- Email: [email protected]
Prediction of long-term mean and mean square pollutant concentrations in an urban atmosphere
Discussions
1240
AUTHOR’S
REPLY
I thank Dr. Cliff Davidson for his comments on my paper. The relative contribution of wet and dry deposition has no influence on the conclusions in my article and has therefore not been considered in detail. From Nord-Austlandet there exist only such meteorological data as can be inferred from weather maps. These maps indicate that the period when the highest content of anthropogenic material appeared was dominated by airmasses that passed central parts of U.S.S.R. some 24 hours earlier. Furthermore, the trajectories were somewhat undirectional during the time interval before the next snowfall. Taking into consideration the short period with anthropogenic deposition, it is reasonable to assume that the better part of the snow contaminants in this layer most probably result from wet deposition. Any detailed study of the relative contribution of wet and dry deposition in this area should also take into consideration
MULTIPLE-SOURCE PLUME MODELS OF URBAN AIR POLLUTION: THEIR GENERAL STRUCTURE* Calder (1977) performs a valuable service to urban pollution modelers by detailing the mathematics of plume diffusion models. His timely and authoritative exposition provides a much-needed mathematical basis for the subject. He correctly points out that “considerable simplification” and “major saving of computer time” result from making use in numerical area-source diffusion models of the observation that plumes are long and narrow. 1 would like to offer two brief comments, hoping by this means to underline Calder’s remarks on this small but, as it turns out, crucial point. First, Calder refers to the “narrow-plume hypothesis”. It may not be clear to all readers that he uses the term “hypothesis” in the strictly mathematical sense. He does not, I feel sure, mean to imply any uncertainty about the physical reality of long, narrow plumes. The very word “plume” is descriptive of the real world and would otherwise be inaccurate. “Point source plumes in the atmosphere tend to be long and narrow, and so the concentration at a point can be influenced only by sources in a narrow, plume-shaped upwind sector” (Gifford and Hanna, 1970). * Calder
K. L. (1977) Atmospheric
Environment
11,
403-414.
Institute of Geophysics Uniuersiry of Bergen 5014 Bergen, Norway
YNWART. GIESSING
REFERENCE Gjessing Y. T. (1977) The filtering effect of snow. Int. Ass. Hydrol. Sci. Publ. 118.
The second point is that it is not only in the area of numerical computations that the narrow-plume idea pays off. As Calder (1969) remarked in his unoublished note. “. . . it does appear likely that an important mathematical simplification may be possible if use is made, as suggested by Gifford, of the known fact that normally the concentration of point-source plumes decreases very rapidly with distance from the plume centerline.” By use of this simplification we have derived a simple analytical area-source diffusion model that has proved quite effective in applications. When we first proposed the narrow-plume simplification, urban modelers tended to be quite skeptical of the neglect of lateral diffusion. Calder’s strong support on this point is thus especially welcome. National Oceanic and Atmospheric Admin. P.O. Box E Oak Ridge,Tn 37830, U.S.A.
FRANKLINA. GIFFORD,JR.
REFERENCES Calder K. L. (1969) A narrow plume simplification for multiple source urban pollution models (unpublished). Gifford F. A. and Hanna S. R. (1970) Urban air pollution modeling. Paper No. ME-320, Proc. 2nd Int. Clean Air Congress, Washington, D.C.
PREDICTION OF LONG-TERM MEAN AND MEAN SQUARE POLLUTANT CONCENTRATIONS IN AN URBAN ATMOSPHERE* The authors of the above paper present comprehensive concentration models for point, line and area sources for use in predicting long-term mean and mean square pollutant concentrations in an urban atmosphere. On page 710 of the paper, the authors state that the line-source concentration model given by Equation (21) must in principle be evaluated ‘Kumar
the contribution from the air-filtering inside the snow volume. Due to variations in air pressure and strong winds above the surface, the air inside the snow volume, say 30-50 vol. %, will constantly be replaced and the snow will act as a mechanical filter for the particulate content of the air. Experiments have shown (Gjessing, 1977) that, say, a 1 m snow layer is not a totally effective filter for aerosols. Therefore, a part of the aerosol content may be transported to deeper layers before it is absorbed or deposited.
S., Lamb R. G. and Seinfeld J. H. (1976)
Atmospheric Environment 10, 707-715.
numerically when the line source is not perpendicular to the wind direction 0. They then present an analytic solution in Equation (22) obtained under the assumption that cryand a, are independent of the distances 8’ along the line source. In this discussion, we show an analytic solution for Equation (21) in which fY and err are dependent on /Y and linearly related to the distance x from the source. To our knowledge, such a solution does not appear in the literature. Figure 1, which is similar to Fig. 2 in the paper by Kumar et al. (1976) shows a continuously emitting finite line source of length L, directed along the /I coordinate at height z’ with one end of the line source at the point (G(= 0, p = 0, z = 2’). The contribution to the concentration at point r from a point source located at point S in Fig. 1 is given by the expression
Discussions
1241
Let:
xs=&{exp[-f$~]
A=-
+erp[ -t(y]} x
S,k
(9)
2nii1;
{exp[ -i($y]}
(1)
where S, is the source strength per unit length of the line source and ii is the mean wind speed. Following Csanady (1973, p. 69) and others, we express uYand u, as
B = [k(z - z’)]’
(10)
C = [k(z + z’)]’
(11)
D=
(12)
X'
tan e - (&OS e)
0 = X’ - p sin e
(13)
Equation (7) can be restated in the form
B+(x’cOte-WcOte+~)z 1W Y ~+(~~c0te-~c0te+D)~
I dw
PO2 Y (14)
or, XL=
-~S:l-“i”“~{exp[-:(~-~+l)] +exp[-j($-~+I)]}d~
(15)
where
Fig. 1. Line source geometry.
F = (Ez + B)/I;
(16)
E=x’cotO+D
(17)
G = (I? + C)/I;
(18)
H = 2E cot O/l;
(19)
I = cot2 e/r;
(20)
Equation (15) can be further rewritten in the form and
uz=l,x=5x kY
x1 = -${exp[
x’=(a+/ItanB)cosO
y=B’+j-B
P=jYcos6+x’tanO-(fi/cose) CDSe
- &)‘]du
(4)
+exp[ -$(I
- g)]l:l-“i”$
(5)
xexp[-;(;-;)i]dw}.
The concentration at the point r produced by the line source is given by
s
-f)]jll-“i”“$
xexp[ -:(i
where k is a constant and I, and I, are the vertical and lateral intensities of turbulence. From the geometry shown in Fig. 1, x=x’-g=x’-/?sinO;
-;(I
(3)
If we let J=&_$)
(22)
K=&?)
(23)
D
x1 =
x,
=
S,k 2nii1;
s 1
xsdB
(6)
0
1
(21)
0 (x’ - /Y sin e)z
the integrals appearing in Equation (21) become
:;-‘5i”:;exp[
-;(;
_ !#m
s (7)
1,(x’ -p
P cos e + X’tan e - (~?/cose) sin e)
dB s
where acote; L.:
P+acote$Lr 4+ac0te>Lp
(8)
(24)
1242 where
-f(& -$))I}
Jo&-g)
(26)
F
1 H -2 x’ - e sin 0 2F
J(
J,=
(27)
Ko=J4(pJ K, =
(30)
--
To conserve space, we have omitted the steps showing the substitution of the original parameters for A, f, G, H and I in Equation (30). It should be noted that Equation (30) can easily be extended to include the limitation to vertical mixing presented by an elevated inversion at height h. We believe the use of Equation (30) when the line source is not at right angles to the wind direction should significantly reduce the computation time from that required by numerical techniques.
H 2G
(29)
From the definition of the error function, the analytic solution is
R. K. DUMBAULD S. F. SATERLIE
H.E. Cramer Company P.O. Box 8049 Salt Lake City, UT 84108, U.S.A.
REFERENCE
Csanady G. T. (1973) Turbulent Diffusion in the Environment. Reidel, Derdricht.
OPTIMUM
DESIGN OF VENTURI SCRUBBERS*
I note that the pressure drop of the liquid droplet collector following the venturi (but part of the total collector) is ignored. It may be negligible compared to the venturi when droplet size is large but when droplet size is decreased by separate atomization, as is suggested, with the concomittant reduction in venturi pressure drop postulated, may not this require a higher pressure drop droplet collector and perhaps the additional pressure drop of a mist eliminator to follow the
THE DURATION OF HIGH SO2 CONCENTRATIONS IN AN URBAN ATMOSPHERE* The representation of the simple model on air quality has been required not only to explain the mechanism of air pollution but also for the establishment of air pollution abatement strategy. In this respect, the study developed by Drufuca and Giugliano will be worthy of attention. Here in this discussion section, I would like to reinforce the availability of their analysis from the view point of theoretical foundation. (1) As for their Fig. 3 and Equation (1) of “Total time T that the instantaneous concentration C is exceeded, for each time series, where e is indicated”. * Drufuca G. and Giugliano M. (1977) Atmospheric Environment 11, 729-736.
droplet collector. Is it possible that optimum design may have to take these factors into account as well as those considered in the paper? ARTHURC.
STERN
Department of Environmental Sciences University of North Carolina Chapel Hill, N.C., U.S.A. * Gael K. C. and Hollands K. G. T. (1977) Atmospheric Environment 11, 837-845.
Assuming that the air pollutant concentration is lognormally distributed, the total time T that the concentration exceeds C is expressed as 5 T= T,,
f(x> d dx, sc where TO,J x and u denote
f(x,u)==xfiaexp
I
-
(InI - Inj$ 2a2 1’
where j, is geometric mean concentration and is related to arithmetic mean ci as follows which is equal to their Equation (6).
1240
AUTHOR’S
REPLY
I thank Dr. Cliff Davidson for his comments on my paper. The relative contribution of wet and dry deposition has no influence on the conclusions in my article and has therefore not been considered in detail. From Nord-Austlandet there exist only such meteorological data as can be inferred from weather maps. These maps indicate that the period when the highest content of anthropogenic material appeared was dominated by airmasses that passed central parts of U.S.S.R. some 24 hours earlier. Furthermore, the trajectories were somewhat undirectional during the time interval before the next snowfall. Taking into consideration the short period with anthropogenic deposition, it is reasonable to assume that the better part of the snow contaminants in this layer most probably result from wet deposition. Any detailed study of the relative contribution of wet and dry deposition in this area should also take into consideration
MULTIPLE-SOURCE PLUME MODELS OF URBAN AIR POLLUTION: THEIR GENERAL STRUCTURE* Calder (1977) performs a valuable service to urban pollution modelers by detailing the mathematics of plume diffusion models. His timely and authoritative exposition provides a much-needed mathematical basis for the subject. He correctly points out that “considerable simplification” and “major saving of computer time” result from making use in numerical area-source diffusion models of the observation that plumes are long and narrow. 1 would like to offer two brief comments, hoping by this means to underline Calder’s remarks on this small but, as it turns out, crucial point. First, Calder refers to the “narrow-plume hypothesis”. It may not be clear to all readers that he uses the term “hypothesis” in the strictly mathematical sense. He does not, I feel sure, mean to imply any uncertainty about the physical reality of long, narrow plumes. The very word “plume” is descriptive of the real world and would otherwise be inaccurate. “Point source plumes in the atmosphere tend to be long and narrow, and so the concentration at a point can be influenced only by sources in a narrow, plume-shaped upwind sector” (Gifford and Hanna, 1970). * Calder
K. L. (1977) Atmospheric
Environment
11,
403-414.
Institute of Geophysics Uniuersiry of Bergen 5014 Bergen, Norway
YNWART. GIESSING
REFERENCE Gjessing Y. T. (1977) The filtering effect of snow. Int. Ass. Hydrol. Sci. Publ. 118.
The second point is that it is not only in the area of numerical computations that the narrow-plume idea pays off. As Calder (1969) remarked in his unoublished note. “. . . it does appear likely that an important mathematical simplification may be possible if use is made, as suggested by Gifford, of the known fact that normally the concentration of point-source plumes decreases very rapidly with distance from the plume centerline.” By use of this simplification we have derived a simple analytical area-source diffusion model that has proved quite effective in applications. When we first proposed the narrow-plume simplification, urban modelers tended to be quite skeptical of the neglect of lateral diffusion. Calder’s strong support on this point is thus especially welcome. National Oceanic and Atmospheric Admin. P.O. Box E Oak Ridge,Tn 37830, U.S.A.
FRANKLINA. GIFFORD,JR.
REFERENCES Calder K. L. (1969) A narrow plume simplification for multiple source urban pollution models (unpublished). Gifford F. A. and Hanna S. R. (1970) Urban air pollution modeling. Paper No. ME-320, Proc. 2nd Int. Clean Air Congress, Washington, D.C.
PREDICTION OF LONG-TERM MEAN AND MEAN SQUARE POLLUTANT CONCENTRATIONS IN AN URBAN ATMOSPHERE* The authors of the above paper present comprehensive concentration models for point, line and area sources for use in predicting long-term mean and mean square pollutant concentrations in an urban atmosphere. On page 710 of the paper, the authors state that the line-source concentration model given by Equation (21) must in principle be evaluated ‘Kumar
the contribution from the air-filtering inside the snow volume. Due to variations in air pressure and strong winds above the surface, the air inside the snow volume, say 30-50 vol. %, will constantly be replaced and the snow will act as a mechanical filter for the particulate content of the air. Experiments have shown (Gjessing, 1977) that, say, a 1 m snow layer is not a totally effective filter for aerosols. Therefore, a part of the aerosol content may be transported to deeper layers before it is absorbed or deposited.
S., Lamb R. G. and Seinfeld J. H. (1976)
Atmospheric Environment 10, 707-715.
numerically when the line source is not perpendicular to the wind direction 0. They then present an analytic solution in Equation (22) obtained under the assumption that cryand a, are independent of the distances 8’ along the line source. In this discussion, we show an analytic solution for Equation (21) in which fY and err are dependent on /Y and linearly related to the distance x from the source. To our knowledge, such a solution does not appear in the literature. Figure 1, which is similar to Fig. 2 in the paper by Kumar et al. (1976) shows a continuously emitting finite line source of length L, directed along the /I coordinate at height z’ with one end of the line source at the point (G(= 0, p = 0, z = 2’). The contribution to the concentration at point r from a point source located at point S in Fig. 1 is given by the expression
Discussions
1241
Let:
xs=&{exp[-f$~]
A=-
+erp[ -t(y]} x
S,k
(9)
2nii1;
{exp[ -i($y]}
(1)
where S, is the source strength per unit length of the line source and ii is the mean wind speed. Following Csanady (1973, p. 69) and others, we express uYand u, as
B = [k(z - z’)]’
(10)
C = [k(z + z’)]’
(11)
D=
(12)
X'
tan e - (&OS e)
0 = X’ - p sin e
(13)
Equation (7) can be restated in the form
B+(x’cOte-WcOte+~)z 1W Y ~+(~~c0te-~c0te+D)~
I dw
PO2 Y (14)
or, XL=
-~S:l-“i”“~{exp[-:(~-~+l)] +exp[-j($-~+I)]}d~
(15)
where
Fig. 1. Line source geometry.
F = (Ez + B)/I;
(16)
E=x’cotO+D
(17)
G = (I? + C)/I;
(18)
H = 2E cot O/l;
(19)
I = cot2 e/r;
(20)
Equation (15) can be further rewritten in the form and
uz=l,x=5x kY
x1 = -${exp[
x’=(a+/ItanB)cosO
y=B’+j-B
P=jYcos6+x’tanO-(fi/cose) CDSe
- &)‘]du
(4)
+exp[ -$(I
- g)]l:l-“i”$
(5)
xexp[-;(;-;)i]dw}.
The concentration at the point r produced by the line source is given by
s
-f)]jll-“i”“$
xexp[ -:(i
where k is a constant and I, and I, are the vertical and lateral intensities of turbulence. From the geometry shown in Fig. 1, x=x’-g=x’-/?sinO;
-;(I
(3)
If we let J=&_$)
(22)
K=&?)
(23)
D
x1 =
x,
=
S,k 2nii1;
s 1
xsdB
(6)
0
1
(21)
0 (x’ - /Y sin e)z
the integrals appearing in Equation (21) become
:;-‘5i”:;exp[
-;(;
_ !#m
s (7)
1,(x’ -p
P cos e + X’tan e - (~?/cose) sin e)
dB s
where acote; L.:
P+acote$Lr 4+ac0te>Lp
(8)
(24)
1242 where
-f(& -$))I}
Jo&-g)
(26)
F
1 H -2 x’ - e sin 0 2F
J(
J,=
(27)
Ko=J4(pJ K, =
(30)
--
To conserve space, we have omitted the steps showing the substitution of the original parameters for A, f, G, H and I in Equation (30). It should be noted that Equation (30) can easily be extended to include the limitation to vertical mixing presented by an elevated inversion at height h. We believe the use of Equation (30) when the line source is not at right angles to the wind direction should significantly reduce the computation time from that required by numerical techniques.
H 2G
(29)
From the definition of the error function, the analytic solution is
R. K. DUMBAULD S. F. SATERLIE
H.E. Cramer Company P.O. Box 8049 Salt Lake City, UT 84108, U.S.A.
REFERENCE
Csanady G. T. (1973) Turbulent Diffusion in the Environment. Reidel, Derdricht.
OPTIMUM
DESIGN OF VENTURI SCRUBBERS*
I note that the pressure drop of the liquid droplet collector following the venturi (but part of the total collector) is ignored. It may be negligible compared to the venturi when droplet size is large but when droplet size is decreased by separate atomization, as is suggested, with the concomittant reduction in venturi pressure drop postulated, may not this require a higher pressure drop droplet collector and perhaps the additional pressure drop of a mist eliminator to follow the
THE DURATION OF HIGH SO2 CONCENTRATIONS IN AN URBAN ATMOSPHERE* The representation of the simple model on air quality has been required not only to explain the mechanism of air pollution but also for the establishment of air pollution abatement strategy. In this respect, the study developed by Drufuca and Giugliano will be worthy of attention. Here in this discussion section, I would like to reinforce the availability of their analysis from the view point of theoretical foundation. (1) As for their Fig. 3 and Equation (1) of “Total time T that the instantaneous concentration C is exceeded, for each time series, where e is indicated”. * Drufuca G. and Giugliano M. (1977) Atmospheric Environment 11, 729-736.
droplet collector. Is it possible that optimum design may have to take these factors into account as well as those considered in the paper? ARTHURC.
STERN
Department of Environmental Sciences University of North Carolina Chapel Hill, N.C., U.S.A. * Gael K. C. and Hollands K. G. T. (1977) Atmospheric Environment 11, 837-845.
Assuming that the air pollutant concentration is lognormally distributed, the total time T that the concentration exceeds C is expressed as 5 T= T,,
f(x> d dx, sc where TO,J x and u denote
f(x,u)==xfiaexp
I
-
(InI - Inj$ 2a2 1’
where j, is geometric mean concentration and is related to arithmetic mean ci as follows which is equal to their Equation (6).