The deposition of dust particles from elevated sources and the vertical wind profile

Atmospheric

Lkironment

Pergamon

Press 1971. Vol. 5, pp. 3118.

Printed in Great Britain.

THE DEPOSITION OF DUST PARTICLES FROM ELEVATED SOURCES AND THE VERTICAL WIND PROFILE K. WOJCIECHOWSKI Instytut Fiilci

Doiwiadczalnej

Uniwersytetu Wroctawskiego, Poland

Wrocfaw, ul. Cybulskiego 36,

(First received 31 January 1970 and in final form 3 August 1970)

Abstract-The influence of the vertical wind profile on dust deposition from an elevated source is investigated. Some new formulae are compared with Sutton’s and Krieb’s formulae. From the comparison it appears that the formulae taking into account the vertical wind profile give a better picture of deposition. INTRODUCTION

THE derivation of formulae for dust deposition S(x,y,z) and for dust concentration C(x,y,z) is one of the main problems concerning air pollution. This problem was investigated by many authors and was first considered by SCHMIDT (1925). The following methods are used by different authors: (i) method based on SUTTON’S (1947) statistical theory and on Schmidt’s hypothesis which says that the deposition of dust particles is influenced by two separate concurrent processes of diffusion and settling; (ii) direct solution of proper diffusion equation (analytically or numerically) Rou>ms (1955), ARRAGOand SHVETS (1963); (“‘) 111 semiempirical BOSANQUET, CAREY and HALTON(1950). This paper deals with first method. Now we consider in some detail the basic assumptions which are used by the authors mentioned above in order to obtain the formulas for dust deposition. The authors, BARON, GEFLHARDand JOHMTONE (1949), KRIEB (1955), Meteorology and Atomic Energy (1955), CSANADY (1958), CLARENEIURG (1960) assumed that due to gravity the axis of smoke cloud has an inclination a to the horizontal

tga=U

U'

(1)

where v is the fall velocity of particles and II is the mean velocity of the wind at the emitter height h. With this assumption, the trajectory of a particle with diameter 4 settling with velocity ~(4) is represented by the straight line H(x) = h -

479 x, u(h)

with the inclination a given by (1). The quoted authors have accepted SUMON’S (1947) formula for gas distribution from a continuous point source at height h and to obtain their formulae for dust concentration they have replaced t by H(X) given by (2). Different assumptions as to the effective source strength were used: (1) the source strength Q is independent on the distance x from the source (Meteorology and Atomic Energy, 1955); (ii) the 41

K. W~J~ECH~W~KI

42

source strength Q is a function of x (BARONet al., 1949; KRIEB, 1955;

CSANADY,

19%;

CLARENBURG,1960).

In this paper we shall examine the influence of a vertical wind profile on deposition of dust particles considering both cases: Q = const. and Q = Q(x). THE TRAJECTORY

OF PARTICLES

AND WIND

PROFILE

In general, the trajectory of a particle is more complicated than given by (2). If we assume that the settling of particles is influenced by the variation of wind u = u(z) with height, then the trajectory of a particle is described by the equations dx - = u(z), dt dz d-r = G, where t denotes time of travel. In this paper we take for u(z) the Roberts expression (SUTTON, 1953) u(z) = u(zl)

(-y2-: z:

(4)

where u(zJ = u1 is the mean wind velocity at the measuring height z1 and n is a parameter depending on the stability of the atmosphere. Solving (3) for u(z) given by (4) under the initial conditions: t = 0, z = h,x = 0, dz/drl,=, = t’, dx/dtl,,, = U(A), we obtain the following trajectory: 2 H(x)=h

x

I--2-_ni;W

[

1

2-42 ,

(3

where w = &W(~). In FIGS. 1 and 2 values of H(x) computed from (2) (n = 0) and from (5) for different n and h = 100 m, w = 0.1 and 0.5 are compared. From this comparison it follows that for both particle velocities used the differences between (2) and (5) increase with increasing atmospheric stability. If we use the mean wind velocity 1( over the height h - z1 in place of u(h) in equation (2)

we obtain the following particle trajectory H(x)

=

h -

( >9 1 -

;

z1

PW -

Zl &I 4

which is again a straight line. This situation is illustrated in FIGS. 1 and 2 by dashed lines.

\\\\ A

The Deposition of Dust Particles from Elevated Sources

100

90

43

60 7-Y

\

\

\

E

6°F

4Cr

I

w=Ol

h -1GO.m

FIG. 1. The trajectories of particles emerging from a stack of height 100 m for different atmospheric stability conditions in the case when the ratio of particle velocity to mean wind velocity at height of stack is equal to O-1.

20

40

60

80

100

r20

140

160

i60

200

X,m

FIG, 2. The trajectories of particles emerging from a stack of height 100 m for diierent atmospheric stabifity conditions in the case when the ratio of particle velocity to mean wind velocity at height of the stack is equal to O-5.

K. WOJCIECHO~SKI

4.4

FORMULAS

FOR DUST DEPOSITION AND DISCUSSION OF THE RESULTS Accepting Sutton’s formula for gas concentration and assuming that we have to replace h in this formula by H(x) given by (5) we obtain the following expressions for dust deposition on the ground S, 6,

Y,

0) = w Q(4.W exp

where

I

-

H2(X) + --& g2 Y

[

II

c,2

,

for Q = Q(x);

(71

Q(0) = Q,

(8)

f(x) = (?r c, c, x2-“)_I,

and

de(x) = dx

- w Q(x) t/Lf(x)] expf-

Hz(x)/Cz2 x2-“),

C, and C, are turbulent diffusion coefficients in x- and z-directions (for h > 25 m, c, = c, = C),

H2(x) g2 f Y ~,2

for Q = const.

,

(9)

The formula corresponding to (7) derived under assumption that the wind velocity does not change with height is that of KRIEB (1955) 2-n nw + .Y s, (x, y, 0) = 2w

h sz CT y, 0).

(W

The formula corresponding to (9) (for Q = const) derived under the assumption that wind velocity does not depend on the stack height 1s that of Sutton (Meteorolog), and Atomic Energy, 1955) y2 c,’

.z

(h - wx)’ T

c

2

.

CW

Krieb’s formula was chosen because it is one of the best formulae derived under assumption (1) for Q = Q(x). The formula derived more recently by CLARENBURG (1960) for wQ 1 is incorrect because of a wrong solution of an equation similar to (8) (WOJCIECHOWSKI, 1967). The comparison of numerical computations of deposition rate of particles for some values of w* and n are given in FIGS. 3-9. The rates of deposition in FIGS. 3 and 5 are given as function of distance from the stack computed from formulae (9) and (9a). From these figures it is evident that for normal atmospheric stability (n = 0.25) the value of maximum deposition obtained from (9) and (9a) is nearly the same for w = 0.1 (small particle velocity, and the position x,,, of maximum deposition computed from (9) is smaller than this computed from (9a). For greater value of particle velocity (o = O-61?) the picture of deposition obtained from (9) differs remarkably from the one obtained from Sutton’s formula (9a). *The sharp peaks in FIGS.3 and 5 are caused by the fact that (dSz/dx),,,., m &. tThe value o = 0.61 corresponds to u = 0.61 m s-l or tq particle dia. equal to 120 pm under condition that u(h) = 1 m s-l (HAWKINS and NONHEBEL, 1955).

The Deposition

of Dust Particles from Elevated Sources

45

X’,m

FIG. 3. The surface deposition calculated from formula (9) (full line) and from Sutton’s formula (9a) (dotted line) for Q = 1 g s-l, h = 50 m, w = 0.1 and n = 0.25.

h= 100 c = 0.08 w=O.I

n = 0.25

1 1000

I

I

I

1100

I

1200

X’, m

FIG. 4. The surface deposition calculated from formula (9) (full line) and from Sutton’s formula (9a) (dotted line) for Q = 1 g s-l, h = 100 m, w = O-1 and n = O-25.

The situation is similar if we compare the results obtained from (7) and (7a) (FIGS. In FIG. 9 the comparison of deposition rates computed from (7) and (7a) is given for stable atmosphere (n = O-5). From these figures it follows that in this case the picture of deposition differs remarkably. This result is in accordance with the physical nature of the atmospheric stability and indicates that the importance of the vertical wind profile increases with increasing n. Moreover, comparison with data of C~ANADY(1964) shows that formula (7) gives a better distance of maximum deposition (xm(l) = 90 m) than formula (7a) (x,,,(‘) = 110 m) (the experimental value of x, was equal to 70 m). Therefore we may conclude that formulae taking into account the vertical wind profile give a better picture of dust deposition than those derived under the assumption that wind velocity is independent of the height of the stack. 6-8).

46

K. WorcncHowsKI

0.!6

I

/ \

I/ i 7

‘\

h-IOC)m c=OO8 \

!

VI y O.IE \ 0 m

\

v=O.61 n :0,25

\

I

\

“0 -

\ \ \s / /

I

I

180

220

200

\2

I

1

240 X,m

260

.

._1

280

300

surface deposition calculated from formula (9) (full line) and from Sutton’s formula (9a) (dotted line) for Q = 1 g s-l, h = 100 m, w = 0.61 and n = 0.25.

FIG. 5.The

I

300

350

I

I

400

450 X,m

500

I

550

1 6CJ

FIG. 6. The

surface deposition calculated from formula (7) (full line) and from Ihieb’s formula (7a) (dotted line) for Q(0) = 1 g s-l, h 2 50 m, o = 0.1 and R = 0.2%

FIG. 7.

The surface deposition calculated from formula (7) (full line) and from Krieb’s formula (7a) (dotted lime) for Q(0) = 1 g s-‘, h = 100 m, w = O-1and n = 0.25.

The Deposition

47

of Dust Particles from Elevated Sources

0.3

h = 50 c - 0.1 I w = 0.1 ” = 0.25

L

I

1

350

3ca

I

I

I

450 X,m

400

550

500

600

FIG. 8. The surface deposition calculated from formula (7) (full line) and from Krieb’s formula (7a) (dotted line) for Q(O) = 1 g s-l, fr = 100 m, w = 0.61 and n = 0.25.

0.6

/

1

-\

‘1

\ ’

% \ \

1 -I

1600

1400

2300

2200

2000

!800

X,m

FIG. 9. The surface deposition calculated from formula (7) (full line) and from Krieb’s formula (7a) (dotted line) in the case of stable atmosphere (n = O-5) for Q(0) = 1 g s-l, h = 100 m and w = 0.1.

-------------------x.

”

zo.5

t

20

60

-_--

---

--__--_----0.7

I

I00

I

I

140

IS0

223

260

hem FIG. 10. The ratio .SI*/sI*

as function of height of the sounx.

: 10

1s

K. WOJCIECHOWSKI

According to Sutton’s assumption, the wind velocity occurring in formulae (7-91, is calculated for the height of source. If we put, however, instead u(h), the mean velocity zi computed as in (6), the results are changed. In order to estimate this change we have calculated the ratio ,V= S, */$: *, where

The results of our calculations are presentedin FIG. 10, and it can be easily seen that only if n = 0 we obtain y = 1. For n > 0 the difference between 3, * and S, * increases as n increases, and also increases slowly with increasing height of the source. Generally one can say that for n > 0,11 < u(h) what leads to the conclusion: Si
spreading from tall source (in Russ.) Probl. Turb. Difi Lower Atm. 15, 47-51. BARON T., GERHARDE. R. and JOHZISTOXE H. F. (1949) Dissemination of aerosol particles dispersed from stacks. Znd. Ertgng Chem. 41, 2403-2408. BOSANQUETC. H., CAREYW. F. and HALTOS E. M. (1950) Dust deposition from chimney stacks, Proc. Inst. &tech. Engng 162, 355-367. CLAFGVBURGL. A. (1960) A study on air pollution, Thesis, Utrecht. CSANADY G. T. (1958) Deposition of dust from industrial stacks. Ausr. J. uppl. Sci. 9, 1-7. CSANADY G. T. (1964) An atmospheric dust fall experiment. /. Armos. Sci. 21, 222-225. HAWKINS J. E. and No>~L G. (1955) Chimneys and dispersal of smoke, J. Inst. Fuel. 28,530-533. KRIEB K. H. (1955) Theoretische Betrachtung zur Frage der Staubausbreitung, Mitt. VGB 37, 700-705. Meteorology and Atomic Energy (1955) Washington, AECU 3066, Chap. VII, paragraph 3. ROUNDSW. JR. (1955) Solution of the two-dimensional diffusion equations. Trans. Am. geophys. Un. 36, 395-405. SCHMIDTW. (1925) Der Massenaustausch in freier Luft und verwandte Erscheinungen. Probleme der Kosmischen Physik. Verlag von Henri Grand, Hamburg. SUTTON 0. G. (1947) The problem of diffusion in the lower atmosphere, Q. J. R. Met. Sot. 73, 257-276. SU-ITON0. G. (1953) Micromereorology, Chap. II. McGraw-Hill, New York. WOJCIECHOWXIK. (1967) The dispersion of dust particles from elevated sources. AcU geophys. Poion. 15, 345-349.