2 I.-

\

I0

i\

o m ~3 .) I.•~ ..J

S

!

•

6

,J /

4

"-\.

~2 0 0

I00

200 DISTANCE

DOWNWIND

300

400

(M)

Fig.9. Trial 6 relative deposition density curves: observed (solid line); predicted by eq.3 (dashed line); predicted by eq.7 (broken dashed line).

With the aforementioned data, the Davidson and Herbach eq.3 and 7 were evaluated {~or relative deposition, DR = D / Q , at selected distances downwind from the source. These results are compared in Fig.4--9 with observed relative crosswind integrated deposits as reported by Hage et al., 1960a. ANALYSIS OF RESULTS

Frequent reference to meteorological variables measured during the six trials will be helpful in understanding the discussion which follows. These variables are presented in Tables I and III. Table VI gives a summary of relative TABLE VI Comparison of departures of location and value of maximum relative deposition as calculated by Davidson and Herbach eq.3 and 7 observed crosswind-integrated deposits for diffusion trials 1--6 Trial

1 2 3 4 5 6

Approximate distance (m) of point of maximum observed deposition from point of predicted maximum by: eq.3 eq.7

Percent deviation from observed maximum deposit of predicted maximum deposit according to:

--45 --30 --5 --I0 0 0

--75% --57% --29% --32% --39% --37%

0 +15 +35 +20 +50 +25

eq.3

eq.7 -26% +20% +100% +100% +69% +66%

436 discrepancies between selected, observed and predicted values of deposition according to the Davidson and Herbach eq.3 and 7. Trials 1 and 2 show the greatest discrepancy between observed and predicted deposits according to eq.3. Values of peak deposition are severely underestimated. These two trials are unique in that they have the smallest vertical vane angle standard deviations, az, when compared to the remaining trials. Note that stability conditions are essentially the same in trials 2 and 4, but apparently, in the case of trial 4, vertical wind fluctuations overrode the stability effect indicated by the lapse rate, AT. The greater variability of the vertical wind dynamically typifies a more turbulent atmosphere, and thus disperses effluent more efficiently through eddy transport. This reasoning is verified by comparing observed deposition in Fig.5 and 7, representing trials 2 and 4, respectively. Note that the observed deposition curve of trial 4 exhibits less peakedness. Trial 4, then, is better classified with trials where more balanced atmospheric conditions prevail. A balanced atmosphere shall be defined as one which approaches the effect of a neutral atmosphere, but takes into account the degree of vertical and horizontal wind fluctuations as well as thermal stability. Trials 1 and 2 are better predicted by eq.7, which neglects atmospheric diffusion. Vane angle fluctuations and given stable lapse rates indicate minimal atmospheric diffusion characteristics. Under the strong inversion conditions of trial 1, peak deposition is underestimated by eq.7. Note that this is the only trial where observed concentrations exceed those predicted by eq.7. This may be due to a subsidence effect, depositing more particles than expected within a narrower region. Observed deposits of trial 2 are more closely represented by eq,7. The remainder of the trials are poorly forecast by eq.7 due to overestimation of either value or point of maximum deposition. Predicted deposits by eq.3 in trial 3 were handled with a considerably greater degree of confidence. This case is near neutral {slightly stable) but again, vertical vane angle fluctuations indicate a degree of turbulence. The combination o f effects represents the ability of the atmosphere to transport mass. Depositions in trials 3 and 4 show a similar pattern indicating that small differences in thermal stability and vertical vane angle fluctuations apparently do not have great practical significance. Trial 5 is unique with respect to the other trials in that wind speeds are much greater. This tends to spread the particles further downstream, move the point of maximum deposition further from the source, and smooth the deposition curve, making it less peaked. The onset of significant deposition and the point of maximum deposit is well forecasted by eq.3. However, the value of maximum deposition is underestimated and is not well accounted for. Trial 6 is the most unstable case represented. The lapse rate is superadiabatic and vertical vane angle fluctuations are second only to trial 5. Although the point of maximum deposition was well accounted for by eq.3, the value of maximum deposit was again underestimated.

437 In each trial, eq.3 predicts too low a value for maximum deposition. With the exception of trial 5, where high wind speed is adequately handled by the equation, eq.3 predicts the onset of significant deposition too close to the source. Both of these errors may be due to the exceptionally large particles emitted during the experiments, which cannot adequately be handled by the equation. There is better agreement between eq.3 and the value of observed deposition as a balanced atmosphere is approached. Trials where extremes of thermal stability and instability occur fail to predict the deposition amount, even though under thermally unstable conditions the point of maximum deposition is well accounted for. In addition, the point of maximum deposition is predicted too close to the source as thermal stability increases. The aforementioned errors involve the stability of the atmosphere, usually accounted for in diffusion equations by the addition of a term such as Deacon's fi (Deacon, 1949). The ~ used by Davidson and Herbach is a fitting term for the wind profile just as ~ is. However, fi may take on negative as well as positive values, according to atmospheric stability. For a comparison of predicted versus observed wind profiles see Fig.3. It is evident that the concept of a balanced atmosphere is not accounted for by eq.3 or 7. Errors in prediction by eq.7 are inherent in the equation from the basis of its derivation. Since atmospheric diffusion effects are neglected, one should not expect total agreement with observed results. The fact that agreement does occur, especially so well in trial 2, is indicative of the dominating influence of particle size and its distribution, over atmospheric diffusion properties such as stability, and eddy diffusivity. Apparently wind speed is the primary atmospheric element that influences the shape of the corresponding deposition curve. POSSIBLE SOURCES OF ERROR

Calculation of Zo is highly dependent upon the value of mean wind speed, which may, in itself, contain errors due to turbulence (Pries, 1970) or to being determined over a point rather than an area. Hage (1961a) compares some results of calculated z o when wind speed is varied. The computed value of u, has been pointed out as being dependent upon the value chosen for Von Karman's constant. These results may be seen in Table V. Due to a decrease of mass mean diameter of the particles from the source, individual values of the crosswind-integrated deposits may be in error. Since all mass deposits were calculated assuming a uniform sized particle those deposits close to the source are calculated too small, while those at greater distances are too large. Because of this error it has been estimated by Hage, Diehl and Dudley (1960a) that the true position of maximum deposit may be 5--10% closer to the source than is indicated. The value calculated for the standard deviation of the logarithm of

438 terminal velocities is larger than that which would be evaluated using only the range of 20--80% of the mass distribution as suggested by Drinker and Hatch {1954}. Inclusion of all data permits a reasonable value of Po to be calculated for evaluation of eq.3. Calculation of terminal velocities assumes a spherical particle. As reported by Hage et al. (1960a), about 85--90% of the glass beads used in the experiments were nearly spherical in shape. Those remaining introduce a slight error because their terminal velocities will not be the same as if they were spheres. ACKNOWLEDGEMENTS

The author wishes to express his gratitude to Dr. Samuel Jaffe for his assistance throughout the course of this study and to Professor William E. Reifsnyder for his critical review of the manuscript. A grant covering all computer costs was provided by the Computer Center, Rutgers -- the State University of New Jersey. REFERENCES Boving, P. A., Maksymiuk, B., Winterfield, R. G. and Orchard, R. D., 1971. Equipment needs for aerial application of microbial insecticides. Trans. Am. Soc. Agric. Eng., 14: 48--51. Calder, K. L., 1949. Eddy diffusion and evaporation in flow over aerodynamically smooth and rough surfaces: A treatment based on laboratory laws of turbulent flow with special reference to conditions in the lower atmosphere. Q. J. Mech. Appl. Math., 2: 153--176. Calder, K. L., 1961. Atmospheric diffusion of particulate material considered as a boundary value problem. J. Meteorol., 18: 413--416. Davidson, B. and Herbach, L., 1964. Two dimensional diffusion of a polydispersed cloud. J. Atmos. Sci., 21: 311--319. Deacon, E. L., 1949. Vertical diffusion in the lowest layers of the atmosphere. Q. J. R. Meteorol. Soc., 75: 89--103. Drinker, P. and Hatch, T., 1954. Industrial Dust, 2nd ed. McGraw-Hill, New York, N.Y., 401 pp. Hage, K. D., 1961a. On the dispersion of large particles from a 15-M source in the atmosphere. J. Meteorol., 18: 534--539. Hage, K. D., 1961b. Description and test of a model for the deposit of particulates from an elevated source. Defense Res. Board Can., Suffield Exp. Sta., Ralston, Alta., Suffield Tech. Pap., 2 1 7 : 1 9 pp. Hage, K. D., Diehl, C. H. H. and Dudley, M., 1960a. On the ground deposit of particles emitted from a continuous elevated point source, Part I: Deposit of nominal 100 p glass microspheres from 15 meters. Defense Res. Board Can., Suffield Exp. Sta., Ralston, Alta., Suffield Tech. Pap., 1 9 7 : 2 0 pp. Hage, K. D., Diehl, C. H. H. and Dudley, M., 1960b. On horizontal flat-plate sampling of solid particles in the atmosphere. Am. Med. Assoc. Arch. Ind. Health., 2 1 : 1 2 4 - - 1 3 1 . Hatch, T. and Choate, S. P., 1929. Statistical description of the size properties of nonuniform particulate substances. J. Franklin Inst., 207: 369--387. Hess, S. L., 1959. Introduction to Theoretical Meteorology, Holt Rinehart and Winston, New York, N.Y., 362 pp. Johnson, J. C., 1954. Physical Meteorology. Wiley, New York, N.Y., 393 pp.

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