Atmospheric dispersion model of removal mechanism with integral term

Atmospheric

dispersion

model:

J. B. Shukla and R. S. Chauhan

Paper AMM1064

Atmospheric dispersion model of removal mechanism with integral term J. B. Shukla and R. S. Chauhan Department of Mathematics, Indian Institute of Technology, Kanpur-208016,

India

(Received January 1986; revised August 1986)

A general mathematical model for atmospheric dispersion is suggested, in the form of an integro-partial differential equation, the integral term characterizing the removal mechanism which may arise due to absorption of air pollutant in fog or rain droplets present in the atmosphere. This model is applied to the study of the effects of instantaneous and delayed removal on dispersion of air pollutant from an elevated time-dependent point source. It has been shown that the concentration of the pollutant along the central line at a given time and point decreases due to the removal process and the concentration of pollutant with a delayed removal process may be greater than that for an instantaneous removal case. Keywords: mathematical moval, time-dependence,

The study of atmospheric dispersion of air pollutants from point, line, and area sources has received a great deal of attention during the last few decades.le9 The effects of the removal process on the dispersal of air pollutants have also been investigated.lG14 When air pollutants are removed by rainout/washout in the environment, the dispersal process becomes very complex and very little attention has been given to understanding this mechanism.12J4-16 In view of this, a growth equation is suggested for studying the removal of air pollutant by rain or fog droplets. In such a case, the unsteady state partial differential equation for dispersion of air pollutant is modified into the form of an integro-differential equation. To explain the effects of the removal process with this model the authors have studied the dispersion of air pollutant from an elevated time-dependent point source which may be applicable in India during the monsoon season.

model, instantaneous point source

equation for air pollutant concentration Cartesian coordinates (x, y, z) as:

model

Consider the dispersion of air pollutant from an elevated time-dependent point source in a stable environment, such that the wind velocity may be assumed to be uniform. In such a case, the three-dimensional transport

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Appl. Math. Modelling,

1987, Vol. 11, April

delayed

re-

C is written in

-kC

(1)

where U is the mean wind velocity, D is the diffusion coefficient (assumed constant), and k the removal rate. When fog or rain droplets are present in the environment, air pollutant is absorbed in the droplet phase and the removal process is not instantaneous. Assuming that the rate of decrease of pollutant, K/at, is proportional to its concentration, C,, in the droplet phase, the transport equation (1) may, in this case, be modified to:

-

Mathematical

and

k,W,

Y,

z,4

(2)

Since the rate of change of the concentration of air pollutant in the droplet phase, X,/at, may increase with the increase of concentration of pollutant C and decrease due to rainout/washout, the governing equation for C, can be written as: 0307-904X/87/02082-07/$03.00 0 1987 Butterworth & Co. (Publishers)

Ltd

Atmospheric dispersion model: J. 6. Shukla and R. S. Chauhan

G -_= at

a& -

ffc,

(3)

where LYE and LYare constant rates of absorption and removal. Now, integrating equation (3) with the condition C, = 0 at t = 0, gives:

-4?rs2D$=

W(t)ass+OfortSO

The last boundary condition implies that the point source has prescribed a time-dependent flux. In this paper, the following forms of W(t) are considered: W(t)=

W&t);

W(t)=

w,; W(t)=

(10)

w,ostat,

and W(t) = 0, t > to Cr(&Y,~,t)=~o

i

’ e-“(‘-~

C(x, y, z, t) dT

(4)

0

where S( .) is the Dirac delta function. The following dimensionless quantities defined:

are now

and the equation (2) then becomes:

(11) -

’ e-a(‘-” C(x, y, z, 7) dT

ffl

(5)

i 0

where the integral term represents the removal of air pollutant by the droplet phase, and o1 = cxoko.It may be noted here that equations (1) and (5) can be combined as follows:

-

‘G(t-

T)C(x,y,z,

i0

T)dT

where u* is the friction velocity. Using these quantities, equations (5)-( 10) can be written in dimensionless form (dropping the bold typeface for convenience) as:

(6)

’ e-“(‘-” C(X, Y, t) dT

-cY

For G(t - 7’)= ka(t - T), where S(.) is the Dirac delta function, equation (6) reduces to equation (1) and for G = ~ie-(y(l-r), reduces to equation (5). Equation (6) may thus be interpreted as a general transport diffusion equation with an integral term representing the removal mechanism. Equation (1) may be designated as a diffusion equation with instantaneous removal term, while equation (5) may represent the case for delayed (slow) removal process. It is noted here that under steady state conditions, equation (3) gives:

C(s,t)=O

t=O

s>o

(13)

C(s,t)=O

s-+m

t20

(14)

-4VS2 a’ _ w(t) as W(t) -

s+o

WC

= Q,a(t);

!!$

WC

tao

(15)

= 1 c

w(t)

y=l

aoko c,=-c. CY

oatat,

(16)

c

In such a case, equations identical, provided that: aoko k=__=’ ff

(12)

0

(1) and (2) would become

=o t>t, where:

Qo=-!$$ ff

c

and the two removal mechanisms would be identical under this condition. Under unsteady conditions, however, these removal mechanisms are different. The dispersion of air pollutant is, therefore, studied for delayed and instantaneous removal processes, when k is greater, less, or equal to al/a. Taking the source as the origin of the coordinate system, the initial and boundary conditions for equations (1) and (5) are written as: C(s, t) = 0 at t = 0 for all s = (x2 + y* + z2)‘j2 > 0 (7) C(s,t)=Oass+afortbO

(8)

Solution Taking the Laplace transform of equation (12) and using initial condition (13) gives:

ac d2C a2c a2c

uZ=ax2+T+1-mC ay a.2

(17)

C is the Laplace transform of C, m =p + aI/@ + a), and p is the Laplace variable. The

where

boundary conditions then become: C(s, p) = 0

s+m

Appl. Math. Modellinn,

1987,Vol.

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83

Atmospheric

dispersion

model:

J. B. Shukla and R. S. Chauhan 0.07

(18) Assuming that the solution of equation (17) is of the form: C=exp

(

$

1

f(s,p)

0.06

0.05

and using equation (18) gives: (19)

Taking the inverse Laplace transform of equation (19), allows the solution of equation (12) satisfying the initial and boundary condition to be written as:

._6 g t g 0.04 8 t c .p & 0.03 E 5

0.02

y+im

y-k

ux

W(P)

0.01

(p + a)(p + b) l/2

-exp Pt+T1 CP+ a> I WC L

1

s

dP

(20) 0

where:

0

2

I

4

3

Dimensionlessdownwinddistance Figure 1 Concentration-distance profile for instantaneous flux at source: (---) no removal, (Y, = 0; (-_) delayed removal, (Y, = 0.1; (--1 instantaneous removal, k= 0.5; U= 3, a = 0.3, Q, = 1

and yis a real positive number such that all the singularities of the integrand lie on the left-hand side of Re(p) = y in the Bromwich cont0ur.l’ The integral in equation (20) can be evaluated and the concentration along the central line for each case can be found as follows: (i) when the flux is instantaneous,

i.e. W(t) = W&t):

a cc

I

1 “1 -M(x, u, t) du = I b"

(23)

From this last expression, it is observed that the concentration of pollutant as t+ w tends to the steady state, as given by the following expression:

C(x,O,O,t)=~{~M(x,u,t)du

+

--

&2(x, u, t) du

(21)

b

C(x, 0, 0, t) = g{exp[

- (T+:)“2x]

where P(x) = exp( Ux/2) (u - a)(u - b)

M(x, u, t) = emu’sin [

(u

-

a)

Similarly, by solving equation (1)) the concentration distribution along the central line in the dimensionless form is given by:

1 1/:

Similarly, the solution of equation (1) is obtained and the concentration along the central line in the dimensionless form is written as:

Cl@,0,074 = E[exp[ -exp(-ut)sin(u

c,(x, ‘2 ‘3 t, = Aexp[q-(T+k)t--$1

-(T+ki”2x]

- bI)li2xdu

(24)

@)3/2

where:

(22) When U = 0 and k = 0, it reduces to a form identical to the one obtained by Carslaw and Jaegar.” (ii) when the flux is constant, i.e. W(t) = W,: C(x,

84

Appl.

o,o, t)

= g[

exp[ - (T+:)“‘x]

Math. Modelling,

1987, Vol. 11, April

b,=;+k

(iii) When the flux is a step function, for 0 < t s to and W(t) = 0 for t > to: C(x, 0, 0, t) = g[exp[

-(~+~)“2x]

i.e. W(t) = W,

Atmospheric

dispersion

model:

J. B. Shukla and R. S. Chauhan 1

Dimensionless downwind distance

Figure 4 ConcentraGon-distance profile for constant flux at source: (---I no removal, (Y, = 0; (--_) delayed removal, a, = 0.1; k-1 instantaneous removal, k = 0.5; U = 3, Q = 0.3 Dimensionless downwind distance

Figure 2 Concentration-distance profile for instantaneous flux at source. Key as for Figure 7 except: (-_) delayed removal, a, 0.04

= 0.15

003

S iz f u 5 v 002 I a, c .P r E 6

006

005

i-s g s

001 004-

5 f .P E 003“E a

0

2

3

4

Dimensionless downwind distance

Figure 5 Concentration-distance source. Key as for Figure4except:

profile for constant flux at (-1 delayed removal, Q, = 0.15

- e”QH(t -

co)]M(x, u, t)du

Dimensionless downwind distance

Figure 3 Concentration-distance profile for instantaneous flux at source. Key as for Figure 7 except: (-_) delayed removal, a, = 0.3

where H(t - to) is defined by: l8

Appl.

Math. Modelling,

1987,Vol.

1 l,ApriI

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Atmospheric 0.04

dispersion

model:

J. B. Shukla and R. S. Chauhan

r

0.03

0.03

c .P 5 S

5 ‘Z z 5 E s 0.02

E 0 002 I u c .o g E -6

I u s .a 5 E a

0.0 I

0.0 I

C

0

Figure 6 Concentration-distance source. Key as for Figure 4 except:

H(t - t*) = 0

to + =

3

4

profile for constant (-_) delayed removal,

flux at a, = 0.3

Figure 7 Concentration-distance profile for stepped flux at source: (---I no removal, (Y, = 0; (-_) delayed removal, (Y, = 0.1; (:.-I instantaneous removal, k= 0.5; to = 0.5, U= 3, (Y= 0.3

td to t

=-

2

Dimensionlessdownwind distance

Dimensionlessdownwind distance

t()
Oo4x

&

1

t> to

In this case, for instantaneous removal, the concentration distribution along the central line, is given in dimensionless form:

C,(x, 070, t) = g{

exp[ - (T+k)li2X]

x [l - H(t - to)] -i

i iemu’sin[(u 1

I

- b#/*x]

x [ 1 - e”W(t - to)] du

(26)

I It is noted that equations (24) for to 2 t.

(25)-(26)

reduce

to (23) and

Results and discussion The expressions for C and Ci given by equations (21)-(26) in the three cases have been computed for V=3,K=0.5, a=0.3, cw,=0.0,0.1,0.15and0.3(i.e. a,/a =o, 4, : and l), and are plotted in Figures l-9. It is observed that for all three types of source, the concentration of pollutant along the central line decreases as the downwind distance from the source increases. When the flux is instantaneous, it is noted from Figures l-3 that, for a given velocity, the concentration of pollutant decreases and the maximum point in the concentration-distance profile moves away from the source as time increases. It can be seen from Figures 4-6 that when the flux is constant, the concentration

86

Appl. Math. Modelling,

1987,Vol.

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2 3 Dimensionlessdownwind distance

Figure 8 Concentration-distance source. Key asfor Figure 7except:

profile for stepped (-_) delayed removal,q

4 flux at = 0.15

of pollutant always increases with time and reaches a steady state. When the flux is given by the step function, however, the concentration of air pollutant increases for t d to and decreases for t > to as t increases (Figures 7-9).

The effect of removal is shown in Figures 1-9. At a given point and time, the effect of removal (delayed or instantaneous) is to decrease the concentration of

Atmospheric

dispersion

model:

J. 6. Shukla and R. S. Chauhan 1

0.04

003 .-S $ 5 E 8

=2 -3 =4

002

I u S ?I .-E D 0.0 I

I 4 Dimensionless downwind distance Figure source.

9 Concentration-distance Key as for Figure 7except:

profile for stepped (-_) delayed removal,

0

flux at a, = 0.3

‘I Dimensionless downwind distance

Figure source:

71 Concentration-distance (-_) t = 0.5; (---) t = 0.75;

profile for constant LY= 0.3, a, = 0.1

To compare the delayed and instantaneous cases the following cases are considered: k>’

\ . Y

I

Figure 70 atsource:

3 2 Dimensionless downwind distance

4

5

Concentration-distance profile for instantaneous flux t=0.5;(---)t=0.75;a=0.3,a,=0.1,Q0=1

air pollutant in comparison for all three types of source.

with the no-removal

case

a

k=’

CY

kc’

flux

at

removal

CY

For k > al/a, it is observed that for all three types of source, the concentration of pollutant for the case of delayed removal is greater than in the case of instantaneous removal (Figures I, 4 and 7). In the case of constant flux, it is noted that even if k d al/~, the concentration of the pollutant for the case of delayed removal is always greater than in the instantaneous removal case (Figures 4-6). However, for the other two types of source when k d ~/LX, the concentration in the delayed removal case can be less, equal, or greater than for the instantaneous removal case, and this behaviour depends upon the downwind distance from the source, time elapsed, etc. (Figures 2-3, 8, 9). The results can be visualized by comparing the removal mechanism in equations (1) and (5), and noting that the integrand in the integral term of equation (5) involves a negative exponential which decreases as either t or LYincreases. The effect of wind velocity on the concentration of air pollutant at different locations and times, can be seen in Figures l&12. It is noted from Figure 10 that, for instantaneous flux, the concentration of pollutant at a fixed instant increases at different locations and the maximum point in the concentration distance profile moves away from the source as wind velocity increases. For constant flux (Figure II), the concentration increases with wind velocity at a particular time and location. When the flux is given by step function, however (Figure 12) the concentration increases with wind velocity for fixed t d to, but when t > to, the increase or

Appl.

Math. Modelling,

1987, Vol. 11, April

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Atmospheric

I

3

2

4

5

Dimensionless downwind distance

Figure 72 Concentration-distance profile for stepped flux at source: (-_)t=0.5;(---)t=0.75;a=0.3,(~,=0.1,f~=0.5

decrease in concentration with respect to wind velocity depends upon the downwind distance from the source.

References elevated 1 Smith, F.B. The diffusion of smoke from a continuous point source into a turbulent atmosphere, J. Fluid Me&., 1957, 2, 50-76 diffusion, Von Nostrand, Princeton, New 2 Pasquill, F. Atmospheric Jersey, I962

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Math. Modelling,

1987,Vol.

11,April

dispersion

model:

J. B. Shukla and R. S. Chauhan

3 Gifford, F.A. and Hanna, S.R. Urban air pollution modelling, Proc. 2nd ht. Clean Air Congr. (eds, Engleend, H.M. and Beary, W. T.), 1971, pp 11461151 4 Heines, T.S. and Peters, L.K. The effect of a horizontal impervious layer caused by a temperature inversion aloft on the dispersion of pollutants in the atmosphere, Atmos. Environ., 1973,7,39-48 5 Heines, T.S. and Peters, L.K. An analytical investigation of the effect of a first order chemical reaction on the dispersion of pollutants in the atmosphere, Atmos. Environ., 1973,7,153-162 6 Lebedeff, S.A. and Hameed, S. Study of atmospheric transport over area sources by an integral method, Atmos. Environ., 1975, 9,333-338 7 Ragland, K.W. Multiple box model for dispersion of air pollutant from area sources, Atmos. Environ., 1973,7,1017-1032 8 Ermak, D.L. An analytical model for air pollutant transport and desposition from a point source, Atmos. Environ., 1977, 11,231-237 9 Karamchandani, P. and Peters, L.K. Analysis of the error associated with grid model representation of point source, Atmos. Environ., 1983,17,927-933 10 Striven, R.A. and Fisher, B.E.A. The long range transport of airborne material and its removal by deposition and washout: I. General considerations, Atmos. Environ., 1975,9,49-58 11 Striven, R. A. and Fisher, B.E.A. The long range transport of airborne material and its removal bv denosition and washout: II. The effect of turbulent diffusion, Atmos.*Et&ron., 1975,9,59-68 12 Fisher. B.E.A. The transnort and removal of sulfur dioxide in a rain system, Atmos. Environ:, 1982,16, 775-783 13 Llewelyn, R.P. An analytical model for the transport dispersion and elimination of air pollutants emitted from a point source, Atmos. Environ., 1983,17,249-256 14 Alam, M.K. and Seinfeld, J.H. Solution of the steady state, threedimensional atmospheric diffusion equation for sulfur dioxide and sulfate dispersion from point sources, Atmos. Environ., 1981, 15, 122140 15 Hales, J.M. Fundamentals of the theory of gas scavenging by rain, Atmos. Environ., 1972,6,635659 16 Slinn, W.G.N. The redistribution of a gas plume caused by reversible washout, 1974,8,233-239 17 Carslaw, H.S. and Jaeger, J.C. Conduction of heat in solids, Clarendon Press, Oxford, 1959, pp 357 18 Carslaw, H.S. and Jaeger, J.C. Operational methods in applied mathematics, Dover Publication, New York, 1941