Atmospheric transport of emissions over urban area sources

I?.pp.

1347.

ArmospkrL Enrironmenr Vol. I343Pergamon Press Ltd. 1978. Prin~cd in Great Britain.

b

ATMOSPHERIC TRANSPORT OF EMISSIONS OVER URBAN AREA SOURCES K. N.

MEHTA

and R.

BALASUBRAMANYAM

Department of Mathematics. Indian Institute of Technology, Delhi, India (First received 20 September 1976 and in final

firm 12 October

1977)

Abstract-The method of variational imbedding has been applied in reducing the problem of solving the nonlinear partial differential equation governing steady-state, two-dimensional advective-diffusive transport of emissions from ground-based area sources to an equivalent initial value problem in ordinary differential equations. Picard’s method has then been used to build up a sequence of successive approximations to density distribution of contaminants for the case when both turbulent diffusivity and mean downwind speed are taken to vary with the vertical distance coordinate according to the empirical power law. Solutions corresponding to a few simpler situations have also been presented in this paper.

1. INTRODUCTION

In a recent paper, Lebedeff and Hameed (1975) have shown that the integral method of hydrodynamic boundary layer theory can be satisfactorily adapted for studying the problem of advective-diffusive transport of contaminants being emitted as a steady flux from area sources situated at the ground level. Though the results obtained by following this approach have been found to be comparable with the exact solution in the case of uniform wind speed, the solutions do not incorporate the effect of variability of diffusion coefficient. In another recent paper (Lebedeff and Hameed, 1976) these authors have extended the integral method to the case of variable diffusivity. An alternative approach based on the method of variational imbedding has been recently proposed by us (Mehta and Balasubramanyam, 1977) for theoretically predicting the concentration of contaminants across the contaminated boundary layer for two types of nonlinear dispersive models characterised by general functional dependence of turbulent diffusivity K and downwind speed U on local instantaneous concentration. In the present paper, the method of variational imbedding has been used in reducing the problem of solving the non-linear partial differential equation governing two-dimensional steady-state advective-diffusive transport of emissions for groundbased area sources to an equivalent non-linear initial value problem in ordinary differential equations. The reduction to a simpler equivalent problem is achieved by introducing the concept of penetration distance (Lebedeff and Hameed, 1975), in this case boundary layer thickness 6(x) (generalized coordinate). Picard’s method has then been used to build up a sequence of successive approximations to density distribution of contaminants for the most general case of variable turbulent diffusivity, downwind speed and spatially dependent source function. Solutions corresponding

to a few simpler situations have also been obtained. In the special case of uniform wind speed and constant diffusivity and source function, the solution obtained by this approach is found to be in complete agreement with the known exact solution (Carslaw and Jaeger, 1959) unlike the corresponding approximate solution obtained by the integral method. 2. FORMULATION

OF THE PROBLEM

Consider steady-state two-dimensional advective-diffusive transport of contaminants being emitted as a steady flux from ground-based area sources in the atmospheric medium x > x0, z > 0 where the x-axis is chosen along the direction of the mean downwind speed V(z) and the z-axis is taken along the vertically upwards direction with z = 0 as the mean ground level; x = x0 is the periphery of the urban area in which the emission sources are situated at z = 0. Under steady-state conditions, the concentration X(x, z) of contaminants at a general point (x, z) of the atmospheric medium satisfies the non-linear partial differential equation :

U(z):= ;

[

K(z)% 1

(1)

in which V(z) and K(z) are the mean downwind speed and turbulent diffusivity respectively and these have been taken to be of the empirical form (Lebedeff and Hameed, 1975), m V(z) = U, ; , K(Z) = K, + K, f “. (2) 0 z1 0 It may be noticed that the diffusion coefficient should be non-zero at the point of emission z = 0 (i.e. K, # 0). This is an essential requirement for the onset of diffusion in the vertical direction starting from the, ground level.

1343

1344

K. N. MEHTAand R. BALASUBRAMANYAM

It is assumed that the contaminants are being emitted as a steady flux at the surface z = 0, i.e. K(z):

= -Q(x),

z= 0

(3)

and that the contaminants remain confined within the boundary layer having thickness distribution 6(x), with no flux across the top boundary of the contaminated layer, i.e. X(x, z) = 0,

z 2 6(x)

from 0 to 6(x), we can write the function J(X, X*) in the form of an integral involving 6(x), i(x) and constant coefficients defined through beta functions. Thus we have, J(X. x*)

J; c(x)[c,(x)?{am+iB(m 6In+2 -:-B(m+2,2p)

(4a)

and z 2 6(x).

at

_K 1cP(P3

We now assume the contaminant profile of the form

x

1-

&

[

concentration

1

1)

6”-‘B(n.2~

- 2)

6”-‘B(n + 1,2p - 2)

- 1) - h”B(n + 1.2~ -1) 1.

(10)

G(x)Q(x)

(7)

which follows readily from (3) on using (2) and (6). Choosing the adjoint function X*(x, I) of the form (Mehta and Balasubramanyam. 1977) -Gil”-‘[l

dx

(6)

PK,

X*(x,z)=c(x)[l

-K,cv

P

in which p can take integral values > 3. This choice for X(x, 2) has been made so as to satisfy the boundary conditions (4a), (4b) and the smoothing condition (5). The two unknown functions c(x) and 6(x) introduced in (6) are connected by the relation =

I

B(l.2~ - 2) - ;B(2,2p

2 = 6(x).

X(.x, ;) = c(u)

1) -i

xB(m+3,2p-1)

This together with Equations (1) and (4b) gives the smoothing condition

c(?()

+c~p~ ZY 6’

6”+‘B(m+2,2p-

(4b)

From physical considerations it is further assumed that the top boundary of the layer varies smoothly in the x-direction. i.e.

?X

1

+ 1,2p)

p+3

?X - = 0, (7z

iX -- = 0

=

-&]

(8)

Here prime denotes derivative with respect to x. From the functional (lo), the Euler’s equation for i.(x), after simplification can be written in the form

/l$+BL= &L +M6”) c

(11)

in which the constants A, B and L, M are given by A = p:‘B(m -? B= $ L = K,P(P M = ?

+ 3,2p - 1).

B(m + 2,2p). -

lW2,2p

- 2).

[p(p - l)B(n + 2,2p - 2)

the Lagrangian which imbeds the descriptive set of Equations (l), (4a, b) and (5) is

-pnB(n

+ 1,2p - l)].

Since

J(x,x*)

=

J:Jyx*pqz)E - ;

K(z)g ‘1

c(x)

II

dz dx

=

4x)QCx) PK,

(9)

where the interval [0,x] is arbitrary and i(x) is an unknown function which need not be determined as will be seen in the later analysis. Substituting for X(x, z), X*(x, z), U(z) and K(z) in (9) and effecting integration partially with respect to z

’

we can write Equation (11) in the form S’ Q' 1 (A+B)-g+Be=-&L+M6”)

(12)

in which Q(x) is the source function introduced in Equation (3). The solution of the non-linear differential Equation

Atmospheric transport of emissions

(12) subject to the initial condition 6(x,) = 0 will yield the thickness distribution of the contaminated boundary layer. It may be observed that such a solution will incorporate the effect of variability of diffusion coefficient through the presence of the term involving M (i.e. K,).

1345

which subject to 6(x,) = 0 has the solution x-x0=-

A+B

a

L

s 0

a”+‘(1 + ~6”)~ d6.

(16)

If Ia61 < 1, Equation (16) yields the following algebraic equation determining the 6(x) distribution : gm+n+*

A+B am+* ---_ m+2 L (

-

3. SOLUTIONS:

SIMPLE

CASES

m+n+2 gm+*n+2

(a) In the case when the source function Q(x) = Q. (constant) and M = 0 (i.e. K, = 0 implying that K = K,, constant), Equation (12) reduces to with 6(x,) = 0

(‘4 + B)6’ = -& which has the solution 6(x) =

1

+ 2)(x - xe) n+l.

A(* [

Substituting concentration

(13)

for 6(x) in (7) gives us the ground level

+

Qo ~(m+2)W,)n:*.

l-

[

pKo

(14) In order to compare with the results obtained by Lebedeff and Hameed (1975), we take p = 3 in the expressions for A, B, L and M occurring in (14) and get

_

3ab* +

s

U,{B(m+3,5)-B(m+2,6)}

m+Z.

l-

=

x

-

(17)

x0.

>

6(x - x0) = 0,

A+B a = L.

Writing this equation in the form 6* = a( 1 - @)- ’ = a( 1 f /?S + fi*S*), 16)< 1.5&-i 2 with a= -(x -x0), a

1

K&n + 2)(x - x,,)z;

. . .

m+2n+2

When E = 0, this equation gives the same solution as the one already obtained in Equation (13) for the case K = K,.It is therefore reasonable to expect that terms involving E in (17) incorporate the effect due to variability of turbulent diffusivity. As an example which illustrates how 6(x) may be determined, we may consider the special case with m = 0, n = 1. By including terms up to the first power of E, Equation (17) reduces to a cubic in 6, 2aeb3 -

X(x, 0) = c(x) =

E2

2c b = 3, the solution of the cubic

may then be approximated quadratic

by the positive root of the

(1 - a/J*)6* - 2a@ - a = 0. This result is more general and is analogous to the result contained in Equation (11) of the paper by Lebedeff and Hameed (1975). Considering the case when downwind speed U(z) is independent of z, i.e. m = 0, Equation (14) reduced to give c(x)

=

J”.454(v 3

2 )I,2 (x - x0)“* 1 0

which is in complete agreement with the known exact solution (Carslaw and Jaeget, 1959):

(c) In the case of x-dependent source function and constant diffusion coefficient (M = 0),Equation (12) takes the form

6’+-

BQ’g=L A+B6"+"l

A+BQ

Writing Y(x) = [S(x)]” +*, the non-linear initial value problem reduces to the linear form y, +

(m+ 2P A+B

w =$ (u,to)“2 (x - x0)1’* unlike the corresponding and Hameed (1975):

c(x) =

result obtained by Lebedeff

6(x,)=0.

Q'

y

=

Q

Lb + 2) A+B'

y(x

)

=

0

'

which has the solution 6(x) =

(m + 2)L A+B{QW-

'j-lo {Q(x))" dx]&

$ (u,~o)l,2 (x - xd1i2.

(b) We now give solution of Equation (12) for the case when the source function Q(x) is constant Q. and K is z-dependent (i.e. K, # 0 which in turn implies that M # 0). In this case Equation (12) becomes

It may be observed that this solution reduces to the solution given in Equation (13) when the source function is taken as constant. 4. GENERAL

SOLUTION

Writing Y(x) = [a(~)]~+*, the problem of finding

K. N.

1346

MEHTA

and R.

the solution of Equation (12) subject to the initial condition 6(x,) = 0 reduces to the non-linear initial value problem y’ =

“+2(1+&T, (1

m+2 -

af(x)Y

= cpk Y), Y(x,) = 0

(19)

BALASUBRAMANYAM

For a source Qo(x) extending over the finite length x = x0 to x = x1, the corresponding ground level concentration for points x > x, can be obtained by adding up the concentrations due to a pair of semi-infinite sources, one of strength Q,,(x) extending from x0 to cc and the other of strength - Qo(x) extending from x, to cc. Making use of Equation (22), the concentration c(x) for x > x 1for such a source of finite length comes out to be

in which A+B

E = ;,

a=L3

f(x)

B Q’(x) = - y. L Q(x)

x CP(--G Y,)dx s XII

= w(x

- x0) = p(x - x,).

(20)

It may be noticed that the first approximation gives the solution given in Equation (13) which corresponds to the case Q(x) = Q. and K = K,; in fact, this solution can be obtained by setting E = 0 andf(x) = 0 in (19). Second approximation : m+2

Y, = __

a

[

(x - x0) +

m+n+z (x -Xo)m+Z

AL qm+2

(21)

With Q(x) prescribed, the integral in Equation (21) can be evaluated to yield completely the second approximation to the solution. The determination of further successive approximations to the solution is quite straightforward. 5. GROUND

LEVEL CONCENTRATION URBAN AREA SOURCES

DUE TO

(7), thereby

c(x) = kQ,,(x)[

is a constant.

4x1 = 1 Cl(X) I=0

where each cl(x) = X,(x,0) K(z)%

and

= = -[Q,(x)

IO

,...)

- Q,-,(x)], s.

Since such a distribution of finite-length sources can be regarded as equivalent to the distribution of sources of semi-infinite extent with strengths QAx) - Ql_ t(x), x > x,, I = 0,1, . . , s, we get, on successive application of the result given by Equation (22)

c(x) = K i

{Qdx) - Q,- ,W}

I=0

{Qdx)- Q~-~(x)I-" {QXX)- Q,_,(x)}“dx

mi2, l-

x > x,.

(25)

In the particular case when the source strengths are constants, Equation (25) reduces to the simpler form

giving

iQoW-" j:.{Q&,r"dx~ (22)

in which

For a typical emission inventory of urban area sources in the form of successive continuous sources of assigned strengths Q,(x) distributed on xI < x < x,+ 1, I = 0,1, . , s, the corresponding ground level concentration c(x) is obtained from

X

The ground level concentration c(x) due to a semiinfinite continuous source of variable strength Qo(x) and extending from x = x0 to x = co is obtained by inserting the expression for 6(x) given by Equation (18) in Equation

m-+2

I=O,1,2

x,)f(x)dx.

(23)

/,

“=I-_-.

m+n+2

x0

x)x,

where

m+2

1 5x(x-/l

-3

x, {Qo(x,}“dx

We can then readily write down the successive approximations by Picard’s method to the solution of the initial value problem described by Equation (19): First approximation : Y, = Y(x,) +

IA 1,

x

x (Q,- Q,-1)(x -

XI&

x > x,.

(26)

With p = 3, this equation gives modified form of the result obtained by Lebedeff and Hameed (1975) in Equation (17) of their paper. Acknowledgements - We wish to thank the reviewers for their criticism and helpful advice.

Atmospheric transport of emissions REFERENCES Carslaw H. Solids, p. LebedelT S. transport

S. and Jaeger J. C. (1959) Conduction o/Heat in 75. Oxford University Press, London. A. and Hameed S. (1975) Study of atmospheric over area sources by an integral method. Atmos-

pheric Environmenr 9, 333-338.

1347

Lebedeti S. A. and Hameed S. (1976) Laws of efRuent dispersion in the steady-state atmosphkric surface layer in stable and unstable conditions. .I. oppl. Met. 15, 326-336. Mehta K. N. and Balasubramanyam R. (1977) Application of the method of variational imbedding to atmospheric diffusion transport. Atmospheric Environment 11, 109-l 12.