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An analytical model for air pollutant transport and deposition from a point source
Atmospherrc Envsonmenf
Vol
11, pp 231-237
Pergamon
Press 1977. Prmted m Great Brdam.
AN ANALYTICAL MODEL FOR AIR POLLUTANT TRANSPORT AND DEPOSITION FROM A POINT SOURCE* L. ERMAK
DONALD
Biomedical and Environmental Research Division, Lawrence Livermore Laboratory, University of California, Livermore, CA 94550, U.S.A. (First received 7 May 1976 and in jinal form 18 August 1976)
Abstract-An atmospheric transport and deposition model is presented for pollutants emitted from an elevated point source over flat terrain. The model is obtained from the analytic solution of the atmospheric diffusion equation with the coefficients of eddy diffusion taken to be functions of downwind distance and the average wind velocity taken to be constant. Ground deposition of the pollutants is accounted for by (1) including a gravitational settling term in the atmospheric diffusion equation and by (2) applying an absorptive boundary condition at the ground surface. In order to facilitate application of the model, the results for the general situation that includes settling and deposition (and where the distribution is no longer Gaussian) are expressed in terms of the Grassian plume parameters, and specifically the crosswind and vertical standard deviation functions U,,(X)and g=,(x)_ Graphs of the pollutant air concentration and ground deposition flux are presented for a number of deposition conditions.
INTRODUCTION Pollutant particles emitted into the atmosphere may be removed by a number of natural processes. One of the primary removal mechanisms is dry deposition onto the surface of the earth as a result of gravitational settling and ground absorption by the soil, vegetation, buildings, or a body of water. Surface deposited pollutants may have a significant impact upon the local ecosystem as the pollutants enter into and travel through the biological pathways. In addition, depletion of pollutant plumes in this manner will affect the pollutant air concentration, especially when deposition occurs over a long distance. This paper addresses the problem of air pollution removal and presents an analytical plume model for the transport and dry deposition of air pollutants. In the absence of any removal mechanisms, the Gaussian plume dispersion model is the basic method used to calculate air pollution concentrations from a point source (Turner, 1970; Carpenter et al., 1971; Morgenstern et al., 1975). Use of the Gaussian plume model began to receive popularity when Pasquill (1961) published his dispersion rates for plumes over open level terrain. Subsequently, Hilsmeier and Gifford (1962) expressed these estimates in a slightly more convenient, although exactly equivalent form, and this socalled PasquillGifford system for dispersion estimates has been widely used ever since. Its application has frequently involved the use of the method suggested by Turner (1964) for determining the appropriate atmospheric stability class from the *This work was performed under the auspices of the U.S. Energy Research and Development Administration under Contract No. W-7405-ENG-48.
observed solar radiation and wind speed. The basic idea of representing the common Gaussian plume for an elevated point source in terms of a K-theory diffusion equation was suggested by Wipperman (1966) and noted by Fortak (1970). The Gaussian plume model has achieved popularity because it is easy to use, plume dispersion data are readily available, and most measured data fit the model reasonably well (Whaley, 1974; Shum et al., 1975). The problem of atmospheric transport when gravitational settling and ground absorption cannot be ignored has been discussed by Calder (1961). In his formulation, both the gravitational settling flux and the ground deposition flux are taken to be proportional to the local air concentration. The proportionality factors are respectively called the gravitational settling velocity and the deposition velocity and are generally not equal to each other. Using this type of formulation, solutions to the atmospheric transport equation have been obtained for a number of cases (Rounds, 1955; Smith, 1962; Heines and Peters, 1974). These solutions have not been used extensively, presumably due either to their complicated nature or the difficulty in applying them under a variety of atmospheric stability conditions. A method for including the effects of surface deposition into a plume transport model, which has received considerable use, is the source depletion approach (Hosker, 1973; Martin et al., 1974; Heffter and Ferber, 1975; Vaughan et al., 1975). This approach is described in detail by Pasquill (1962) and essentially treats ground deposition as a perturbation to the Gaussian plume dispersion model. The shape of the vertical plume profile is assumed to be unaltered by the deposition process and the constant
231
232
DONALD L. ERMAK
source strength is replaced by a vtrtual decreasing source strength. This virtual source strength is derived from an integral form of the continuity equation and the assumption that the deposition rate is proportional to the pollutant air concentration at ground level. The result is a plume which diminishes exponentially with downwind distance while retaining the original Gaussian shape of the undepleted plume. The purpose of this work is to develop an analytical plume model for the atmospheric dispersion of pollutants which treats pollutant deposition in a more physically-realistic manner than the source depletion approach and yet retains the ease of application associated with Gaussian plume type models. The model presented here treats dry deposition m the manner proposed by Calder (1961) where both the gravitational settling flux and the ground deposition flux are taken to be proportional to the local air concentration. Using this assumption and the basic assumptions of the Gaussian plume model, the atmospheric transport equation is solved and analytic expressions are obtained for the pollutant air concentration and ground deposition flux. MATHEMATICAL
FORMLLATION
OF iHE
PROBLEM
The situation to be analyzed is that of the pollutant dispersion and dry deposition from an elevated point source which 1s contmuously emitting a non-reacting pollutant. It is assumed that the terrain is flat, the average wind velocity is constant, and the atmosphere is unlimited in the vertical direction. A Cartesian coordinate system is used with the x-axis oriented in the direction of the constant wind, the )I-axis in the horizontal crosswind direction, and z-axis oriented in the vertical crosswind direction. Pollutant transport is assumed to be governed by the atmospheric advectiondiffusion equation EC
_= at
a /
ii.YKX1_.-
ac (.Y
I;cc.i’K,dC ay
8~
time t-x/U. Consequently. only the steady state solution to the constant source problem needs to be considered. Second, the diffusion coeffictents in the J- and z-directions are taken to be functions of only the downwind distance from the source. Smce the pollutant particles travel downwind at a constant speed. this 1s equivalent to assuming that the crosswind diffusion coefficients are functions of the downwind flight time. An analogous situation arises in the theory of Brownian motion where the diffuston coefficient of the Brownian particles is a function of time (Chandresekhar, 1943). The equation to be solved IS therefore the steady state form of the atmospheric advectiondiffusion equation and is
As previously stated, the diffusivitles K, and KZ are functions of only the downwind distance and are therefore independent of the height 2. Furthermore. the settling velocity Wz IS also taken to be a function of only the downwind distance .Y and its functional form is to be that of KZ. At large downwind distances it is assumed that both the vertical diffusrvrty K=(x) and the settling velocity W,(s) approach the limiting values K and W respectively. The mathematical description of the problem IS completed by the boundary conditions which accompany equation (2). They are (2a) A continuous point source at (O,O, h) of constant strength Q. C(0 . 4' 7) = 1-
Qb(y)6(- /I) -
Lf
(2b and c) The pollutant concentration approaches zero far from the source in the lateral duections. C(x. * %. -_)= 0. (2d) The pollutant concentration high above the source height
approaches
zero
C(s. .1’./,) = 0 Here C is the pollutant concentration m the air at any location (x, I’, z) and time t; K,, K,, and K, are the coefficients of eddy diffusivity in the x-, y-, and z-directions, respectively; U is the constant average wind speed; and W, is the pollutant particle gravitational settling velocity and is positive in the downward negative z-direction. Equation (1) is simplified by using two basic assumptions of the Gaussian plume approach. First, the wind speed is assumed to be sufficiently large so that diffusive transport in the x-direction can be neglected in comparison to advection by the wind. This restraint makes the x-component of the pollutant particle velocity a constant and equal to the average wind speed U. The pollutant concentratton at any time t and a distance downwind x from the source IS therefore proportional to the source strength at the
(2e) Pollutant depositton onto the ground occurs at a rate proportional to the local au concentration.
The deposition velocity c’will depend upon such factors as the type and size of the pollutant particles. the roughness of terrain and the type of ground cover. and the meteorological conditions. A review of the deposition velocities for a number of pollutants over a variety of ground surface conditions is given by Rasmussen et al. (1975). Using Stokes law, the terminal gravitational settling velocity is (3)
233
An pollutant transport and deposition from a point source
where p is the particle density, g is the gravitational acceleration, d is the effective particle diameter, and q is the atmospheric dynamic viscosity (Green and Lane, 1964). In order to facilitate application of the results, the solution to equation (2) will be expressed in terms of the Gaussian plume parameters for the standard deviation of the plume width and height, cry and ur. These parameters are detined in terms of their respective diffusion coefficients to be X dx’ K(x). (4) s0 In this way they can be used to describe the nonGaussian plume which occurs in the general situation when settling and deposition are included. Empirical values for o,, and 0, are available for a variety of meteorological and terrain conditions. In regions of open country, Turner (1970) and Carpenter et al. (1971) have presented dispersion coefficients for various emission heights. McElroy (1969) and Bowne (1974) give dispersion coefficients for urban and suburban regions. While these data are generally in tabular or graphical form, analytic expressions which approximate the data are easily made. TRANSPORT
AND
DEPOSITION
Heines and Peters (1974) for pollutants which are absorbed at the earth’s surface can be obtained by setting the gravitational settling velocity equal to zero and using diffusion coefficients which are power functions of the downwind distance. The pollutant flux in the vertical direction at any downwind location is given by JJx,y,z)
= -Kz;
- W,C.
Using boundary condition (2e), the pollutant deposition flux onto the earth’s surface, -.lJx, y, 0), can be obtained directly from the pollutant air concentration at the ground level, C(x, y, 0), given by equation (5). Another useful measure of pollutant deposition is the net deposition rate which is defined as N(x)=
-
LX’ m dy’ Jr&‘, Y: 0). s0 I -m
The net deposition rate tant deposited per unit the downwind distance integration in equation
MODEL.
2&exp Y
exp
(7)
is the total amount of pollutime between the source and x. Carrying out the indicated (7), N(x) is seen to be
The procedure used to solve equation (2) is given in the Appendix. For a point source located at (0, 0, h) and emitting at a constant rate Q, the steady state downwind pollutant concentration is
C(x, Y, 4 =
(6)
(8) where VI = V-jW and V, = V-W This expression is indeterminate for the special case where W = V For this condition N(x) can be determined by either setting Wequal to V in equation (5) and then integrating according to equation (7), or by taking the limit of equation (8) as W approaches I! In either case the result is
2
V,(z + h)* + V:af K
2K
v, -++ I’ 02
erfc 2i/*K
z+h
2’13,
(5)
where rr,, = o,,(x) and oZ = a,(x) are defined by equation (4) and VI = V-i W Several of the models mentioned in the introduction can be obtained from equation (5) as special cases. In the trivial deposition case where both the gravitational settling velocity and the deposition velocity are zero, equation (5) reduces to the well known Gaussian plume model. When gravitational settling is neglected and the vertical diffusion coefficient K, is constant, equation (5) is equivalent to a solution presented by Rounds (1955). The results of
+ g)exp{!$}erfc{$$
(9)
DISCUSSION The model results are assumed to be capable of describing a range of deposition phenomena which can be separated into a number of general classes.
234
DONALD
L. ERMAK
While these classes are arbitrarily defined, they are useful in relating the model results to actual physical situations. a. W= V= 0. The trivial case is applicable to gaseous or small particles (generally 70.1 pm dia.) under conditions where deposition can be neglected b. W= 0, V > 0. This class applies to gaseous or small particles where the effects of gravitational settling can be neglected; however, deposition does occur. Deposition could be due to absorption onto vegetation or soil surfaces or be due to absorption by a body of water. c. W = V> 0. In this class deposition is due entirely to gravitation settling. This behavior is typical of the larger particles (5 50 pm dia.). d. I/> W> 0. Here deposition is enhanced beyond that due to gravitational settling. Enhancement is usually a function of the ground surface roughness. Generally this class applies to intermediate size particles ( z 0.1-50 pm dia.). e. W > V 2 0. When the deposition velocity is less than the gravitational settling velocity, deposited particles are being returned to the atmosphere. A physical example would be a dust storm. The most common classes of deposition phenomena of those described above are classes (ad). The following discussion will address classes (bc) over a range of deposition velocities. The trivial case W = V = 0 can be included in both of these classes and this will be done where applicable. For the same deposition velocity r/; the results for class (d), where V > w will be between those for classes (b-c). For the purpose of illustration it is useful to express the model results in terms of dimensionless variables. Making the assumption that K, = K, for simplicity, the new system of variables to be used is
r(x) =
&
Using these variables. the dimensionless is C(r, 4, s) c(r, 4,s) = ~ Cll
j(r, 4) = (0 + w)c(r, q, 0) and the dimensionless
K(x’)dx’; q(y) = f : s(z) = ;
0
V’O-J
11I
dq’.j(r’, y’).
1.0
V=l v=o.5
rate is given by
(b)
v=2
i
(12)
i
r
r
Fig. 1. The dimensionless ground-level concentration downwind
distance,
(13)
With this system of units the case w = 0 corresponds to the situation where the gravitation settling velocity is zero (W = 0) and the case w = c’ corresponds to the situation where the gravitational settling velocity and the deposition velocity are equal (W = V). The dimensionless ground-level concentration along the plume center-line is given by c(r, 0,O). Graphs of c(r, 0,O) are presented in Figs. l(a and b) for the two cases w = 0 and w = v over a range of u values. In the absence of gravitational settling (Fig. la), the ground level concentration is greatest when u = 0 and it decreases monotonically to zero when ~1= X. This reduction in the ground level concentration is a direct result of ground deposition. For the values of u considered here, the peak in c(r,O,O) occurs between r = 0.1 and r = 0.2. As the value of u is increased the location of this peak moves closer to the source. When w = v (Fig. 1b), the peak in c(r, 0,O) also moves toward the source with increases in u; however, the value of c(r,O,O) at the peak increases with increases in u. This is a result of gravitational settling which tends to move the entire plume toward the earth as it travels downwind. Figure 2 shows the vertical plume profile for the case w = 0 at three downwind distances. Four profiles are shown in each graph corresponding to four values of u. In Fig. 2(a) the plume is only a short distance downwind and the plume has not spread enough to have touched the ground. Consequently. all four
s
@I=0
net deposition
n(r) =
x
1.0
(11)
(See Appendix for the explicit expression for c(r, q, s).) The dimensionless flux onto the ground is
,=g;v=~(2v- W);C,=$. (a)
concentration
along the plume center-line (a) w = 0, (b) w = u
vs the dimensionless
Air pollutant transport and deposition from a point source
,
1
1
,
,
,
,
,
,
,
,
,
,
,
,
,
w=o
0.5
1.0
,
(
(a)
t-=0.04
0
,
1.5
2.0
C(O.O4,O,S) 4
,
I
,
I,
I,
I
I,,
I,,
,
(b)
w=o 3
vI
2
1
235
A similar treatment of the vertical plume profile for the case w = u is presented in Fig. 3. The dominant feature shown in Fig. 3(a) is the descent of the plume as a result of gravitational settling. The plume continues to descend until the maximum concentration occurs at the earth’s surface. This condition with the maximum concentration at s = 0 continues as the plume moves further downwind. The dimensionless deposition flux along the plume center-line, j(r, 0), is plotted in Fig. 4 for the two cases w = 0 and w = 0 over a range of deposition velocities. In both cases the peak value of j(r,O) increases as the deposition velocity is increased. When the gravitational settling velocity is zero (Fig. 4a), there is a limiting value for jpealr of 0.689 which occurs at r = 0.125 when u = co. Therefore, for large values of u, there is only a negligible increase in the deposition flux due to increases in the deposition velocity. This is not so for the case w = I’ (Fig. 4b). As the value
0
0
0.1
0.2
0.3
0.4 2
c(O.2,O.s)
1
0
0
0.2
0.4
c 4L,
,
,
)
I
0.6
0.8
(0.1.0,s) I
I
,
I
,
I
,_I
(b) -
0 0.01
0.05
0.1
c(l.O,s)
Fig. 2. The vertical plume profile for the case w = 0 at the dimensionless downwind distances (a) r = 0.04, (b) r = 0.2, and (c) r = 1. The scale of the horizontal axis differs for each I value.
plumes are essentially identical. In Fig. 2(b) the plume is beginning to touch the ground and deposition is beginning to occur. Due to ground deposition the plume concentration is depleted near the earth’s surface. At higher elevations the plume profile is not affected and all four plumes are nearly identical. As the plume moves downwind, the reduction in the plume concentration due to deposition moves higher up the plume profile and the maximum concentration continues to occur at an elevation above the earth’s surface. When the source depletion approach to ground deposition is used, the shape of the vertical plume profile is independent of the deposition velocity (Pasquill, 1962) and is the same as that shown in Fig. 2 for v = 0.
0
0.04
0.08
0.10
0.12
0.14
0.16
c (0.4,O.S) 5s
0
0.04
0.08
0.12
Fig. 3. The vertical plume profile for the case w = v at the dimensionless downwind distances (a) r = 0.1, (b) r = 0.4, and (c) r = 0.8. The scale of the horizontal axis differs for each r value.
DONALD L. EKMAK
236
r
Fig.
4.
r
The dimensionless deposition flux along the plume center-line vs the dimensionless downwind distance, (a) w = 0. (b) w = L‘
of u is increased, the value of jpeak increases without limit and occurs closer to the source. The dimensionless net deposition rate has a maximum value of n(co) = 1 so that n(r) is the ratio of the amount of pollutant deposited on the ground between the source and the downwind distance r and the total amount of pollutant released. Figure 5 presents graphs of n(r) for the two cases w = 0 and w = u over a range of u values. In general, as the deposition velocity is increased, deposition onto the ground occurs closer to the source. Consequently, a smaller ground surface area is contaminated but at a higher concentration. Figure 5 shows a significant difference between the two classes w = 0 and w = u in the rate at which the final l&20”/, of pollutant is deposited upon the ground. For comparable values of u, the rate is much faster when w = u. This is due to the gravitational settling velocity which tends on the average to transport the pollutant particles at a constant rate toward the ground where they may be removed. In the absence of gravitational settling, only the atmospheric diffusion process acts to bring the pollutant particles into contact with the ground. For transport over long (a)
‘.OOl
distances, the random nature of the diffusion process makes it much slower than gravitational settling. CONCLUSION
An atmospheric transport and deposition model has been presented for pollutants emitted from an elevated point source over flat terrain. The model is analytical and treats gravitational settling and dry deposition in a more physically-realistic manner than the source depletion approach. In order to facilitate application of the results to a variety of atmospheric stability conditions, the plume dispersion parameters have been expressed as general functions of the downwind distance as is done in the Gaussian plume model. The model results for two classes of deposition phenomena were discussed. They were: deposition when the gravitational settling velocity is zero and deposition when the gravitational settling velocity and the dry deposition velocity are equal. The value of these velocities was shown to have significant effects upon the ground-level pollutant concentration, the vertical plume profile, and the pollutant deposition flux onto the ground surface. (b)
t
I;lg. 5. The dimensionless
net deposition
rate vs the dimensionless (b) w = ~1
downwind
distance,
(a) w = 0,
237
Air pollutant transport and deposition from a point source REFERENCES
Bowne N. E. (1974) Diffusion rates. J. Air PoZLut. Control Ass. 24, 832-835. Calder K. L. (1961) Atmospheric diffusion of particulate material, considered as a boundary value problem. J. Meteor. 18, 413-416. Carpenter S. B., Montgomery T. L., Leavitt J. M., Coibough W. D. and Thomas F. W. (1971) Prinapal plume dispersion models: TVA power plants. J. Air Pollut. Control Ass. 21(S), 491495. Carslaw H. S. and Jaeger J. C. (1959) Conduction of Heat m Solids, 2nd Edn.. pp. 358-359. Oxford Univ. Press, London. Chandrasekhar S. (1943) Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15. l-89. Fortak H. G. (1970) Numerical simulation of temporal and spatial distributions of urban air pollution concentration. Proceedings of Symposium on Multiple-Source Urban Diffusion Models, Air Pollution Control Office Publication AP-86. Green H. L. and Lane W. R. (1964) Par~ica~ate Clouds: Dusts, Smokes, and Mists, 2nd Edn., pp. 67-73. Van Nostrand, New York. Heffter J L. and Ferber G. J. (1975) A regional-continental scale transport, diffusion, and deposition model. Part II: Diffusion-Deposition Models. pp. 17-21, NOAA ERL AL-50. Heines T. S. and Peters L. K. (1974) The effect of ground level absorption on the dispersion of pollut~ts in the atmosphere. Atmospheric environment 8, 1143-l 153. Hilsmeier W. and Gifford F. (1962) Graphs for estimating atmospheric dispersion. USAEC, Division of Technical Information, ORD-545. Hosker R. P., Jr. (1973) Estimates of dry deposition and plume depletion over forests and grass&d. Annual Reuort ATDL- 106. un. 23 l-2.58. Environmental Research Laboratories, Gak Ridge, Tennessee. Martin J. A. Jr., Nelson C. B. and Cuny P. A. (1974) A computer code for calculating doses, population doses, and ground depositlons due to atmospheric emissions of radionuclides. pp. 5-7, USEPA EPA-520/l-74-004. McElroy J. L. (1969) A comparative study of urban and rural dispersion. J. appt. Met. 8, 19-31. Morgenstern P., Morgenstern L. N., Chng K. M., Barrett D. H. and Mears C. (1975) Modeling analysis of power plants for compliance extensions in 51 air quality control regions. J. Air Pollut. Control Ass. 25(3), 287-291. Pasquill F. (1961) The estimation of dispersion of windborne material. Meteorol. Mag. 90. 3349. Pasquill F. (1962) Atmospheric Diffusion, pp. 231-235. Van N&strand, New York.. __ .. Rasmussen K. H.. Taheri M.. and Kabel R. L. (197% Global emissions ‘and natural processes for ;emo;al of gaseous pollutants Wat. Air SoiE Pollut. 4. 33-64. Rounds W., Jr. (1955) Solutions of the two-dimensional diffusion equations. Trans. Am. Geophys. Union 36, 395405. Shum Y. S., Loveland W. D. and Hewson E. W. (1975) The use of artificial activable trace elements to monitor pollutant source strengths and dispersion patterns. f. Air Pollut. ControE Ass. 25(1l), 1123-I 128. Smith F. B. (1962) The problem of deposition in atmospheric diffusion of particulate matter. J. Atmos. Sci. 19, 429-434. Turner D. B. (1961) A diffusion model for an urban area. .I. appl. Met. 3. X3-91. Turner D. B. (1970) ~rkbook of Atmosoheric Disnersion Es~z~fes. U‘SEPS AP-26. ” _ Vaughan B. E.. Abel K. H.. Cataldo D. A.. Halls J. M.. H& C. E., ‘Rancitelli L.. A., Routson R: C., Wild& R. E. and Wolf E. G. (1975) Review of potential impact
on health and environmental quality from metals entering the environment as a result of coal utilization. Energy Program Report, pp. 16-18, Pacific Northwest Laboratories, Richland, Washington. Whaley H. (1974) The derivation of plume dispersion parameters from measured three-dimensional data. Atmas~heric Environment 8, 281-290. Wipperman F. K. (1966) On turbulent diffusion in an arbitrarily stratified atmosphere. J. appl. Met. 5(S), 640-645.
APPENDIX The atmospheric transport problem of equation (2) can be expressed in a form which is more readily solvable by applying a few simplifying procedures. Using the separation of variables technique with the assumption that the solution can be expressed in the form C(x, y, z) = A(x, y) * B&c, z), equation (2) can be separated into two independent equations. These two equations can be simplified by a change of variables from the (x, y, z) system to the dimensionless variables (r, 4, s) as defined in the discussion section of the text. The equation for B. can be further reduced in complexity by expressing B, as BO(r, s) = B(r, s) *exp[ - w(s - 1) - w2r]. After completing these transformations, the two resultant equations with their associated boundary conditions are
(A-1) (a) at rl = 0, A = Cij2 6(q) (b and c) at 9 = + CO,A = 0 aB(r,, s)
d’B(r,,
s)
as2
ar,
(A-2)
’
(a) at rz = 0, B = Ch’z 6(s - 1) (b) at s = 0, awas = VB (c)ats=
co,B=O
and where rl a j K,(x’)dx’
and
r2 a: j li,(x’)dx’.
Equations (A-l and A-2) are now expressed in the standard form of the one-dimensional diffusion equation with r corresponding to time and q and s corresponding to position. Therefore, the solution to equation (A-l) describes the diffusion of material from an instantaneous point source in free space, that is & - a, co), and equation (A-2) describes the diffusion of material from an instantaneo~ source in the region x[O, co) with absorption at the boundary s = 0 according to the radiation boundary condition. The solution to these two equations may be obtained using Laplace transform techniques (Carslaw and Jaeger, 1959). Combining the results, the steady state solution to the atmospheric transport and deposition problem with a continuous point source is C
1 x(fIxp[yi +
C, = 4n(r,
*exp $ r2)11* [
.exp[-w(s
- 1) - w2r2].
exp[y]
- u(4zr2)li2 exp[v(s + 1) + v2r2] x erfc[vr:”
+ $11.
(A-3)
Vol
11, pp 231-237
Pergamon
Press 1977. Prmted m Great Brdam.
AN ANALYTICAL MODEL FOR AIR POLLUTANT TRANSPORT AND DEPOSITION FROM A POINT SOURCE* L. ERMAK
DONALD
Biomedical and Environmental Research Division, Lawrence Livermore Laboratory, University of California, Livermore, CA 94550, U.S.A. (First received 7 May 1976 and in jinal form 18 August 1976)
Abstract-An atmospheric transport and deposition model is presented for pollutants emitted from an elevated point source over flat terrain. The model is obtained from the analytic solution of the atmospheric diffusion equation with the coefficients of eddy diffusion taken to be functions of downwind distance and the average wind velocity taken to be constant. Ground deposition of the pollutants is accounted for by (1) including a gravitational settling term in the atmospheric diffusion equation and by (2) applying an absorptive boundary condition at the ground surface. In order to facilitate application of the model, the results for the general situation that includes settling and deposition (and where the distribution is no longer Gaussian) are expressed in terms of the Grassian plume parameters, and specifically the crosswind and vertical standard deviation functions U,,(X)and g=,(x)_ Graphs of the pollutant air concentration and ground deposition flux are presented for a number of deposition conditions.
INTRODUCTION Pollutant particles emitted into the atmosphere may be removed by a number of natural processes. One of the primary removal mechanisms is dry deposition onto the surface of the earth as a result of gravitational settling and ground absorption by the soil, vegetation, buildings, or a body of water. Surface deposited pollutants may have a significant impact upon the local ecosystem as the pollutants enter into and travel through the biological pathways. In addition, depletion of pollutant plumes in this manner will affect the pollutant air concentration, especially when deposition occurs over a long distance. This paper addresses the problem of air pollution removal and presents an analytical plume model for the transport and dry deposition of air pollutants. In the absence of any removal mechanisms, the Gaussian plume dispersion model is the basic method used to calculate air pollution concentrations from a point source (Turner, 1970; Carpenter et al., 1971; Morgenstern et al., 1975). Use of the Gaussian plume model began to receive popularity when Pasquill (1961) published his dispersion rates for plumes over open level terrain. Subsequently, Hilsmeier and Gifford (1962) expressed these estimates in a slightly more convenient, although exactly equivalent form, and this socalled PasquillGifford system for dispersion estimates has been widely used ever since. Its application has frequently involved the use of the method suggested by Turner (1964) for determining the appropriate atmospheric stability class from the *This work was performed under the auspices of the U.S. Energy Research and Development Administration under Contract No. W-7405-ENG-48.
observed solar radiation and wind speed. The basic idea of representing the common Gaussian plume for an elevated point source in terms of a K-theory diffusion equation was suggested by Wipperman (1966) and noted by Fortak (1970). The Gaussian plume model has achieved popularity because it is easy to use, plume dispersion data are readily available, and most measured data fit the model reasonably well (Whaley, 1974; Shum et al., 1975). The problem of atmospheric transport when gravitational settling and ground absorption cannot be ignored has been discussed by Calder (1961). In his formulation, both the gravitational settling flux and the ground deposition flux are taken to be proportional to the local air concentration. The proportionality factors are respectively called the gravitational settling velocity and the deposition velocity and are generally not equal to each other. Using this type of formulation, solutions to the atmospheric transport equation have been obtained for a number of cases (Rounds, 1955; Smith, 1962; Heines and Peters, 1974). These solutions have not been used extensively, presumably due either to their complicated nature or the difficulty in applying them under a variety of atmospheric stability conditions. A method for including the effects of surface deposition into a plume transport model, which has received considerable use, is the source depletion approach (Hosker, 1973; Martin et al., 1974; Heffter and Ferber, 1975; Vaughan et al., 1975). This approach is described in detail by Pasquill (1962) and essentially treats ground deposition as a perturbation to the Gaussian plume dispersion model. The shape of the vertical plume profile is assumed to be unaltered by the deposition process and the constant
231
232
DONALD L. ERMAK
source strength is replaced by a vtrtual decreasing source strength. This virtual source strength is derived from an integral form of the continuity equation and the assumption that the deposition rate is proportional to the pollutant air concentration at ground level. The result is a plume which diminishes exponentially with downwind distance while retaining the original Gaussian shape of the undepleted plume. The purpose of this work is to develop an analytical plume model for the atmospheric dispersion of pollutants which treats pollutant deposition in a more physically-realistic manner than the source depletion approach and yet retains the ease of application associated with Gaussian plume type models. The model presented here treats dry deposition m the manner proposed by Calder (1961) where both the gravitational settling flux and the ground deposition flux are taken to be proportional to the local air concentration. Using this assumption and the basic assumptions of the Gaussian plume model, the atmospheric transport equation is solved and analytic expressions are obtained for the pollutant air concentration and ground deposition flux. MATHEMATICAL
FORMLLATION
OF iHE
PROBLEM
The situation to be analyzed is that of the pollutant dispersion and dry deposition from an elevated point source which 1s contmuously emitting a non-reacting pollutant. It is assumed that the terrain is flat, the average wind velocity is constant, and the atmosphere is unlimited in the vertical direction. A Cartesian coordinate system is used with the x-axis oriented in the direction of the constant wind, the )I-axis in the horizontal crosswind direction, and z-axis oriented in the vertical crosswind direction. Pollutant transport is assumed to be governed by the atmospheric advectiondiffusion equation EC
_= at
a /
ii.YKX1_.-
ac (.Y
I;cc.i’K,dC ay
8~
time t-x/U. Consequently. only the steady state solution to the constant source problem needs to be considered. Second, the diffusion coeffictents in the J- and z-directions are taken to be functions of only the downwind distance from the source. Smce the pollutant particles travel downwind at a constant speed. this 1s equivalent to assuming that the crosswind diffusion coefficients are functions of the downwind flight time. An analogous situation arises in the theory of Brownian motion where the diffuston coefficient of the Brownian particles is a function of time (Chandresekhar, 1943). The equation to be solved IS therefore the steady state form of the atmospheric advectiondiffusion equation and is
As previously stated, the diffusivitles K, and KZ are functions of only the downwind distance and are therefore independent of the height 2. Furthermore. the settling velocity Wz IS also taken to be a function of only the downwind distance .Y and its functional form is to be that of KZ. At large downwind distances it is assumed that both the vertical diffusrvrty K=(x) and the settling velocity W,(s) approach the limiting values K and W respectively. The mathematical description of the problem IS completed by the boundary conditions which accompany equation (2). They are (2a) A continuous point source at (O,O, h) of constant strength Q. C(0 . 4' 7) = 1-
Qb(y)6(- /I) -
Lf
(2b and c) The pollutant concentration approaches zero far from the source in the lateral duections. C(x. * %. -_)= 0. (2d) The pollutant concentration high above the source height
approaches
zero
C(s. .1’./,) = 0 Here C is the pollutant concentration m the air at any location (x, I’, z) and time t; K,, K,, and K, are the coefficients of eddy diffusivity in the x-, y-, and z-directions, respectively; U is the constant average wind speed; and W, is the pollutant particle gravitational settling velocity and is positive in the downward negative z-direction. Equation (1) is simplified by using two basic assumptions of the Gaussian plume approach. First, the wind speed is assumed to be sufficiently large so that diffusive transport in the x-direction can be neglected in comparison to advection by the wind. This restraint makes the x-component of the pollutant particle velocity a constant and equal to the average wind speed U. The pollutant concentratton at any time t and a distance downwind x from the source IS therefore proportional to the source strength at the
(2e) Pollutant depositton onto the ground occurs at a rate proportional to the local au concentration.
The deposition velocity c’will depend upon such factors as the type and size of the pollutant particles. the roughness of terrain and the type of ground cover. and the meteorological conditions. A review of the deposition velocities for a number of pollutants over a variety of ground surface conditions is given by Rasmussen et al. (1975). Using Stokes law, the terminal gravitational settling velocity is (3)
233
An pollutant transport and deposition from a point source
where p is the particle density, g is the gravitational acceleration, d is the effective particle diameter, and q is the atmospheric dynamic viscosity (Green and Lane, 1964). In order to facilitate application of the results, the solution to equation (2) will be expressed in terms of the Gaussian plume parameters for the standard deviation of the plume width and height, cry and ur. These parameters are detined in terms of their respective diffusion coefficients to be X dx’ K(x). (4) s0 In this way they can be used to describe the nonGaussian plume which occurs in the general situation when settling and deposition are included. Empirical values for o,, and 0, are available for a variety of meteorological and terrain conditions. In regions of open country, Turner (1970) and Carpenter et al. (1971) have presented dispersion coefficients for various emission heights. McElroy (1969) and Bowne (1974) give dispersion coefficients for urban and suburban regions. While these data are generally in tabular or graphical form, analytic expressions which approximate the data are easily made. TRANSPORT
AND
DEPOSITION
Heines and Peters (1974) for pollutants which are absorbed at the earth’s surface can be obtained by setting the gravitational settling velocity equal to zero and using diffusion coefficients which are power functions of the downwind distance. The pollutant flux in the vertical direction at any downwind location is given by JJx,y,z)
= -Kz;
- W,C.
Using boundary condition (2e), the pollutant deposition flux onto the earth’s surface, -.lJx, y, 0), can be obtained directly from the pollutant air concentration at the ground level, C(x, y, 0), given by equation (5). Another useful measure of pollutant deposition is the net deposition rate which is defined as N(x)=
-
LX’ m dy’ Jr&‘, Y: 0). s0 I -m
The net deposition rate tant deposited per unit the downwind distance integration in equation
MODEL.
2&exp Y
exp
(7)
is the total amount of pollutime between the source and x. Carrying out the indicated (7), N(x) is seen to be
The procedure used to solve equation (2) is given in the Appendix. For a point source located at (0, 0, h) and emitting at a constant rate Q, the steady state downwind pollutant concentration is
C(x, Y, 4 =
(6)
(8) where VI = V-jW and V, = V-W This expression is indeterminate for the special case where W = V For this condition N(x) can be determined by either setting Wequal to V in equation (5) and then integrating according to equation (7), or by taking the limit of equation (8) as W approaches I! In either case the result is
2
V,(z + h)* + V:af K
2K
v, -++ I’ 02
erfc 2i/*K
z+h
2’13,
(5)
where rr,, = o,,(x) and oZ = a,(x) are defined by equation (4) and VI = V-i W Several of the models mentioned in the introduction can be obtained from equation (5) as special cases. In the trivial deposition case where both the gravitational settling velocity and the deposition velocity are zero, equation (5) reduces to the well known Gaussian plume model. When gravitational settling is neglected and the vertical diffusion coefficient K, is constant, equation (5) is equivalent to a solution presented by Rounds (1955). The results of
+ g)exp{!$}erfc{$$
(9)
DISCUSSION The model results are assumed to be capable of describing a range of deposition phenomena which can be separated into a number of general classes.
234
DONALD
L. ERMAK
While these classes are arbitrarily defined, they are useful in relating the model results to actual physical situations. a. W= V= 0. The trivial case is applicable to gaseous or small particles (generally 70.1 pm dia.) under conditions where deposition can be neglected b. W= 0, V > 0. This class applies to gaseous or small particles where the effects of gravitational settling can be neglected; however, deposition does occur. Deposition could be due to absorption onto vegetation or soil surfaces or be due to absorption by a body of water. c. W = V> 0. In this class deposition is due entirely to gravitation settling. This behavior is typical of the larger particles (5 50 pm dia.). d. I/> W> 0. Here deposition is enhanced beyond that due to gravitational settling. Enhancement is usually a function of the ground surface roughness. Generally this class applies to intermediate size particles ( z 0.1-50 pm dia.). e. W > V 2 0. When the deposition velocity is less than the gravitational settling velocity, deposited particles are being returned to the atmosphere. A physical example would be a dust storm. The most common classes of deposition phenomena of those described above are classes (ad). The following discussion will address classes (bc) over a range of deposition velocities. The trivial case W = V = 0 can be included in both of these classes and this will be done where applicable. For the same deposition velocity r/; the results for class (d), where V > w will be between those for classes (b-c). For the purpose of illustration it is useful to express the model results in terms of dimensionless variables. Making the assumption that K, = K, for simplicity, the new system of variables to be used is
r(x) =
&
Using these variables. the dimensionless is C(r, 4, s) c(r, 4,s) = ~ Cll
j(r, 4) = (0 + w)c(r, q, 0) and the dimensionless
K(x’)dx’; q(y) = f : s(z) = ;
0
V’O-J
11I
dq’.j(r’, y’).
1.0
V=l v=o.5
rate is given by
(b)
v=2
i
(12)
i
r
r
Fig. 1. The dimensionless ground-level concentration downwind
distance,
(13)
With this system of units the case w = 0 corresponds to the situation where the gravitation settling velocity is zero (W = 0) and the case w = c’ corresponds to the situation where the gravitational settling velocity and the deposition velocity are equal (W = V). The dimensionless ground-level concentration along the plume center-line is given by c(r, 0,O). Graphs of c(r, 0,O) are presented in Figs. l(a and b) for the two cases w = 0 and w = v over a range of u values. In the absence of gravitational settling (Fig. la), the ground level concentration is greatest when u = 0 and it decreases monotonically to zero when ~1= X. This reduction in the ground level concentration is a direct result of ground deposition. For the values of u considered here, the peak in c(r,O,O) occurs between r = 0.1 and r = 0.2. As the value of u is increased the location of this peak moves closer to the source. When w = v (Fig. 1b), the peak in c(r, 0,O) also moves toward the source with increases in u; however, the value of c(r,O,O) at the peak increases with increases in u. This is a result of gravitational settling which tends to move the entire plume toward the earth as it travels downwind. Figure 2 shows the vertical plume profile for the case w = 0 at three downwind distances. Four profiles are shown in each graph corresponding to four values of u. In Fig. 2(a) the plume is only a short distance downwind and the plume has not spread enough to have touched the ground. Consequently. all four
s
@I=0
net deposition
n(r) =
x
1.0
(11)
(See Appendix for the explicit expression for c(r, q, s).) The dimensionless flux onto the ground is
,=g;v=~(2v- W);C,=$. (a)
concentration
along the plume center-line (a) w = 0, (b) w = u
vs the dimensionless
Air pollutant transport and deposition from a point source
,
1
1
,
,
,
,
,
,
,
,
,
,
,
,
,
w=o
0.5
1.0
,
(
(a)
t-=0.04
0
,
1.5
2.0
C(O.O4,O,S) 4
,
I
,
I,
I,
I
I,,
I,,
,
(b)
w=o 3
vI
2
1
235
A similar treatment of the vertical plume profile for the case w = u is presented in Fig. 3. The dominant feature shown in Fig. 3(a) is the descent of the plume as a result of gravitational settling. The plume continues to descend until the maximum concentration occurs at the earth’s surface. This condition with the maximum concentration at s = 0 continues as the plume moves further downwind. The dimensionless deposition flux along the plume center-line, j(r, 0), is plotted in Fig. 4 for the two cases w = 0 and w = 0 over a range of deposition velocities. In both cases the peak value of j(r,O) increases as the deposition velocity is increased. When the gravitational settling velocity is zero (Fig. 4a), there is a limiting value for jpealr of 0.689 which occurs at r = 0.125 when u = co. Therefore, for large values of u, there is only a negligible increase in the deposition flux due to increases in the deposition velocity. This is not so for the case w = I’ (Fig. 4b). As the value
0
0
0.1
0.2
0.3
0.4 2
c(O.2,O.s)
1
0
0
0.2
0.4
c 4L,
,
,
)
I
0.6
0.8
(0.1.0,s) I
I
,
I
,
I
,_I
(b) -
0 0.01
0.05
0.1
c(l.O,s)
Fig. 2. The vertical plume profile for the case w = 0 at the dimensionless downwind distances (a) r = 0.04, (b) r = 0.2, and (c) r = 1. The scale of the horizontal axis differs for each I value.
plumes are essentially identical. In Fig. 2(b) the plume is beginning to touch the ground and deposition is beginning to occur. Due to ground deposition the plume concentration is depleted near the earth’s surface. At higher elevations the plume profile is not affected and all four plumes are nearly identical. As the plume moves downwind, the reduction in the plume concentration due to deposition moves higher up the plume profile and the maximum concentration continues to occur at an elevation above the earth’s surface. When the source depletion approach to ground deposition is used, the shape of the vertical plume profile is independent of the deposition velocity (Pasquill, 1962) and is the same as that shown in Fig. 2 for v = 0.
0
0.04
0.08
0.10
0.12
0.14
0.16
c (0.4,O.S) 5s
0
0.04
0.08
0.12
Fig. 3. The vertical plume profile for the case w = v at the dimensionless downwind distances (a) r = 0.1, (b) r = 0.4, and (c) r = 0.8. The scale of the horizontal axis differs for each r value.
DONALD L. EKMAK
236
r
Fig.
4.
r
The dimensionless deposition flux along the plume center-line vs the dimensionless downwind distance, (a) w = 0. (b) w = L‘
of u is increased, the value of jpeak increases without limit and occurs closer to the source. The dimensionless net deposition rate has a maximum value of n(co) = 1 so that n(r) is the ratio of the amount of pollutant deposited on the ground between the source and the downwind distance r and the total amount of pollutant released. Figure 5 presents graphs of n(r) for the two cases w = 0 and w = u over a range of u values. In general, as the deposition velocity is increased, deposition onto the ground occurs closer to the source. Consequently, a smaller ground surface area is contaminated but at a higher concentration. Figure 5 shows a significant difference between the two classes w = 0 and w = u in the rate at which the final l&20”/, of pollutant is deposited upon the ground. For comparable values of u, the rate is much faster when w = u. This is due to the gravitational settling velocity which tends on the average to transport the pollutant particles at a constant rate toward the ground where they may be removed. In the absence of gravitational settling, only the atmospheric diffusion process acts to bring the pollutant particles into contact with the ground. For transport over long (a)
‘.OOl
distances, the random nature of the diffusion process makes it much slower than gravitational settling. CONCLUSION
An atmospheric transport and deposition model has been presented for pollutants emitted from an elevated point source over flat terrain. The model is analytical and treats gravitational settling and dry deposition in a more physically-realistic manner than the source depletion approach. In order to facilitate application of the results to a variety of atmospheric stability conditions, the plume dispersion parameters have been expressed as general functions of the downwind distance as is done in the Gaussian plume model. The model results for two classes of deposition phenomena were discussed. They were: deposition when the gravitational settling velocity is zero and deposition when the gravitational settling velocity and the dry deposition velocity are equal. The value of these velocities was shown to have significant effects upon the ground-level pollutant concentration, the vertical plume profile, and the pollutant deposition flux onto the ground surface. (b)
t
I;lg. 5. The dimensionless
net deposition
rate vs the dimensionless (b) w = ~1
downwind
distance,
(a) w = 0,
237
Air pollutant transport and deposition from a point source REFERENCES
Bowne N. E. (1974) Diffusion rates. J. Air PoZLut. Control Ass. 24, 832-835. Calder K. L. (1961) Atmospheric diffusion of particulate material, considered as a boundary value problem. J. Meteor. 18, 413-416. Carpenter S. B., Montgomery T. L., Leavitt J. M., Coibough W. D. and Thomas F. W. (1971) Prinapal plume dispersion models: TVA power plants. J. Air Pollut. Control Ass. 21(S), 491495. Carslaw H. S. and Jaeger J. C. (1959) Conduction of Heat m Solids, 2nd Edn.. pp. 358-359. Oxford Univ. Press, London. Chandrasekhar S. (1943) Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15. l-89. Fortak H. G. (1970) Numerical simulation of temporal and spatial distributions of urban air pollution concentration. Proceedings of Symposium on Multiple-Source Urban Diffusion Models, Air Pollution Control Office Publication AP-86. Green H. L. and Lane W. R. (1964) Par~ica~ate Clouds: Dusts, Smokes, and Mists, 2nd Edn., pp. 67-73. Van Nostrand, New York. Heffter J L. and Ferber G. J. (1975) A regional-continental scale transport, diffusion, and deposition model. Part II: Diffusion-Deposition Models. pp. 17-21, NOAA ERL AL-50. Heines T. S. and Peters L. K. (1974) The effect of ground level absorption on the dispersion of pollut~ts in the atmosphere. Atmospheric environment 8, 1143-l 153. Hilsmeier W. and Gifford F. (1962) Graphs for estimating atmospheric dispersion. USAEC, Division of Technical Information, ORD-545. Hosker R. P., Jr. (1973) Estimates of dry deposition and plume depletion over forests and grass&d. Annual Reuort ATDL- 106. un. 23 l-2.58. Environmental Research Laboratories, Gak Ridge, Tennessee. Martin J. A. Jr., Nelson C. B. and Cuny P. A. (1974) A computer code for calculating doses, population doses, and ground depositlons due to atmospheric emissions of radionuclides. pp. 5-7, USEPA EPA-520/l-74-004. McElroy J. L. (1969) A comparative study of urban and rural dispersion. J. appt. Met. 8, 19-31. Morgenstern P., Morgenstern L. N., Chng K. M., Barrett D. H. and Mears C. (1975) Modeling analysis of power plants for compliance extensions in 51 air quality control regions. J. Air Pollut. Control Ass. 25(3), 287-291. Pasquill F. (1961) The estimation of dispersion of windborne material. Meteorol. Mag. 90. 3349. Pasquill F. (1962) Atmospheric Diffusion, pp. 231-235. Van N&strand, New York.. __ .. Rasmussen K. H.. Taheri M.. and Kabel R. L. (197% Global emissions ‘and natural processes for ;emo;al of gaseous pollutants Wat. Air SoiE Pollut. 4. 33-64. Rounds W., Jr. (1955) Solutions of the two-dimensional diffusion equations. Trans. Am. Geophys. Union 36, 395405. Shum Y. S., Loveland W. D. and Hewson E. W. (1975) The use of artificial activable trace elements to monitor pollutant source strengths and dispersion patterns. f. Air Pollut. ControE Ass. 25(1l), 1123-I 128. Smith F. B. (1962) The problem of deposition in atmospheric diffusion of particulate matter. J. Atmos. Sci. 19, 429-434. Turner D. B. (1961) A diffusion model for an urban area. .I. appl. Met. 3. X3-91. Turner D. B. (1970) ~rkbook of Atmosoheric Disnersion Es~z~fes. U‘SEPS AP-26. ” _ Vaughan B. E.. Abel K. H.. Cataldo D. A.. Halls J. M.. H& C. E., ‘Rancitelli L.. A., Routson R: C., Wild& R. E. and Wolf E. G. (1975) Review of potential impact
on health and environmental quality from metals entering the environment as a result of coal utilization. Energy Program Report, pp. 16-18, Pacific Northwest Laboratories, Richland, Washington. Whaley H. (1974) The derivation of plume dispersion parameters from measured three-dimensional data. Atmas~heric Environment 8, 281-290. Wipperman F. K. (1966) On turbulent diffusion in an arbitrarily stratified atmosphere. J. appl. Met. 5(S), 640-645.
APPENDIX The atmospheric transport problem of equation (2) can be expressed in a form which is more readily solvable by applying a few simplifying procedures. Using the separation of variables technique with the assumption that the solution can be expressed in the form C(x, y, z) = A(x, y) * B&c, z), equation (2) can be separated into two independent equations. These two equations can be simplified by a change of variables from the (x, y, z) system to the dimensionless variables (r, 4, s) as defined in the discussion section of the text. The equation for B. can be further reduced in complexity by expressing B, as BO(r, s) = B(r, s) *exp[ - w(s - 1) - w2r]. After completing these transformations, the two resultant equations with their associated boundary conditions are
(A-1) (a) at rl = 0, A = Cij2 6(q) (b and c) at 9 = + CO,A = 0 aB(r,, s)
d’B(r,,
s)
as2
ar,
(A-2)
’
(a) at rz = 0, B = Ch’z 6(s - 1) (b) at s = 0, awas = VB (c)ats=
co,B=O
and where rl a j K,(x’)dx’
and
r2 a: j li,(x’)dx’.
Equations (A-l and A-2) are now expressed in the standard form of the one-dimensional diffusion equation with r corresponding to time and q and s corresponding to position. Therefore, the solution to equation (A-l) describes the diffusion of material from an instantaneous point source in free space, that is & - a, co), and equation (A-2) describes the diffusion of material from an instantaneo~ source in the region x[O, co) with absorption at the boundary s = 0 according to the radiation boundary condition. The solution to these two equations may be obtained using Laplace transform techniques (Carslaw and Jaeger, 1959). Combining the results, the steady state solution to the atmospheric transport and deposition problem with a continuous point source is C
1 x(fIxp[yi +
C, = 4n(r,
*exp $ r2)11* [
.exp[-w(s
- 1) - w2r2].
exp[y]
- u(4zr2)li2 exp[v(s + 1) + v2r2] x erfc[vr:”
+ $11.
(A-3)