- Email: [email protected]
A note on atmospheric dispersion with chemical reaction
Armospheric
En~ironmenr
Vol. 11. pp. 853-856.
A NOTE
Pergamon
Press 1977. Printed in Great Britain.
ON ATMOSPHERIC DISPERSION CHEMICAL REACTION RICHARD 4. NUNGEand K. R.
WITH
VAIDYANATHAN
Department of Chemical Engineering, Clarkson College of Technology, Potsdam, NY 13676, U.S.A. (Received
1 November 1976)
Abstract-aeneralized dispersion theory is used to obtain an analytical solution tive-diffusion equation for dispersion of a reactive pollutant in the atmosphere. obtained for the ground level concentration as a function of time and spatial tinuous crosswind line source at different elevations and for various values of rate under inversion conditions.
NOMENCLATURE concentration turbulent diffusivity of mass coefficients in equation (16) dimensionless inversion height dispersion coefficients defined by equation dimensionless source strength, mass/time/length @ID,, time aIn0 wind speed in the .x-direction WWe downwind coordinate zlH vertical coordinate KH=/D,o Dirac delta function c/c0 first-order reaction rate constant t D:,/H’
to the unsteady advecNumerical results are coordinates for a conthe first-order reaction
Yaglom (1971) for a continuous source. point concentration is described by
ac z
ac
+
a
~(4da, = z&L(z)
Thus
the
ac z
+
S(t, x, z; zl) - kc,
(18)
1. INTRODUCTION Earlier, the generalized dispersion theory was used to develop an analytical solution to the unsteady, advective-diffusion equation for an empirical eddy diffusivity model of dispersion in the atmosphere from an instantaneous source (Nunge, 1974a, b). Solutions were obtained for the point concentration distribution under various conditions. Later, Nunge and Subramanian (1977) applied the theory to a continuous area source of an inert pollutant. The purpose of this note is to report an extension of the continuous source problem to include the effect of a first-order, homogeneous, irreversible chemical reaction.
(1) where the source function S for a line source at z = zi may be written as 5(r,x,z;z,) = m(r) 6(x) 6(z - 21).
(2)
Here x is the mean wind direction, z is the height above ground and 6 is the Dirac delta function. For comparison with previous papers in this series, it is assumed in writing Equation (1) that downwind diffusion can be ignored compared with downwind advection. Also, there is no pollutant in the atmosphere initially, the ground is impermeable to the pollutant and there is no penetration of the pollutant through the inversion layer at z = H. Although more realistic wind and eddy diffusivity profiles may be used, to obtain particular results comparable to those in the earlier papers, it is further assumed that the mean wind is steady and linear in z and the eddy diffusivity, D,, is constant at D,,. In addition, although time-dependent sources are readily handled, for simplicity the emission rate per unit length, m, is considered constant at m,. If time-dependent sources are treated, it may be necessary to include the downwind diffusion term in Equation (1). Under the above assumptions, the model solved here may be written as
ac z
+
uoz
ac
F
ax
=
a% Dzoazz
+
m&c) &z - ~1) -
KC
(3)
2. THE PHYSICAL PROBLEM c(0, x, z) = 0 The particular physical problem chosen for study is the infinite cross-wind line source such as has been used to model a highway. The local time-averaged (or ensemble average) concentration of pollutant is assumed to satisfy a two-dimensional version of the advective-diffusion equation given by Monin and
D,, ;
(6 x, 0) = Dr,,
2 (t, x, H)
f&m,3 =0. 853
(4) = 0
(5)
Equation (6) results from there being a finite amount of pollutant in the atmosphere at all finite times.
3. METHOD
OF SOLtiTlOh
In view of the previous development of the theory (Nunge. 1974a; Nunge and Subramanian, 1977) and other applications of the generalized dispersion theory to similar continuoLIs source problems (Gill and Sankarasumbramanian. 1972). the method of solution will only be outlined here; full details of the solution are given elsewhere (Vaidyanathan, 1976). The solution of Equations (3~(6) may be formulated as the time integral of the solution of an initial value problem by using superposition (Courant and Hilbert. 1962); thus
where
(8) with l+b,(O, x, z) = d(x) 6(2 - z,)
(9 (10) (11)
The problem is solved in dimensionless form generality; the dimensionless groyps chosen are
for
Equations (12)-(15) are a special case of the problems solved in genera1 previously (Nunge. 1974a). using the generalized dispersion theory, wherein + is represented by
with $,, = j ’ $(t* X, Z) dZ 0 and
The key to the generalized dispersion theory is determining the gir and Ki coefficients in equations (I 6) and (18). Equation (12) is first integrated over 2 and then Equation (16) is substituted for II/. The result is of the form given by Equation (18), so that equations for the Ki as integrals over Z of the gk coefficients are obtained. The gk functions are found by solving the equations resulting from substituting Equation (16) into Equation (12) and setting the coefficients of ak$ Ji)Xk equal to zero. As discussed previously (Nunge, 1974b), it is necessary to retain only terms of g2 and K? in the infinite series represented by equations (16) and (18) to obtain convergence over a wide range of z, X and Z. 4 lVUMERlCAL
RESULTS
From the analysis above, the functional of the dimensionless concentration is 0 = i)(~,s.z;z,,~).
where c0 is the reference concentration 2~1~~~~~ chosen such that O- I as X + -L for It = 0 at all finite times. Equations (@(I 1) in dimensionless form become
(12) l/40, x, Z) = 6(X) 6(< - 2,)
(13)
(17)
dependence
(19)
Predictions of the point concentration, 0, have been obtained over a wide range of values of the independent variables 7, X, Z for /j = 0, I. 10 and 101) with Z, = 0.1,0.2,0.4, 0.6 and 0.8. Extensive plots of these results are given elsewhere (Vaidyanathan, 1976). 4.1 Sfeady state For the most part, one is concerned with the concentration levels which occur at ground level. As representative of the results obtained. Fig. 1 presents the dimensionless ground level concentration vs X for a dimensionless source elevation of 0.6 with fi as a parameter. Examination of this figure shows that as X -+ i/-, O+ 1.O when there is no reaction (/I = 0). For 0 = I. there is only a small effect on the ground level concentration compared to the case of no reaction until downwind distances exceeding X = 0.1 have been reached. For sources close to the ground,
A note on atmospheric dispersion with chemical reaction I
I
’
855
Z, = 0.1 and 0.2, increasing p to 10 still has only relalow level emission elevation in a neutral atmosphere; the value of K for the reaction of SOz with 0, forming SO, was taken from Gartrell et al. (1963), which may be somewhat low for dry atmospheric conditions. For Z, = 0.2 and /? = 1.0, the maximum dimensionless ground level concentration for these conditions is ~10 at X ‘v 1 x 10e3. Furthermore, use of the definitions of the dimensionless groups and
z,=o..6
\
2m0 cg = _ uoH reveals that a value of m,, = 1.8 g SO2 (m s)- * will yield a ground level concentration of 365 pg mm3 at x = 66.7 m downwind of the source. 4.2 Transient behavior IO‘<
IO =
IO
10-l
X Fig. 1. Steady-state dimensionless ground level concentration as a function of dimensionless downwind distance
for various values of the dimensionless reaction parameter with Z1 = 0.6. zi = 0.1 and 0.2, increasing /l to 10 still has only relatively small effects on the ground level concentration close to the source, but greater differences appear as the downwind distance from the source increases. The general effect of increasing /3 is to decrease the ground level concentration for all values of X. As particular example of the application of these steady-state results, suppose it is desired to find the maximum allowable emission rate per unit length of SOz using the primary standard of 365 pgrne3 of SO, as the average ground level concentration for a 24 h period. Consider the physical situation in which u0 = 5 m s-l, H = 2OOm, Dz,, = 3m2s-‘, 0.6
I
I
I 2,
=0.6
One of the major advantages of the generalized dispersion theory approach is in its ability to handle transient behavior in a natural manner. That is, the behavior of the system leading up to the steady-state results displayed in Fig. 1 is part of the solution of the model equations, and the time required to reach the steady state at each downwind location may be estimated. Figure 2 presents an example of the results obtained; here the ground level concentration is plotted vs downwind distance at various values of the dimensionless time, ‘t, with Z, = 0.6 and /3 = 10. For these conditions, the steady state (z = cc.) yields the highest ground level concentrations at the ground for all downwind distances. It also shows that a dimensionless time of 0.2 is required to reach the steady state over most of the downwind distances of interest. If the example given in 4.1 is altered slightly to a source height zi = 120 m, then this figure indicates that about 45min are required to reach the steady state for the parameters selected. However, other realistic values of the parameters may be chosen for which the time to reach steady state is prolonged to several hours (Vaidyanathan, 1976).
0.5
REFERENCES 0.4
G x_ 0.3 IQ 0.2
0.1
0 IO-' X
Fig. 2. Dimensionless ground level concentration as a function of dimensionless downwind distance for various values of dimensionless time with Z, = 0.6 and B = 10.
Courant R. and Hilbert D. (1962) Methods ofMathematical Physics, Vol II, p. 204. Interscience, New York. Gartrell F. E., Thomas F. W. and Carpenter S. B. (1963) Atmospheric oxidation of SO, in coal-burning power plant plumes. Am. ind. Hyg. Ass. J. 24, 113-120. Gill W. N. and Sankarasubramanian R. (1972) Dispersion of non-uniformly distributed time-variable continuous sources in time-dependent flow. Proc. R. Sot. A, 327, 191-208. Monin A. S. and Yaglom A. M. (1971) Statistical Fluid Mechanics, Vol. 1, chapter 10. M.I.T. Press, Cambridge. Nunge R. J. (1974a) Application of an analytical solution for unsteady, advective-diffusion to dispersion in the atmosphere-I. Theory. Atmospheric Environment 8. 969-983. Nunge R. J. (1974b) Application of an analytical solution for unsteady, advective-diffusion to dispersion in the atmosphere-II. Results. Atmospheric Environment 8, 9X4-1001.
856
RICHARD J. NUNGE
and K. R.
Nunge R. J. and Subramanian R. S. (1977) Atmospheric dispersion of gaseous pollutants from a continuous source-a model of an industrial city. American Institute of Chemical Engineers Symposium Series, No. 165, Vol. 73, 10-24.
VAIDYANAIHAN
Vaidyanathan K. R. (1976) Atmospheric dispersion of gaseous pollutants from a continuous line source with a first-order, homogeneous, irreversible chemical reaction. M.S. thesis, Clarkson College of Technology.
En~ironmenr
Vol. 11. pp. 853-856.
A NOTE
Pergamon
Press 1977. Printed in Great Britain.
ON ATMOSPHERIC DISPERSION CHEMICAL REACTION RICHARD 4. NUNGEand K. R.
WITH
VAIDYANATHAN
Department of Chemical Engineering, Clarkson College of Technology, Potsdam, NY 13676, U.S.A. (Received
1 November 1976)
Abstract-aeneralized dispersion theory is used to obtain an analytical solution tive-diffusion equation for dispersion of a reactive pollutant in the atmosphere. obtained for the ground level concentration as a function of time and spatial tinuous crosswind line source at different elevations and for various values of rate under inversion conditions.
NOMENCLATURE concentration turbulent diffusivity of mass coefficients in equation (16) dimensionless inversion height dispersion coefficients defined by equation dimensionless source strength, mass/time/length @ID,, time aIn0 wind speed in the .x-direction WWe downwind coordinate zlH vertical coordinate KH=/D,o Dirac delta function c/c0 first-order reaction rate constant t D:,/H’
to the unsteady advecNumerical results are coordinates for a conthe first-order reaction
Yaglom (1971) for a continuous source. point concentration is described by
ac z
ac
+
a
~(4da, = z&L(z)
Thus
the
ac z
+
S(t, x, z; zl) - kc,
(18)
1. INTRODUCTION Earlier, the generalized dispersion theory was used to develop an analytical solution to the unsteady, advective-diffusion equation for an empirical eddy diffusivity model of dispersion in the atmosphere from an instantaneous source (Nunge, 1974a, b). Solutions were obtained for the point concentration distribution under various conditions. Later, Nunge and Subramanian (1977) applied the theory to a continuous area source of an inert pollutant. The purpose of this note is to report an extension of the continuous source problem to include the effect of a first-order, homogeneous, irreversible chemical reaction.
(1) where the source function S for a line source at z = zi may be written as 5(r,x,z;z,) = m(r) 6(x) 6(z - 21).
(2)
Here x is the mean wind direction, z is the height above ground and 6 is the Dirac delta function. For comparison with previous papers in this series, it is assumed in writing Equation (1) that downwind diffusion can be ignored compared with downwind advection. Also, there is no pollutant in the atmosphere initially, the ground is impermeable to the pollutant and there is no penetration of the pollutant through the inversion layer at z = H. Although more realistic wind and eddy diffusivity profiles may be used, to obtain particular results comparable to those in the earlier papers, it is further assumed that the mean wind is steady and linear in z and the eddy diffusivity, D,, is constant at D,,. In addition, although time-dependent sources are readily handled, for simplicity the emission rate per unit length, m, is considered constant at m,. If time-dependent sources are treated, it may be necessary to include the downwind diffusion term in Equation (1). Under the above assumptions, the model solved here may be written as
ac z
+
uoz
ac
F
ax
=
a% Dzoazz
+
m&c) &z - ~1) -
KC
(3)
2. THE PHYSICAL PROBLEM c(0, x, z) = 0 The particular physical problem chosen for study is the infinite cross-wind line source such as has been used to model a highway. The local time-averaged (or ensemble average) concentration of pollutant is assumed to satisfy a two-dimensional version of the advective-diffusion equation given by Monin and
D,, ;
(6 x, 0) = Dr,,
2 (t, x, H)
f&m,3 =0. 853
(4) = 0
(5)
Equation (6) results from there being a finite amount of pollutant in the atmosphere at all finite times.
3. METHOD
OF SOLtiTlOh
In view of the previous development of the theory (Nunge. 1974a; Nunge and Subramanian, 1977) and other applications of the generalized dispersion theory to similar continuoLIs source problems (Gill and Sankarasumbramanian. 1972). the method of solution will only be outlined here; full details of the solution are given elsewhere (Vaidyanathan, 1976). The solution of Equations (3~(6) may be formulated as the time integral of the solution of an initial value problem by using superposition (Courant and Hilbert. 1962); thus
where
(8) with l+b,(O, x, z) = d(x) 6(2 - z,)
(9 (10) (11)
The problem is solved in dimensionless form generality; the dimensionless groyps chosen are
for
Equations (12)-(15) are a special case of the problems solved in genera1 previously (Nunge. 1974a). using the generalized dispersion theory, wherein + is represented by
with $,, = j ’ $(t* X, Z) dZ 0 and
The key to the generalized dispersion theory is determining the gir and Ki coefficients in equations (I 6) and (18). Equation (12) is first integrated over 2 and then Equation (16) is substituted for II/. The result is of the form given by Equation (18), so that equations for the Ki as integrals over Z of the gk coefficients are obtained. The gk functions are found by solving the equations resulting from substituting Equation (16) into Equation (12) and setting the coefficients of ak$ Ji)Xk equal to zero. As discussed previously (Nunge, 1974b), it is necessary to retain only terms of g2 and K? in the infinite series represented by equations (16) and (18) to obtain convergence over a wide range of z, X and Z. 4 lVUMERlCAL
RESULTS
From the analysis above, the functional of the dimensionless concentration is 0 = i)(~,s.z;z,,~).
where c0 is the reference concentration 2~1~~~~~ chosen such that O- I as X + -L for It = 0 at all finite times. Equations (@(I 1) in dimensionless form become
(12) l/40, x, Z) = 6(X) 6(< - 2,)
(13)
(17)
dependence
(19)
Predictions of the point concentration, 0, have been obtained over a wide range of values of the independent variables 7, X, Z for /j = 0, I. 10 and 101) with Z, = 0.1,0.2,0.4, 0.6 and 0.8. Extensive plots of these results are given elsewhere (Vaidyanathan, 1976). 4.1 Sfeady state For the most part, one is concerned with the concentration levels which occur at ground level. As representative of the results obtained. Fig. 1 presents the dimensionless ground level concentration vs X for a dimensionless source elevation of 0.6 with fi as a parameter. Examination of this figure shows that as X -+ i/-, O+ 1.O when there is no reaction (/I = 0). For 0 = I. there is only a small effect on the ground level concentration compared to the case of no reaction until downwind distances exceeding X = 0.1 have been reached. For sources close to the ground,
A note on atmospheric dispersion with chemical reaction I
I
’
855
Z, = 0.1 and 0.2, increasing p to 10 still has only relalow level emission elevation in a neutral atmosphere; the value of K for the reaction of SOz with 0, forming SO, was taken from Gartrell et al. (1963), which may be somewhat low for dry atmospheric conditions. For Z, = 0.2 and /? = 1.0, the maximum dimensionless ground level concentration for these conditions is ~10 at X ‘v 1 x 10e3. Furthermore, use of the definitions of the dimensionless groups and
z,=o..6
\
2m0 cg = _ uoH reveals that a value of m,, = 1.8 g SO2 (m s)- * will yield a ground level concentration of 365 pg mm3 at x = 66.7 m downwind of the source. 4.2 Transient behavior IO‘<
IO =
IO
10-l
X Fig. 1. Steady-state dimensionless ground level concentration as a function of dimensionless downwind distance
for various values of the dimensionless reaction parameter with Z1 = 0.6. zi = 0.1 and 0.2, increasing /l to 10 still has only relatively small effects on the ground level concentration close to the source, but greater differences appear as the downwind distance from the source increases. The general effect of increasing /3 is to decrease the ground level concentration for all values of X. As particular example of the application of these steady-state results, suppose it is desired to find the maximum allowable emission rate per unit length of SOz using the primary standard of 365 pgrne3 of SO, as the average ground level concentration for a 24 h period. Consider the physical situation in which u0 = 5 m s-l, H = 2OOm, Dz,, = 3m2s-‘, 0.6
I
I
I 2,
=0.6
One of the major advantages of the generalized dispersion theory approach is in its ability to handle transient behavior in a natural manner. That is, the behavior of the system leading up to the steady-state results displayed in Fig. 1 is part of the solution of the model equations, and the time required to reach the steady state at each downwind location may be estimated. Figure 2 presents an example of the results obtained; here the ground level concentration is plotted vs downwind distance at various values of the dimensionless time, ‘t, with Z, = 0.6 and /3 = 10. For these conditions, the steady state (z = cc.) yields the highest ground level concentrations at the ground for all downwind distances. It also shows that a dimensionless time of 0.2 is required to reach the steady state over most of the downwind distances of interest. If the example given in 4.1 is altered slightly to a source height zi = 120 m, then this figure indicates that about 45min are required to reach the steady state for the parameters selected. However, other realistic values of the parameters may be chosen for which the time to reach steady state is prolonged to several hours (Vaidyanathan, 1976).
0.5
REFERENCES 0.4
G x_ 0.3 IQ 0.2
0.1
0 IO-' X
Fig. 2. Dimensionless ground level concentration as a function of dimensionless downwind distance for various values of dimensionless time with Z, = 0.6 and B = 10.
Courant R. and Hilbert D. (1962) Methods ofMathematical Physics, Vol II, p. 204. Interscience, New York. Gartrell F. E., Thomas F. W. and Carpenter S. B. (1963) Atmospheric oxidation of SO, in coal-burning power plant plumes. Am. ind. Hyg. Ass. J. 24, 113-120. Gill W. N. and Sankarasubramanian R. (1972) Dispersion of non-uniformly distributed time-variable continuous sources in time-dependent flow. Proc. R. Sot. A, 327, 191-208. Monin A. S. and Yaglom A. M. (1971) Statistical Fluid Mechanics, Vol. 1, chapter 10. M.I.T. Press, Cambridge. Nunge R. J. (1974a) Application of an analytical solution for unsteady, advective-diffusion to dispersion in the atmosphere-I. Theory. Atmospheric Environment 8. 969-983. Nunge R. J. (1974b) Application of an analytical solution for unsteady, advective-diffusion to dispersion in the atmosphere-II. Results. Atmospheric Environment 8, 9X4-1001.
856
RICHARD J. NUNGE
and K. R.
Nunge R. J. and Subramanian R. S. (1977) Atmospheric dispersion of gaseous pollutants from a continuous source-a model of an industrial city. American Institute of Chemical Engineers Symposium Series, No. 165, Vol. 73, 10-24.
VAIDYANAIHAN
Vaidyanathan K. R. (1976) Atmospheric dispersion of gaseous pollutants from a continuous line source with a first-order, homogeneous, irreversible chemical reaction. M.S. thesis, Clarkson College of Technology.