Dissemination of Aerosol Particles Dispersed from Stacks THOMAS BARON, E. R. CERHARD, AND H. F. JOHNSTONE University of Illinois, Urbana, Ill.
L
A n approximate method has been developed for the estimation of deposition of aerosol particles from point sources based on the statistical methods proposed by Sutton. I t is assumed that concentration profiles above the ground may be expressed as the sum of two profiles, one resulting from diffusion from the source and the other from diffusion from a n image of the source below the ground. Sutton's equations are modified to take into account the finite settling velocity of aerosol particles. The rate of deposition is controlled by the concentration adjacent to the ground and by the true settling rate through the stagnant layer. The results are presented in the form of graphs showing the fraction deposited as a function of distance from the source, height of the source, diameter of the particle, and atmospheric conditions.
the height above ground level. Their equation for the diffusion of gases from a continuous point source is h
w2
I n view of the assumption concerning the effective diffusivity, this equation is at best semiempirical. A more fundamental approach is given by Sutton, who based his work on the statistical theory of turbulence developed by Taylor (4). Sutton's fundamental assumption concerns the form of the correlation coefficient. Because of the available theoretical and empirical information concerning this coefficient, assumptions concerning its form can be made with a greater degree of certainty than assumptions about the functional dependence of the effective diffusivity on the position coordinates. Sutton's equation for diffusion of gases from a continuous point source is
T
HE theory of deposition of aerosols is of interest in connec-
tion with problems of atmospheric pollution and in the dissemination of insecticidal and chemical warfare agents. Although no exact theory of atmospheric diffusion exists, approximate equations for diffusion in the lower atmosphere have been developed by Bosanquet and Pearson (1)and by Sutton (3). The former authors have solved the differential equation for diffusion, assuming that the effective diffusivity is proportional to
The concentration on the ground ( z = 0) becomes
- - -hl_ _ _ v2
2w c0 = iTCuCsUX2-
cEzx2-'L
~ u z x z - ~
e
Distance from Source, 1000 Yards
Figure 1. Effect of Stack Height and Meteorological Conditions on Deposition of Aerosols Particle size 6 0 ~
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atmoiphcric turbulencp, such as lapse, neutral (zero temperature gradient), and inversion conditions, by appropriate use of the factor n. The value of n is determined ba fitting the theoretical
6). I4
-
6.12
n -
profile, u U I z2-n, t o the observed variation of wind velocity with height From this, 8utton found t h a t n is approxi0.10 mately ‘/5 f o y a normal lapse rate, s about ‘/4 for zero or small ternpera8 P for strong ture gradient, and about 0.08 inversion. t Experimental investigations under 0 P normal meteorological conditions corre006 sponding to a zero temperature gradient have shown t h a t the horizontal spreading coefficient Cz, is greater than the vertical coefficient C, near the surface n = 0.25 of the earth, For isotropic turbulence, however, at heights above 25 meters, C , equals 67 ., and decreases with height according to a n empirical law. Since the variation of the spreading coefficients with height is small, an average value Distance from Source, I000 Yards over the distanre z = 0 t o z = h may be assumed. Reasonable approximations of Figure 2. Effect of Atmospheric Conditions on Deposition of 5Op Particles the value of C, for lapse and inversion Stack height 100 yards. Wind velocity 5 xnilea per hour conditions relative t o those found OF neutral conditions are made bv Sutton. Bosanquet’s coefficient for vertical spreading, p , was obtained 111the present work, Sutton’s approach is modified for the purfrom a study of the variation of wind velocity with height. An pose of describing the diffusion and settling of aerosol clouds. An average value of 0.05 was assumed to be generally applicable, example of the application of this method to the dispersion of The value of the horizontal coefficient, q, was obtained from experiinsecticidal aerosols from continuous line sources on the ground ments on the mean deviation in the horizontal direction. This has been given by Johnstone, Winsche, and Smith (2% was taken a8 a constant value of 0.08.. Sutton shows, however, Equations 1 and 3 give results of the same general characterist h a t the value of his coefficient C, varies from 0.21 to 0.07 and tics, the essential difference being in the power of z. BosanquetPs t h a t C, varies from 0.12 to 0.07 for neutral conditions, dcpending equation is for average conditions, and average diffusion coefEa n the stack height. For lapse conditions, C , varies from 0.21 to cients are used. Sutton’s equations can be used for varying ~
-
s-
0Q4i-
-r\
Dlrtrnce born Source, 1000 Yards
Figure 3. Effects of Stack Height and Meteorological Conditions on Deposition of Aerosols Particle size 2 0 ~
. I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
November 1949
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0.12 and for large inversion it varies from 0.05 t o 0.035 for stack heights up to 100 meters. Bosanquet and Pearson were unable t o extend their mathematical treatment t o the diffusion of particles with a finite settling velocity. They suggest, however, the following reasonable modification of Equation l:
The only theoretical justification of this equation is that i t reduces to Equation 1 for negligible settling velocities. Near the surface of the earth there is an effective stagnant film in which the effect of turbulent diffusion is negligible and through which aerosol particles pass by free settling only. The rate of deposition of the aerosol is Distance from Source, 1000 Yards the product of the concentration of the aerosoI cloud adjacent t o the Figure 4. Effect of Particle Size on Deposition of Aerosols stagnant film and the free settling Stack height 100 yards. n = 0.17. Wind velocity 5 miles per hour COV,. velocity through this film-Le., Thus the fraction of the total cloud MODIFICATION OF SUTTON'S EQUATION FOR that is deposited per unit distance becomes in terms of Bosanquet's equation AEROSOLS U8
dg
=
t)"'
'
218
us( 1.78 __-
puzC'
(5)
px
+%/UP)
I
x
e
- -h
l
The boundary conditions of the problem are such t h a t the concentration profile in a gas cloud can be expressed as the sum of two profiles, one resulting from diffusion from the source, and the other from an image of the source below the ground which allows for reflection by the plane z = 0. Sutton's equation can be modified t o take into account the finite settling velocity of aerosol particles. The vertical distance of fall due t o this settling is superimposed on the vertical spreading of the cloud and is given
l
/
D
0
1
/
" I
2 Distance from Source, 1000 Yards
Figure 5. Effect of Particle Size on Ground Level Concentration at Center Line of Cloud Stack height 50 yards.
Lapse conditions.
Wind velocity 5 miles per hour
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! ! 2. U
lfathematically, this is equivalent to the translation of
the axis a t each point doFvnwind, so that
2'
=
2
$-
' U
2.
Also, to
allow for the deposition of the aerosol a t the plane z = 0, the strength of the image must be multiplied by a factor 01 which is a function of the fraction deposited a t any point downwind from the source. =2s a result of thrqe modifications, Equation 2 becomes
(7
=
ET HCgCcUX2 -
+ a?
e
-
and
As a first approximation it may be assumed that the concentration a t ground level is the same R S that given by Equation 3, -*here 01 = 1. Then,
T o obtain a second approximation, the strength of the image, a , may be determined from a material balance stating that the
undeposited fraction remains air-borne. Thus,
(9)
Substituting from Equation 6 and integrating for dy from t,o m we get
+
-m
COMPARISON OF EFFECTS ON DEPOSITION OF AEROSOLS
size, and In order to show the effectof stack height, atmospheric conditions on the deposition of aerosols, the following conditions were assumed: (1)a mean vvind velocity of 5 miles per hour, ( 2 ) Stokes' law settling for small spheres with a specific gravity of 1.0 in air a t 70" F. Values of the meteorological constant n were taken as 0.17, 0.25, and 0.60, respectively, for strong lapse, neutral, and strong inversion conditions. The spreading coefficients C, and C, corresponding t o these condi(6) tions were taken from Sutton'? paper. These extreme values were chosen to bracket the field in order to show the maximum effect of variation in meteorological conditions. It is realizcd that using a constant wind velocity mag not always be consistent with the meteorological conditions chosen. Furthermore, the surrounding topographical conditions may greatly influence the concentration profile and the rate of deposition. It should be emphasized, therefore, that the results nould hold only for certain ideal conditions. We are only interested here in making comparisons t o show the effect of stack height on the deposition of particles of varying sizes under varying atmospheric conditions. The calculations were made for two stack heights, 50 and 100 yards, and for particles 5 , 10, 20, and 50 microns in diameter. The results obtained for the deposition of aerosols from a point source are shown in Figures 1 to 4. The cumulative per cent deposited from 50 and 100 yard stacks for lapse, neutral, and inversion conditions for a cloud of 50-micron particles is shown in Figure 1. Atmospheric turbulence has considerable effect on the per cent deposited. Rfore deposition occurs close to the stack for the highly turbulent conditions (large lapse) than for inversion conditions. Approximately 30% of the cloud deposits for neutral and lapse conditions before any appreciable deposition occurs for inversion. The total per cent deposited, however, is greater for the less turbulent ronditions a t distances greater than 30 stack heights downwind. This follows because of the rapid spreading and dilution of the cloud for highly turbulent conditions. An important feature of these curves is t h a t for large inversion a cloud of 50-micron particles deposits as a whole within a narrow range for 50- and 100-yard stacks. For a 100yard stack a particle falling freely in 10) still air would reach the ground a t 3000 yards from the stack. Because of the vertical spreading of the cloud under inversion conditions, however, 50% of the cloud is deposited at this point. The slope of the curves for the cumulative per cent deposited gives the per cent deposited per yard. These curves are shown in Figure 2 for 50-micron particles from a 100-yard stack for varying conditions of turbulence and are compared with the value obtained from Bosanquet's equation. Whereas, for a gas cloud, the value of the maximum concentration is practically independent of the turbulence of the atmosphere, for aerosols, the maximum rate of deposition, expressed as per cent deposited per yard, increases considerably with decreasing turbulence. For 50micron particles the maximum deposition for inversion is approximately four times as large as for neutral conditions for the 100yard stack. This difference, however, is found to decrease with decreasing particle size, the ratio being only 2 for a cloud of 20micron particles. It is further observed t h a t with decreasing turbulence, the point of maximum deposition shifts farther downE
For a gas cloud 01 = 1. For aerosols it may have a positive or a negative value, depending on the rate of free settling in the atmosphere. On t h e right side of Equation 10, 01 may be removed from the integral, as i t is a function of x only. However, on the left side of the equation an average value of 01 between 0 and x must be used since the functional variation of 01 with x is not known. Dividing through by
orav.
CY
and assuming - = 1, a n approxiO1SV.
mate value of aav.can be obtained. This value of ehY.can be used in the left side of Equation 10 to calculate the cumulative fraction deposited as a function of x. The slope of this deposition curve gives the fraction deposited per unit distance. This can then be used t o determine the value of CY and the concentrations of the aerosol at any point x.
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O.OOE
Q m
ex
0.00:
t 3
2 0.001 s5
2 I
.-6 0.00: L
E
L
e
U
s5 a002 I
-!
w
s
8 0.001 (
I
2
3
4
5
6
7
8
9
io
Distance from Source, 1000 Yards
Figure 6. Ground Level Concentration at Center Line of CIoud Stack height 100 yards.
Lapse conditions.
Wind velocity 5 miles per hour
wind from the source. The distance of the point of maximum show the effect of the size of particles in the cloud on the condeposition from the source is slightly greater for the smaller parcentration under lapse condition for 50- and 100-yard stacks. ticle sizes. The results obtained from Bosanquet’s equation For the 50-yard stack, the maximum concentration of 50-micron agree closely with those calculated for lapse conditions but differ particles is greater than that for a gas cloud of the same strength considerably from those calculated for inversion. under the same conditions, but the maxima for the 5-, lo-, and Figure 3 shows the effect of height of discharge on the deposi20-micron particles are less than for the gas cloud. For the 100tion per yard for 20-micron particles under different atmospheric conditions. For both lapse and neutral conditions the value of the maximum deposition is halved by doubling the stack height for such a cloud. For lapse conditions, the maximum deposition occurs a t approximately 20 stack heights downwind; for neutral conditions it occurs a t 30 to 40 stack heights; and for inversion, a t 160 stack heights. Again, the results predicted by Bosanquet agree reasonably well with those from the more complete analysis for lapse conditions (although the rate of maximum deposition is somewhat lower), but the two equations differ considerably for neutral and inversion conditions. Figure 4 shows the effect of the size of particles in the cloud on the per cent deposited per yard for lapse conditions and for a 50-yard stack. Here, as expected, the per cent deposited per yard is greatly increased by increasing the particle size. Similar results were obtained for neutral conditions. Figures 5, 6, 7, and 8 show the Distance from Source, 1000 Yards aerosol concentrations a t ground Figure 7. Effect of Atmospheric Conditions on Ground Level Concentrations level a t the center line of the cloud at Center Line of Cloud ( z = 0, y = 0). Figures 5 and 6 Stack height 50 yards. Particle size 209
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I I I
0 ox)
I I I I
I
E
.
I
I I
,-
I
- 100 yd.stack
I
---
5 0 y d . stock
Distance f r o m s o u r c e ; 1000 y a r d s
Figure 8. Effect of Stack Height on Ground Level Concentration at Center Line of Cloud Particle size 20e
yard btack, however, the inaximurn value of the concentration lor both the 20- and 50-mirron particles i. greater than for the gas cloud, In the case of an aerosol, the concentration a t ground level a t any distance, on the one hand, tends to decrease because of deposition. On the other Rand, the effect of free settling in the air tends to increase the concentration at the giound The relative importance of thehe two effects determines whether the value of the maximum concentration of an aerosol thud a t ground level i~ greater or less Lhan t h a t for a gas cloud under similai conditions. Figure 7 shows the effect of atmoqpheric turbulence on the ground level concentration of 20-micron particles for a 50->a1d stack. The 5hape of the curves is similar to those for the rate o i deposition, since decreasing turbulence shifts the point of ma+ mum concenti ation downwind and increases its value. Figure 8 s h o ~ the s effect of stack height on the conceiltiations for 20-micron particles under vaiious iai mospheiic conditions. Increasing the stack height decreases the maximum concenl rarion a n d shifts its distance finifhrr downwind from the source
CONCLUSIONS
buttonPsequation for the diffusion of gas cIouds, 1% hen modified for aerosols, gives a reasonable picture of the deposition of aerosol particles. Values calculated from Bosanquel ’ s equation agrce closely with those from Sutton’s theory for lapse conditions, but the two methods differ considerably for neutral and inversion conditions, indicating that a n average value for the vertical ypreading coefficient predicts too low x rate of deposition of aerosol particles, Whereas the value of the inaximuni concentration of a gas cloud is independent of atmospheric turbulence, a decrease in turbulence greatly increases the maximum rate of deposition and the maximum concentration of an aerosol cloud ai, ground level. A decrease in turbulence shifts the point of maximum deposition and ground concentration downwind from the source. 4 n increase in stack height decrrasei: the rate of deposition and
g o u r d c oricentration, especially ai I LIP lnLtxmum values, and shifts the points of maximum deposjtion and Concentration downwind, This effect is greater for the srnallest particles and for the lowest atmospheric turbulence. For 8 gas cloud, the maximum concentration is inversely proportiopal to t h e square of the stack height, but for an aerosol cloud the niaxamum depositiori i s approwiinafrly inversely proportional to the ~ T R height, P ~ NOMENCLATURE L‘ = fi = Cy = C, = h, =
coriceiitra8tionof smoke or gab, mg. per cubic yard concentration a t ground level (P = O ) , mg. per cubic yat-d
horizontal spreading coefficient, ( y a ~ d s ) ~ / 2 vertical spreading coefficient, i y a , r d ~ ) ”a f stack height, yards II = a meteorological consta,nt, p = vert,ical diffusion coefficjenl o = horizontal diffiusion coefficient u mean wind velocity (assiiinrd j i i variant w i t h height), yards per second ob = settling velocity of aermol yar-cicie, p a r d s per second. W = strength of source, m g . per second ,cs y, z = space coordinates with orjgin at base af stack; z , C ~ O W Y I -wind; g , crosswind; and Y . 17 N correction coefficient for image 9 = fraclimi of rloud deposit,wl pry tiriii: dintarice 3
LITERATURE CITED (I) Hoaaiiquet, c‘. K., and ‘Pearson, .7. L . . ’I’iam. Paraday Soc., 32, 1.249 (1936)
I?) Johnstone, H. F., SVinscIw, W. lL. Smith, I,. S V ~ pChem. Revs., 44, 353 (1949). (3) Sutton, 0 . G., Quart. J . Roy. J f e t . Sor;., 73, 257, 426 (1947)* 14) Taylor, G o I., Proc. Lond. 1Mnth.. Soc.. 20, .I96 (1921). Rrrcxrvr:u Jima 3 0 , 1948. This p&yr;reonljaii~td p w t of t h e result8 of ~ I investigation on stack gases at t h e Dniversity o f Illinois Engineering Experiment Station. Previous pepem i n thts series have appeared in LND. ENR.CHEX., 27, 587, 659 (1935); 24, YE%, 3396 (1937); 80, IO1 t, 1 R3RJ : 32, 10.37 (1 940) ; 89, 808 (1947:.
J
