November  1983
                                                     ' — i r   ^
                                                      ' / s
          PLUME CONCENTRATION ALGORITHMS WITH

DEPOSITION, SEDIMENTATION, AND CHEMICAL TRANSFORMATION
      ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
          OFFICE OF RESEARCH AND DEVELOPMENT
         U.S. ENVIRONMENTAL PROTECTION AGENCY
     RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711

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          PLUME CONCENTRATION ALGORITHMS WITH

DEPOSITION, SEDIMENTATION, AND CHEMICAL TRANSFORMATION
                          by
                    K. Shankar Rao
     Atmospheric Turbulence and Diffusion Division
    National Oceanic and Atmospheric Administration
              Oak Ridge, Tennessee 37830
                  IAG-AD-13-F-1-707-0
                    Project Officer

                   Jack H. Shreffler
          Meteorology and Assessment Division
      Environmental Sciences Research Laboratory
     Research Triangle Park, North Carolina 27711
      ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
          OFFICE OF RESEARCH AND DEVELOPMENT
         U.S. ENVIRONMENTAL PROTECTION AGENCY
     RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711

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                                  DISCLAIMER
     This report has been reviewed by the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency, and approved for pub-
lication.  Approval does not signify that the contents necessarily re-
flect the views and policies of the U.S. Environmental Protection Agency,
nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.

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                                   ABSTRACT
     A gradient-transfer model for the atmospheric transport, diffusion,
deposition, and first-order chemical transformation of gaseous and particu-
late pollutants emitted from an elevated continuous point source is formulated
and analytically solved using Green's functions.  This analytical plume model
treats gravitational settling and dry deposition in a physically realistic
and straightforward manner.  For practical application of the model, the eddy
diffusivity coefficients in the analytical solutions are expressed in terms of
the widely-used Gaussian plume dispersion parameters.  The latter can be speci-
fied as functions of the downwind distance and the atmospheric stability class
within the framework of the standard turbulence-typing schemes.

     The analytical plume algorithms for the primary (reactant) and the
secondary (product) pollutants are presented for various stability and mixing
conditions.  In the limit when deposition and settling velocities and the
chemical transformation rate are zero, these equations reduce to the well-
known Gaussian plume diffusion algorithms presently used in EPA dispersion
models for assessment of air quality.  Thus the analytical model for estimating
deposition and chemical transformation described here retains the ease of appli-
cation associated with Gaussian plume models, and is subject to the same basic
assumptions and limitations as the latter.

     A new mathematical approach, based on mass budgets of the species, is
outlined to derive simple expressions for ground-level concentrations of the
primary and secondary pollutants resulting from distributed area-source emissions.
These expressions, which involve only the point-source algorithms for the well-
mixed region, permit one to use the same program subroutines for both point
and area sources.  Thus the area-source concentration equations developed in
this report are simple, efficient, and accurate.

     The new point-source algorithms are applied to study the atmospheric transport
and transformation of SCL to SO,, and deposition of these species.  Calculated
variations of the ground-level concentrations are presented and discussed.  The
results of a sensitivity analysis of the concentration algorithm for the secon-
dary pollutant are given.  The specification of gravitational settling and depo-
sition velocities in the model is discussed.

     The work described in this report was undertaken to develop concentration
algorithms for the Pollution Episodic Model (PEM).  This report was submitted
by NOAA's Atmospheric Turbulence and Diffusion Division in partial fulfillment
of Interagency Agreement No. AD-13-F-1-707-0 with the U. S. Environmental
Protection Agency.  This work, covering the period September 1981 to March 1983,
was completed as of March 31, 1983.
                                      111

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                                   CONTENTS


Abstract 	  iii
Figures  	   vi
Symbols and Abbreviations 	 viii
Acknowledgements 	   xi

     I.  Introduction 	    1

     2.  Literature Survey 	    4

     3.  The Gradient-Transfer Deposition Model	    7
               Mathematical formulations 	    8
               Analytical solutions 	   11
               Parameterization of concentrations 	   28
               Well-mixed region 	   35
               Plume trapping 	   40
               Summary of point source concentration equations 	   42
               Surface deposition fluxes 	   43
               Area sources 	   43

     4.  Results and Discussion 	   60
               Sensitivity analyses 	   61
               Ground-level concentrations 	   67
               Chemical transformation rate 	   71
               Effects of atmospheric stability 	   74

     5.  Summary and Conclusions 	   79

References 	   82

Appendix

     Settling and Deposition Velocities 	   84

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                                FIGURES


Number
 1    Schematic diagram for area-source algorithms showing (a) a grid
      square with emissions and a grid square with receptor, and the
      distances; (b) a cross-section of the calculation grid square,
      and the incoming and outgoing normalized fluxes of pollutant .... 45

 2    Schematic diagram showing a single grid square with emissions
      and four downwind calculation grid squares with receptors, and
      the distances used in area-source algorithms 	 48

 3    Schematic diagram showing a single grid square with receptor
      and four upwind grid squares with emissions, and the dis-
      tances used in area-source algorithms 	 50

 4    Variation of the three major terms in the point-source
      algorithm for the GLC of the secondary pollutant.  A
      non-zero concentration results from the small positive
      imbalance between the terms	 62

 5    Variation of the weighting function, in the point-source
      algorithm for the GLC of the secondary pollutant, as a
      function of the effective source height 	 64

 6    Variation of the weighting function, in the point-source
      algorithm for the concentration of the secondary pollutant,
      as a function of the receptor height 	 65

 7    Variation of the weighting function, in the point-source
      algorithm for the GLC of the secondary pollutant, as a
      function of the deposition velocity of the primary
      pollutant 	 66

 8    Variation of the weighting function, in the point-source
      algorithm for the GLC of the secondary pollutant, as a
      function of its deposition velocity 	 68

 9    Variation of the calculated GLC of the primary pollutant
      for different values of the parameter V.../U, and W  = 0 	 69

10    Variation of the calculated GLC of the secondary pollutant
      for different values of the parameter Vj2/U, and VL = VL = 0  .... 70

11    Comparison of the calculated GLC of the secondary pollutant
      when it is made of~(a) particles with V,2 = W_ = 1 cm/s
      or VJ2/U = 5 x 10   , and (b) gas with V^ = 1 cm/s and

      W2 = 0 or V12/U = 10"2 	 72

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                           FIGURES (Continued)
Number                                                                Page


 12    Variation of the calculated GLC of the secondary pollutant
       for an arbitrary variation of the chemical transformation
       rate by two orders of magnitude 	 73
 13    Variation of the calculated GLC of the primary pollutant
       for an arbitrary variation of the chemical transformation
       rate by two orders of magnitude 	 75
 14    Variation of the calculated GLC of the primary pollutant as
       a function of the P-G atmospheric stability class 	 76
 15    Variation of the calculated GLC of the secondary pollutant
       as a function of the P-G atmospheric stability class 	 78
                                   VI1

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                           SYMBOLS AND ABBEEVIATIONS

SYMBOLS

C-, C2         mean concentrations of primary and secondary pollutants
               for point source

C.j , C.2       mean concentrations of primary and secondary pollutants
               for area source

D. , D?         surface deposition fluxes of primary and secondary pollutants

F- , F2         Weighting functions in secondary pollutant concentration
               algorithms

G, , G2, G_     Green's functions

H              effective height of source

k              chemical transformation rate

K , K          eddy diffusivities in y and z directions

L              height of the inversion lid

L , L          length scales of concentration distribution
               in y and z directions

p              probability density of concentration distribution
               in y direction

Q1 , Q2         source strengths or emission rates of primary
               and secondary pollutants

q- , q2         probability densities of concentration distributions
               of primary and secondary pollutants in z direction

U              mean wind speed
                   deposition velocities of primary and secondary
               pollutants
V              V   - W /2
 11             dl    1/Z

V12            Vd2 - V2

Vi3            vn - (wi
                                     Vlll

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V              V   - W
V21            Vdl   Wl
V              V   - W
 22             d2    2

W1, VL         gravitational settling velocities of primary and
               secondary pollutant particles

x, y, z        horizontal downwind, horizontal crosswind,
               and vertical coordinates

x              downwind distance at which cr  = 0.47L
 m                                         z

a , a          Gausssian dispersion parameters in y and
               z directions

t              characteristic time scale of chemical transformation
                           NONDIMENSIONAL QUANTITIES
g..             crosswind diffusion function
82             x < x
vertical diffusion function for point source when
  < x
  =  m
gi-, ,   g~2      89 m°dified (for deposition and chemical transformation)
               for primary and secondary pollutants

g_             vertical diffusion function for point source in plume-
               trapping region (x  < x < 2 x )

gi^ ,  g'        g.- modified (for deposition and chemical transforma-
               tion) for primary and secondary pollutants

g,             vertical diffusion function for point source in
               well-mixed region (x > 2 x )

g! 1 ,  gj .       g, modified (for deposition and chemical transforma-
               tion) for primary and secondary pollutants
H              H/V2~a
                     Z
               L/V2 a
               Vdl/U  '   Vd2/U
vn , v12      vn/u  ,   v12/u
                                       IX

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v13            v13/u
v21 , v22      v21/u  ,   v22/u
   , w2        Wj/u   ,  w2/u
x, z           x/V2 CT , z/«/2 a
                     Z        ^




y              y/V2 a
 c
               ratio of molecular weight of secondary pollutant

               to molecular weight of primary pollutant
ABBREVIATIONS






ATDL           Atmospheric Turbulence and Diffusion Laboratory



EPA            Environmental Protection Agency




GLC            Ground-level concentration



KST            Atmospheric stability class index



P-G            Pasquill-Gifford

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                               ACKNOWLEDGEMENTS
     This report was prepared for the Office of Research and Development,




Environmental Sciences Research Laboratory of the U.S. Environmental




Protection Agency to support the needs of the EPA's Office of Air Quality




Planning and Standards in urban particulate modeling.  This work was accomplished




under interagency agreements among the U.S. Department of Energy, the National




Oceanic and Atmospheric Administration, and the EPA.  The author is grateful to




Dr. Jack Shreffler of ESRL for the opportunity to do this work, and for his




interest and patience.  The author expresses his appreciation to the following




members of the Atmospheric Turbulence and Diffusion Laboratory:  Director Bruce




Hicks for his understanding and support, Dr. Ray Hosker for many useful sugges-




tions during the course of this work; Ms. Louise Taylor for making the model test




runs and plotting the results on computer; and Ms. Mary Rogers for her expert




technical typing and patient revisions.  Special thanks are due to Dr. Frank




Gifford for his interest and useful discussions.
                                      XI

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                                   SECTION 1








                                 INTRODUCTION








     Pollutant gases and suspended particles released into the atmosphere are




transported by the wind, diffused and diluted by turbulence, and removed by




several natural processes.  An important removal mechanism is dry deposition of




pollutants on the earth's surface by gravitational settling, eddy impaction,




chemical absorption, and other effects.  Another significant removal mechanism




is chemical transformation in the atmosphere.  Depletion of airborne pollutant




material by these natural processes affects pollutant concentrations and




residence times in the atmosphere.  Moreover, the product of a chemical




reaction may be the pollutant of primary concern, rather than the reactant




itself.  Surface deposition of acidic and toxic pollutants may adversely impact




on local ecology, human health, biological life, structures, and ancient




monuments.  Furthermore, large concentrations of particulate products resulting




from chemical reactions may lead to significant deterioration of atmospheric




visibility.  It is important, therefore, to obtain reliable estimates of the




effects of dry deposition and chemical transformation.








     This report presents an analytical plume model for diffusion, dry




deposition, and first-order chemical transformation of gaseous or particulate




pollutants released from an elevated continuous point source, based on gradient-




transfer or K-theory.  The method solves the atmospheric advection-diffusion

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equation subject to a deposition boundary condition.   The model includes




similar and complementary sets of equations for the primary (reactant)




and the secondary (product) pollutants.   These equations are analytically




solved using Green's functions.








     In order to facilitate practical application of the model to air pollution




problems, the K-coefficients are expressed in this report in terms of the




widely-used Gaussian plume dispersion parameters which can be easily obtained




from standard turbulence-typing schemes.  The parameterized diffusion-deposi-




tion-transformation algorithms for various atmospheric stability and mixing




conditions are simplified, and presented as analytical extensions of the well-




known Gaussian plume-diffusion algorithms presently used in EPA models for air




quality assessment.








     For practical application of the model to urban air pollution problems,




a new mathematical approach, based on mass balance considerations, is developed




to derive simple expressions for ground-level concentrations of the reactant




and the product pollutants resulting from distributed area-source emissions.




These novel expressions for area sources involve only the point-source algorithms




of the well-mixed region, thus allowing one to use the same program subroutines




for both point and area sources.  In the limit when deposition rates approach




zero, the concentrations calculated by these expressions agree with the corres-




ponding values given by the area-source algorithms without deposition, currently




used in urban air pollution models.








     This report gives a brief review of the literature on gradient-transfer




models with chemical transformation.  Details of the mathematical formulations

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and analytical solutions of the present model with deposition, sedimentation,




and chemical transformation are given, and the parameterized concentration




algorithms for the primary and the secondary pollutants are listed.   Calculated




variations of the ground-level concentrations and results of a sensitivity




analysis are presented and discussed.   Some guidance is provided for the




specification of the settling and deposition velocities in the model.

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                                   SECTION 2








                               LITERATURE SURVEY








     Applied air pollution models used in industry and regulation are generally




based on the Gaussian plume formulation.  These models have been extensively




modified over the years to include pollutant removal mechanisms such as dry




and wet deposition, and chemical decay.  Rao (1981) gave a brief review of the




existing methodologies of Gaussian diffusion-deposition models, including a




comprehensive literature survey of gradient-transfer (K-theory) models.  For




the latter case, Rao (1981) also gave the mathematical formulations (for a non-




reactive pollutant), analytical solutions, parameterized concentration algorithms




for various atmospheric stability and mixing conditions, and expressions for net




deposition and suspension rates of the pollutant.








     In this section, we briefly review the literature on K-theory or Gaussian




models for chemically reactive pollutants.  Only a first-order chemical trans-




formation is considered, and expressions for both the primary (reactant) and




secondary (product) pollutants are given in the references cited.








     Heines and Peters (1973) studied the diffusion and transformation of




pollutants from a continuous point or infinite line source.  The effect of




a temperature inversion aloft was also included through multiple eddy reflec-




tions.  The eddy diffusion coefficients were assumed to be power functions of




the downwind distance.  The expression for concentration of the secondary

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pollutant was derived from a simple component mass balance.  The concentrations




of the reactant and product were presented in terms of dimensionless plots.




Deposition and sedimentation were not considered for either species.








     Rao (1975) adapted the analytical solution of Monin (1959) to study the




dispersion, deposition, and chemical transformation of the SO- plume from a




power plant stack represented by an elevated continuous point source.  The




eddy diffusivities were expressed in terms of the Gaussian dispersion para-




meters.  A constant first-order transformation rate of SO- to SOT was assumed.




Concentrations of both species were calculated, and compared with observations




at several downwind receptors.  Izrael, Mikhailova, and Pressman (1979) used




Monin1s (1959) instantaneous source solution to estimate the long-range transport




of sulfur dioxide and sulfates, assuming constant eddy diffusivities and non-




equal deposition velocities for the two species.








     Ermak (1978) described a multiple point-source dispersion model which




considered a chain of up to three first-order chemical transformations.  The




source-depletion approach (e.g., Van der Hoven, 1968) was used to include ground




deposition.  The gradient-transfer plume model of Ermak (1977) was used to




incorporate the effects of gravitational settling of particles.  For the latter




case, however, the chemical-transformtion option could not be used, and the




effects of an inversion layer were assumed to be negligible.








     Lee (1980) gave analytical solutions of a gradient-transfer model similar




to that described in this report, in terms of constant eddy diffusivity coeffi-




cients.  These solutions apply only to gases or very small particles emitted




from an elevated continuous point source, since the gravitational settling

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effects were not considered.   A first-order chemical transformation was con-


sidered, and direct emission of the secondary pollutant was assumed to be zero.


Lee used this model, which included wet deposition processes, to study the

                                                     =                        2
atmospheric transport and transformation of SO- to SO,, assuming K  = K  = 5 m /s.


Details of the analytical solutions were not available.

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                                   SECTION 3








                    THE GRADIENT-TRANSFER DEPOSITION MODEL








     This section gives the mathematical formulations and analytical solutions




of the gradient-transfer (K-theory) model for the most general case that includes




transport, diffusion, deposition, sedimentation, and first-order chemical




transformation of pollutants.  We consider two chemically-coupled gaseous or




particulate pollutant species; the primary (species-1 or reactant) pollutant




is assumed to transform into the secondary (species-2 or reaction product)




pollutant at a known constant rate.  For generality, the two species are




assumed to have known non-equal deposition and settling velocities.








     First, we derive analytical solutions for concentrations of the two




pollutant species emitted from an elevated continuous point source.  Then




we express the K-coefficients in these solutions in terms of the widely used




Gaussian plume dispersion parameters.  The resulting expressions are parame-




terized, simplified, and presented as extensions of the Gaussian plume algorithms




currently used in EPA air quality models for various atmospheric stability and




mixing conditions.  Further simplifications of the new algorithms are indicated




for gaseous or fine suspended particulate pollutants with negligible settling,




and for ground-level sources and/or receptors.  Limiting expressions of the




algorithms are derived for large particles, when gravitational settling is

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the dominant deposition mechanism.   Finally, we utilize these new point source




concentration algorithms to derive  expressions for the concentrations of the




two species due to emissions from area sources, using an innovative approach




based on mass balance considerations.








     No assumptions are made here regarding the nature of the pollutant




species.  The formulations and the  solutions are, therefore, general enough




to be applicable to any two gaseous or particulate pollutants that are




coupled through a first-order chemical transformation.  Either of the two




species may be a gas, or particulate matter with a known average size.




Molecular weights of the two species are assumed to be known.  Direct emission




of the secondary pollutant is permitted from both point and area sources.  A




direct emission of secondary pollutant, if present, may contribute to its




concentration significantly more than the chemical transformation.








     In the absence of a chemical coupling, expressions for concentrations of




two chemically-independent pollutants, each subject to deposition and/or




sedimentation, can be obtained as degenerate cases of the concentration




algorithms for the general case.  The notation used in this section is similar




to that of Rao (1981), which is consistent with the notation presently used




in the user's guides for EPA's atmospheric dispersion models.








MATHEMATICAL FORMULATIONS








     We consider the steady state form of the  three-dimensional atmospheric




advection-diffusion equation for the primary pollutant (denoted by subscript  1)




with deposition, sedimentation, and first-order  chemical transformation:

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          D 3C /3x = K  32C /3y2 + K  32C_/3z2 + W. 3C../3Z - C./T        (la)
              x       y    x        z    i        xx       x  c
Here, x, y, z are the horizontal downwind, horizontal crosswind, and vertical



coordinates, respectively; U is the constant average wind speed, and W  is the



gravitational settling velocity (taken as positive in the downward or negative



z-direction) of the primary pollutant particles, C- is the primary pollutant



concentration at (x, y, z), K  and K  are constant eddy diffusivities in the



crosswind and vertical directions, respectively, and t  = 1/k  is the time scale
                                                      C      t-


associated with the chemical transformation which proceeds at a given rate k .



The last term of the equation, - CL/T , represents the chemical sink, or loss



of the primary pollutant due to transformation.
     For a continuous point source, with an emission rate or strength Q1 of



pollutant species-1, located atx=0, y=0, z=H, the initial and boundary



conditions are given by







               C1(0, y, z) = QX/U • 6(y) • 6(z-H)                       (Ib)







               C.(x, ±00, z) = 0                                         (Ic)
               Cj(x, y, oo) = 0                                          (le)







In the initial condition (Ib), which is the limiting form of the mass continuity



equation at the source, 6 is the Dirac delta function.   Boundary condition  (Id)

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states that, at ground-level, the sum of the turbulent transfer of pollutant



down its concentration gradient and the downward settling flux due to the



particles'  weight is equal to the net flux of pollutant to the surface resulting



from an exchange between the atmosphere and the surface; V,..  is the deposition



velocity which characterizes this exchange of the primary pollutant.  When



deposition occurs, from Eq. (Id) the turbulent flux at the surface (z=0) is



given by
               -w'cj = KZ - acyaz  = (vdl - wp •  c1  > o,







which is in agreement with the standard micrometeorological notation.  This



obviously requires that V,  > W  > 0.  If 0 < V,  < W ,  then the direction of



the turbulent flux at the surface is reversed, implying re-entrainment of the



particles from the surface into the atmosphere.  Thus, the deposition boundary



condition (Id), originally suggested by Monin (1959) and discussed by Calder



(1961), adequately describes the exchange between the atmosphere and the sur-



face.  Eq. (Id) is analogous to the so-called 'radiation' boundary condition



used in the theory of heat conduction (see, e.g., Carslaw and Jaeger, 1959,



p. 18) to describe the temperature distribution in a body from the boundary



of which heat radiates freely into the surrounding medium, when the latter is



at zero degrees temperature.







     The corresponding formulations for the secondary pollutant (designated



by subscript 2) can be written as follows:
          U 3C./3x = K  32C0/3y2 + K  82C./3z'i + W0 9C0/3z + Y C./T     (2a)
              2       y2        z2        i   2.         1C
                                     10

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          C2(0, y, z) = Q2/U - 6(y) - 6(z-H)                            (2b)
          C,(x, ±», z) = 0                                              (2c)
                                                                        (2d)
          C0(x, y, «) = 0                 '                              (2e)
In Eq. (2a), VL is the gravitational settling velocity of the secondary




pollutant, and y is the ratio of its molecular weight to that of the primary




pollutant; the term yd-/!  represents the chemical source for the secondary




pollutant.  Eq. (2b) describes the direct emission of species-2 from the point




source, located at x=0, y=0, z=H, with an emission rate or strength 0~.  Eq. (2d)




is the deposition boundary condition, and V,^ is the deposition velocity for the




secondary pollutant.  For generality, we assume here V,_ ^ V,- and W~ ^ W...








ANALYTICAL SOLUTIONS








     The solution of Eqs. (1) and (2) can be expressed as







                   C1(x, y, z) = Qj/U  • p(x,y) • q-^x.z)               (3)





                   C2(x, y, z) = Qj/U  - p(x,y) • q2(x,z)               (4)











where p, q1, and q_ are probability densities of the concentration distributions.




It should be noted that the concentration of the secondary pollutant, C_, is
                                     11

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expressed in terms of the emission rate of the primary pollutant, Q , in Eq. (4).



This is due to the likelihood that, for many practical applications of the



model, the direct emission rate Q_ of the secondary pollutant may be zero;



C~ would then be zero if Q? were used instead of Q  in Eq. (4), thus incorrectly



ignoring the non-zero contribution to C_ by the chemical source.  The proba-



bility density p(x,y) of the concentration distribution in the horizontal



crosswind direction is unaffected by deposition, sedimentation, and chemical



transformation.  Therefore, it is identical for both species.







     Substituting Eq. (3) in Eq. (1) and using the separation of variables



technique, two independent systems of equations and boundary conditions in



p and q, can be obtained as follows:







          U 3p/3x = K  32p/3y2  ,       o
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     The analytical solution of (5) can be written as
                    p(x,y) =
                             g1(x,y)
  r   i      I  -z-2 u
g (x,y) = exp - ^—
             I     y
                                                                        (7)
L   =  2
                                  x/U
where L  is a length scale characteristic of the horizontal crosswind diffusion,

and g.(x,y) is a nondimensional function.  This is one of the fundamental

solutions of the diffusion or heat conduction equation (e.g., see Carslaw and

Jaeger, 1963, p. 107); p(x,y) represents the probability that a particle released

from a source of unit strength located at x=0, y=0 will be at a crosswind loca-

tion y after travelling a distance x downwind with a speed of U.
     Equation (6) cannot be solved in its present form because of the sedimenta-

tion term, W- dq^/dz, and the chemical sink term, -q,/! , in the differential

equation.  In order to remove these terms, we apply the following transformation:
                                    exp( -p1 - ~-
                                                 c
                                                                        (8a)
where
WjCz-H)
2K
z
-L * ,
4K U
z
                                             1/2
                                                                        (8b)
is a nondimensional parameter representing the effect of sedimentation of the

primary pollutant particles on the primary pollutant concentration, and q,(x,z)
                                     13

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is the transformed new variable.  Substituting Eq. (8) in Eq. (6) and simplifying,



we obtain the following:





          U 8q /3x = K  92q-/3z2 ,      0 < x , z < »                    (9a)
              JL       Z    J.


                          [ W (z-H) 1

          q^O.z)  =  exp   -—	 | •  6(z-H)  =  $ (2)                (9b)

                          I    z    J




                                                                         (9c)




          qCx,*) = 0                                                    (9d)



where




          hl = Vll/Kz >      Vll = Vdl - V2                            (9e)






     The homogeneous boundary condition (9c) expresses the relationship  between



the variable q. and its normal gradient at the surface.  This type of equation



is usually referred to as a boundary condition of the third kind; (A more



general form of this boundary condition is 3q../3z - h1q1 = f_.  The Dirichlet



boundary condition of type q. = f1 and the Neumann boundary condition of type



3q1/3z = f_, are boundary conditions of the first and second kind, respectively.



Here, f   f , and f., are functions of x given at the boundary, z = 0, of the region
       •1-   *•       -J


in which the solution is sought).  Equations (9a, c, d) constitute a homogeneous



boundary-value problem of the third kind.  The solution of this problem, also



known as its Green's function G(x,z,4), can be obtained by Laplace transform



methods (Carslaw and Jaeger, 1963, p. 115), and written as follows:
                         z
                             ex? { "^V } + exp {  "(£?i    }
                                       Z    '              Z
                                                                         (10a)
                                      14

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where
L   =  2
 z          z
                                                         (10b)
is a length scale characteristic of the vertical diffusion.  Equation  (10)



describes the stationary diffusion in a semi-infinite region from a unit source



located at x=0, z=|.  The subscript 1 of the Green's function G corresponds to



the subscript of h appearing in Eqs. (9c) and (lOa).
     The solution of Eq. (9) can be now written in terms of its Green's function



(see, e.g., Tychonov and Samarskii, 1964), as follows:
                              00
   J
                                     • 0^,2,4) d4,
where, from Eq. (9b),
                                       W, (4-H)
                                                        (11)
                                                                         (12)
Substituting (10) and (12) in (11), and noting that
           /•
           -*0
f(4)
        = f(a),
where f(|) is any arbitrary function of f, term-by-term integration of  (11)



gives
               qi(x,z) = r
     exp
                       f            ?        \
          - L h • exp  \ h (z+H) + h^ K x/U  f • erf c

             21       l  X         i  z     j
                                                               x/U
                                                            x/U
                                               (13)
                                     15

-------
     In order to simplify the equations, and considerably  reduce the difficulty

in typing them, we define the following nondimensional parameters:



                 _ (z-H)2U              _   (z+H)2!!

               r ~  4KZX       '     S  ~   4KZX
                                     ______
                               + h. ,/SIi/U                               (14)
Noting that


          exp (h^z+H) +  h2K2x/U) = exp(^-s) = A


Eq. (13) can be now written as
                   =  -  [e"r  +  e"S  •  (1 - a]                        (15)
From Eqs. (8) and (15), the solution of Eq.  (6) can be written  as


                    qi(x,2)  =  g'2l I L2   ,                              (16a)


where
                        -p^ - x/Ulc)  • [ e"r <• e"S  •  (1  - c^)  J          (16b)
is a nondimensional function.  The prime indicates modification of function

g_(x,z) of the primary pollutant  (denoted by  the  second  subscript 1)  for the

effects of deposition, sedimentation, and chemical transformation.   In the

absence of these effects, i.e., when or, = p.  =  0  and  k  =  0  or T  = °°, Eq. (16b)

reduces to
                                      16

-------
          g21(x,z) = e"r + e~s = g2(x,z),                                (17)








as in the familiar Gaussian plume model.  The notation used here is an extension



of Rao's (1981) notation for one pollutant, and is consistent with the notation



presently used in the user's guides for EPA air quality models.








     We now turn our attention to the solution of Eq. (2).  Substituting



Eqs. (3) and (4) in Eq. (2) and using the separation of variables technique,



two independent systems of equations and boundary conditions in p and q«




can be obtained.  The p-equations and their solution are the same as those



given in Eqs. (5) and (7).  The q--equations can be written as follows:
     U 3q2/3x = Kz 32q2/3z2 + W£ 3q2/3z + Y q^T,, >   0 < x  , z <
     q2(0,z) = (yQj - 6(z-H) = t|>2(z)                                    (18)
     q2(x,») = 0








     In order to eliminate the sedimentation term, W_ 3q_/3z, from the differen-




tial equation, we use the following transformation:
                         q2(x,z) = q2(x,z) • exp(-p2)                    (19a)
                                     17

-------
where
                    P2U,z) =
W2(z-H)
^z
W^x"
4K U
z
                                                   1/2
                                                                         (19b)
is a nondimensional parameter representing the effect of sedimentation of the



secondary pollutant particles on the secondary pollutant concentration, and



q_(x,z) is the transformed new variable.  Substituting Eqs. (8) and  (19) in


Eq. (18), we obtain the following transformed equations:
U ZqJBx = K  32i,/az2 + y -i • exp
    £       Z    £         L
                            c
q2(0,z) =
                                   J-    + x/Utc)
                                                             )  ,
                                   0 < x , z < »




                             W,(z-H) |

                     • exp l -55	 [ - 6(z-H)
                                                                         (20a)
                                                                         (20b)
                                                                         (20c)
where
          q2(x,o») = 0
             =V/K         V   =V
               V127 z  '       12   Vd2
                                                               (20d)
                                                               (20e)
     Equation (20a) is a nonhomogeneous partial differential equation  for  the



secondary pollutant, coupled to Eq.  (9) for the primary pollutant  through



the chemical source term,
          f(x,z) =  (Y/Tc)-q1(x,z)-exp  J  -(p2-  {J2 + x/Utc
                                                               (21)
                                      18

-------
Unlike Q~, this chemical source is not a constant; in addition to  x  and  z,



it depends on H, U, K , V  , W1, W0, y, and t,.  As shown by Budak et al.  (1964),
                     Z   u 1    1   ^          C                     	 	


the analytical solution of Eq. (20) may be directly obtained as follows:
                            00
            G2(x,z,|)
               q2(x,z) =





                    X

               +  J   dx' J  fCx',4) • G2(x-x',z,|) d|
                                              (22)
Here ijL and f are the functions defined in Eqs. (20b) and  (21), respectively.



The first term in Eq. (22) gives the contribution of the initial condition,



and the second term gives the contribution of the inhomogeneity in Eq.  (20a).



G2(x,z,|) is the Green's function of the homogeneous differential equation,



subject to the boundary condition (20c) of the third kind; the Green's  function



is given by
          G2(x,z,|) = £-
exp
f -(2-£)2u 1  +     I -(z+€)2u  I
1   4K x   f  + exp 1   4K x    J
                         c
                            exp

                        •Jn
          4K x
                                             (23)
G?(x-x',z,4), needed to solve Eq. (22), is the associated source function,



obtained by replacing x in the Green's function G_(x,z,£) of Eq. (23) with x-x'
                                     19

-------
Though straightforward in principle, the solution given by Eq. (22) is difficult




to derive in practice.  This is because the contribution of the inhomogeneity,




given by the second part of Eq. (22), involves evaluation of several integrals




whose integrands are complicated functions of £.  Some of these integrations are




mathematically intractable, thus rendering this direct application of the




Green's function solution to the inhomogenous differential equation fruitless.








     In order to remove the inhomogeneity from the differential equation (20a),




we introduce the following transformation:
          q2(x,z) = q3(x,z) - y q1(x,z)                                 (24)
where
.z)  =  qjCx.z)  •  exp j  -(^- ^ + x/UTc)  j
                                                                        (25)
and q»(x,z) is a new (unknown) variable.  Substituting (24) in Eq.  (20a)




and separating the variables, we obtain two independent homogeneous differential




equations in q- and q_.  The initial and boundary conditions to solve the




differential equation for q.. are obtained by substituting Eq. (25)  in Eqs.




(9b,c,d).  The initial and boundary conditions for the q^-equation  can be




obtained by substituting Eq. (24) in Eqs. (20b,c,d), and using the  initial




and boundary conditions for q- to simplify the resulting expressions.  The




final expressions for the two sets of equations thus obtained can be written




as follows:
                                      20

-------
          U
                                            0  <  X  ,  Z < «
                                 (26a)
                   =  exp
                       W2(z-H)
                         2K
6(z-H) =
(26b)
                                                                         (26c)
                                                                         (26d)
where
and
          h3 = V13/Kz   '      V13
          U 3q,./3x = K  32q /3z2  ,        0  <  x  ,  z  <
              j       Z    j
                                                                    (26e)
                                                                    (27a)
          q3(0,z) = (Q2/QJ  + Y)  * exp
                                   W2(z-H)
                                     2K
            6(z-H) =
(27b)

(27c)
             = 0
                                                                         (27d)
where
*3(X) = - t  I hl * "2      2K
                                                                         (28)
is the inhomogeneity in the boundary condition  (27c).   Thus,  the transfor-
mation (24), which is used to remove the  inhomogeneity in the differential
equation (20a), transforms the homogeneous boundary condition,  Eq.  (20c), into
                                     21

-------
a nonhomogeneous boundary condition of the third kind.   Nevertheless,  it will



be shown later that,  unlike the nonhomogeneous  differential  equation set (20),



we can solve Eq. (27) with relative ease,  if q  (x,0)  in Eq.  (28)  is given  from



the solution of Eq.  (26).
     The solution of Eq.  (26) can be now expressed as



                                     °»

          qi(x,z) = exp(-x/UTc)  -  J   ^(4)  •  G3(x,z,|)d4,
                                                          (29)
where $,(|) is obtained from the initial condition (26b),  and the  Green's



function G~ is given by
          G3(x,z,4) = ±-
 -  2v/
exp

                .
                *
                                     | -(z+|)2U \
                                     Her- J
                                                                        (30)
After term-by-term integration, the final form of the solution,  Eq.  (29),



can be written as
where
:,z) = exp(-x/UT ) • f-  [e"r+ e'S-
               »-    Ju
                     z
                                                                        (31)
 "3 = Lz h3
                              erfc
                      2VK x/U
                         z
and r and s are defined in (14)
                                                                        (32)
                                     22

-------
     Now we can proceed to solve Eq.  (27).  First, we  set  z=0  in Eq.  (31)



and substitute the resulting equation for q.(x,0) in Eq.  (28).   After



simplification, the final expression for the inhomogeneity,  <{»~(x),  in the



boundary condition (27c) is given by
A fir}
 gives  the contribution
                                                  JO


of the inhomogeneity in the boundary condition  (27c), to the solution q_(x,z)
                                     23

-------
     The evaluation of q.,, is straightforward.  The final expression can be



written as
     q3A(x,z)  =  O^/Qj + Y )  • £- [e"r  +  e"s   •  (1 - c^)]           (35a)
where
                         *2

          "2  =  Lz h2  e *  erfc ^2                                     (35b)
To solve for > first we integrate with respect to r)  in Eq.  (34c) ,  substitute



for ~(x') from Eq. (33), and considerably rearrange and simplify the resulting



equation.  The final expression for qOT> can be written  as
                                     Ji>




                                (V   - V  )

               i(x, z)  = - v ^ 2l    22  '    (x^)                    (36a)
where
               V9i = VHI " Wi  =  Vn  "  V2
                21    dl    1      11       1                             (36b)




               V   =V   -W   =V    -  W /?
               V22   Vd2   W2     V12     W2^
and  F (x,z) is a function given by
   1  rx     l             j   -z2U        H^      x'
=  .  ,              . exp  1  4K  (x_xf) - -4^^,  - UT

                                z            z       c
            I  fX     1 	
            -  I               •

            "  Jo  Vx'(x-x')
                                       e    erfc
                                       45          1
                                        3   <*~e~  i
     \l - VrtK (x-x')/U  h_  e'°  erfc  |.  i  dx'                  (36c)
     L       z           2             5 J
                                      24

-------
Here,                 H + 2h0 K x'/U
                   -        *  z
                                                                         (36d)
                       2 VK x'/U
                           Z
                     z + 2h0 K   (x-x')/U

               ,  =  	 2  2     	

                3      2 ,/K (x-x')/U
                           Z
are aondimensional functions.  The integration with  respect  to  x1  in Eq.  (36c)


cannot be done analytically.  Therefore, F..(x,z) needs  to  be numerically


evaluated.
     The expression for q (x,z), from Eqs.  (34a),  (35a),  and  (36a),  can be now
written as
                 i3(x,z)  = (Q2/Q1 + Y)  ' £-   [ e"r  + e"s  •  (1  -  a^ J
                              (v21 -  v  )
                         - Y  -^	^  • F7(x,z)                       (37)
This is the analytical solution of Eq.  (27).  From Eqs.  (24),  (31),  and (37),


we obtain


           q2(x,z)  = (Q^Qj + Y) •  £- [ e"r + e"S  •  (1  -  a2
                      exp(-x/U-Cc)  •  ~  [ e"r + e"S
                      (V21 - v  )
                    Y    * K        •  F1(x,z)                            (38)
                            z
                                     25

-------
This is the analytical solution of Eq. (20).  Finally, from Eqs. (19) and (38),




the analytical solution of Eq. (18) can be written as






                    q2(x,2)  =  g22 / Lz                                (39a)






where
g^2(x,z) =  e
                           (Q/Q  + Y)  •    e~r +  e~s

                                1
                   exp(-x/UTc) -  {  e"r + e"S
             (V21 - V22)(Lz/Kz). FI(X,Z)
                                                                        (39b)
is a nondimensional function.  The prime indicates the modification of function




g-(x,z) of the secondary pollutant (denoted by the second subscript 2) for




the effects of deposition, sedimentation, and chemical transformation.








     The physical meaning of the various terms in Eq. (39b) is as follows:  The




first term has contributions from Q?/Q-, and y; the Qo/Qn part of this term accounts




for the direct emission of the secondary pollutant, and the Y part of this




term assumes that all the primary pollutant is transformed into the secondary




pollutant.  The second term corrects this spurious assumption by subtracting




the equivalent of the primary pollutant still available for transformation at




any given location (x,z).  The third (last) term accounts for the effect of




differences in the removal rates of the two pollutant species by deposition




and sedimentation; F-(x,z) in this term essentially represents a weighting




function such that 0 < FI < 1, as will be shown later.  It can be easily seen




the contribution of this term is zero when V?  = V2_ > 0, i.e., when the




deposition and sedimentation velocities of the two pollutants are equal.
                                     26

-------
     In the absence of a chemical coupling between the two species, all the



terms with y will be identically zero, and Eq. (39b) reduces to
          g22(x,z)  =
                                                               (39c)
which is the concentration algorithm for a chemically-independent, non-



reactive pollutant produced only by direct emission from a source of strength



Q,,.  For this case alone, we can rewrite Eqs. (4) and (39c) as






               C2(x,y,z)  =  Q2/U  •  p(x,y)  •  q2(x,z)
     g22(x,z)
                         =  e
                               *
                                             "S
                                                                         (40a)
where p and q_ are given by Eqs. (7) and (39a), respectively.  This expression



for gI2 is similar in form to that of g*  given in Eq. (I6b) for the primary



pollutant.
     For W  = W  = 0 and Q  = 0, Eq. (39b) reduces to
g22(x,z) =
(e"r
                       "S
                                                  - exp(-x/TJTc)}
                      -s
                                                           FI(X,Z)
                                                              (40b)
This equation agrees with the corresponding expression given by Lee  (1980),



except that the latter paper incorrectly shows 2H/K  instead of L /K  in
                                                   Z             22


the last term.
                                     27

-------
For Vdl =
              and
                                    = W, Eq. (39b) reduces to
>22(x,z)  =
                          •  e"r + e"s (1 - a)
                        Q2/Q1 +  *  I1 * exp(-s/Utc) }
                                                              (40c)
This agrees with the expression given by Rao (1975), who assumed equal non-zero



deposition and sedimentation velocities for the two species for simplicity.



For V  = 0 and W = 0, Eq. (40c) reduces to
          g22(x,z) = (e~r + e"S)
                          VQ1  + Y {l -  exp(-x/UTc)|
                                                                    (40d)
which agrees with the expression given by Heines and Peters (1973), who



considered only chemical transformation and derived this result based on a



simple component mass balance.  Thus, we can obtain all the known  solutions



for simpler problems from the analytical solution (39b) of the general problem



considered here.







PARAMETERIZATION OF CONCENTRATIONS





     In order to facilitate the practical application of the analytical



solutions, the eddy diffusivities K  and K  are expressed here in  terms of



a  and a  , the standard deviations of the crosswind and vertical Gaussian
 y      z'


concentration distributions,  respectively, as follows:
                    K  -
                     y   2  dx
                               TT
                          „  _ U 	z
                          J\  — 7T ' j '
                           z   2  dx
                                                                    (41)
Thus, for Fickian diffusion, K  and K  can be expressed by the  relations
                              y      ^
                                      28

-------
                    K  = a2  • U/2x   ,  K  = a2  • U/2x,                   (42)
                     y    y          '   z    z     '   *                   v  '




in order to utilize the vast amount  of empirical data on the  Gaussian plume



parameters cr  and or  available in the literature for a  variety of meteorological



and terrain conditions.  An excellent review and summary of these data can be



found in Gifford (1976).  Equation (42), in combination with  the Gaussian



assumption (see, e.g., Gifford, 1968), forms the basis  for the practical



plume diffusion formulas that are found in the  applications literature.
     In order to parameterize the expressions  for  concentrations  under



various stability conditions and to considerably reduce  the  difficulty in



typing the equations, we adopt the following nondimensionalization scheme:



All velocities are nondimensionalized by U, the constant mean wind speed.



The horizontal downwind distance x and all vertical height quantities  are
nondimensionalized by V2 CT  ; the transformation time  scale  T   is  nondimen-
                          Z                                 C



sionalized by -/2 a /U.  The horizontal crosswind  distance y is nondimen-





sionalized by      ''jn'jO/^1
                dl    dl              d2    d2



               yv                     A


                  W= W /IT             W  = W /TT
                -   f* - / \J       y      O   **O/ *"^







               V   =V   - V II     V   =V    -W/2
               *ni    JT    " t / **  >  "10   "JO   " o/ *•
                11    dl    1        12    d2     2
               V13 = Vll - (W1 - V/2
               V   =V   -W       V   =V   -W
               V21   Vdl   *1    '  V22   Vd2   W2
               x = x/V2 a        ,  z = z/V2 a
                         Z                   A
               H = H/V2 a        ,  L = L/V2~ a
                         Z                   Z
                  = TcU/V2 az    ,  y = y/V2" ay
                                     29
                                                                         (43)

-------
where the capped quantities denote the nondimensionalized variables, and


L is the mixing height.






     From Eqs. (3), (7), (8), (14), (16), (42), and (43), we can now rewrite



the expression for the primary pollutant concentration as follows:
                           Q      g      g'

               (^(£,7,2) =iT'r'r^                            (44a)
                                   y      z
               gl(x,y) = exp(-y2)                                       (44b)
               ,z) = exp(-p2 - x/Tc)  [ e"r + e"s • (1 - ce              (44c)
               L  = v2n cr   ,  L  = V27t a                               (44d)


where


               r = (z - H)2  ,  s = (z + H)2
                2 = 2Wx(z - H) x + W^ x2                                (44e)
                            x el erfc
               4X =  z + H + 2 Vn x




     Equation (44) clearly shows that concentration C  depends on the ratios



VL/U and V  /U, not on W  and V   per se.  Thus, the effect on Cj of large



values of W.. and V,- at high wind speeds is the same as that of small values



of W  and V,1 at low wind speeds, provided W_/U and V,./U remain constant.



The effect of deposition can be seen as a multiplication of the contribution
                                     30

-------
of the image-source or reflection term, e   , by a factor -1 <  (1  - or  )  <  1.

In the absence of chemical decay (t  = °°) , Eq. (44) agrees with the expression

for concentration given by Rao (1981) .  In the trivial case of negligible
                               *•      /*,
deposition and chemical decay (W = 0, V, = 0, t  = °°) , this equation  reduces

to the well-known Gaussian plume model with g'  = g_, where g_(x,z) is  defined

in Eq. (17).
     For a ground-level receptor (z=0), the concentration algorithm  for  the

primary pollutant reduces to
                        = exp(-p2 - H2 - x/tc)  •  (2 - aj                (45a)
where                    p2 = - 2 V  x H + W^ x2
                         or1 = 4Vn V1X x  e   erfc ^                     (45b)
                         I, = H + 2
Further simplifications of this equation are possible by setting H =  0  for
                          «s         ^\     A
ground-level sources, and W = 0 and V   = V,  for gases and small particles

with negligible settling.
     From Eqs. (4), (7), (19b), (32), (35b),  (39),  (42), and  (43),  the

parameterized expression for the secondary pollutant concentration  can be

written as follows:
                                     31

-------
   -   ~    gl     81
C2(x,y,z) = yi  •  ^  -
                                         8
                                          22
                                         L
                                   y      z
                                                                        (46a)
          >22(x,z) = e
                       3
                         V)   e"r  +  e"s
                                            (1 - «2) j
               -Y exp(-x/tc) J  e"r
                        "s
-Y 4/n
                        21    22'
                                        F,(x,z)
                                                         (46b)
where g1(x,y), L , L , r, and s are defined in Eq. (44), and the remaining
       I        y   2


quantities are as follows:





               P  = 2 W  (z - H) x + W  x2
                        V12 x  e   erfc
42 = z + H + 2 V12 x
                                                                         (46c)
                            x e   erf c
               63 = z + H + 2 V13 x
From Eqs. (36c), (36d), (42), and (43), the nondimensional function FI(X,Z)



in Eq. (46b) can be parameterized as
                                     32

-------
                    1 - 2 V13 x  vnt  e    erf c
                                           I2
r      *        i	  ^s          l
 1 - 2 V,0  x  vW-t) e    erfc |_   dt              (46d)
L       12                        5 J
where               |,

                     *                                                  (46e)
and t = x'/x is the normalized integration variable.






     Equation (46b) calculates gl» as the sum of contributions from the three




major terms on the right-hand side of this equation.  The physical signifi-




cance of these terms has been already explained in the discussion following




Eq. (39b).   In the absence of a direct emission of the secondary pollutant




(i.e., Q- = 0), a delicate balance exists between these three terms, as shown




in the next section.  The third term, which arises due to the differences in




the deposition rates of the two pollutant species, becomes important at large




x and, therefore, cannot be ignored.  The weighting function in this term




F~(x,z), given by Eq. (46d) , should be evaluated by numerical integration




to a sufficiently high degree of accuracy.








     For a ground-level receptor (z=0), the concentration algorithm for the




secondary pollutant reduces to:
                                     33

-------
                    (x,0) =  e
                           H2
            (Q2/Q! + Y)   e"11^  (2 - 02)
                          -Y  exp(-H2 - x/t) • (2 -
  -Y
                                             x •
                                                       (47a)
where
               ~9   ~
= -2VLxH + Wrx
                               12
                                  x  e   erfc
                     a3  = Wt V13 x  e   erf c |3
                                                   (47b)
                        = H + 2 Vu x ,    43 = H + 2 V13 x
and
*-0) = ~n /  7^=
      71 •'o  ./rrn
                             o Vt(l-t)
                  exp
   1  - VJF (4-
                                              s
                                          )  e 4 erf c
                                     e J  erfc  E,    dt
                                                   (47c)
where
   e,  = H/VF-»- 2 V., x v'
                                          13
                          ^5 = 2 V12 S
                                                                           (47d)
                                      34

-------
Further simplifications of Eq. (47) are possible by setting H = 0 for ground-



level sources, and W  = W,, = 0, V 3 = V,., and V - = V,2 when both species




are gases or small particles with negligible settling.








WELL-MIXED REGION








     Under unstable or neutral atmospheric conditions, when the plume travels



sufficiently far away from the source, the pollutant is generally well-mixed



by atmospheric turbulence, resulting in a uniform vertical concentration



profile between the ground and the stable layer aloft at a height L.  This



concentration, which is independent of source height as well as the receptor



height, can be calculated as the average concentration in a mixed layer of



depth L.  Following Turner (1970), we assume that the plume is well-mixed



for x > 2 x , where x  is the downwind distance at which o (x )=0.47 L.
      =    or        m                                    z  m







     The primary pollutant concentration in the well-mixed region can be



calculated as follows:
   .   .    Ql   81   8
C (x v z1 =—• — • —
SU,y,z;   u    L    L
                  y
               g1(x,y) = exp (-y )
                      •  r
                        •Jri
                                      41
                                                    °°
                         dz  =  i-
                                     H=0
       (48a)







       (48b)









    dz (48c)
H=0
                                     35

-------
Substituting for gi..(x,z) from Eq. (44) and carrying out the integration  in



Eq. (48c) , we obtain the following:
For Vdl * W1 or V21 # 0,
                                     I2
       = exp(-62 - x/tc) [(Vn/V21) e X erfc ^
                                               erfc p                    (48d)
where     £_ = 2 V-_ x   and   B  = W.. x.







This algorithm applies to gases or small particles.   This  equation is  indeter-

            A     t     oS


minate when V,- = W_ or V_. = 0.  For this  case,  g!..(x)  can be  determined by




setting W  = V,  in the expression for g'   in Eq.  (44)  and then integrating




as indicated in Eq. (48c).  Alternately, one can  take the  limit of Eq.  (48d)




as W  —> V.s.  Thus, we obtain the limiting expression for gj-(x) for




large particles as follows:







    *.     A     A

FnT V   =W  nr-V   =0
sue v,1 — w_ or v_- — u,
                  - x/tc)     (1 *  2  |)  e    erfc ^   -   2 ^/^T         (48e)
where     4, =27,,  x = V,,  x  =  W,  x.
          31       11       dl       1
Equations  (48d)  and  (48e)  
-------
     The secondary pollutant concentration in the well-mixed region can be


calculated as  follows:



                            QI      8,     gA2
               c2(x,y,z)  = tr  *  r  *  r^                             <49a)
                      ">00                   /»03
           /»00                   /»O3

           f r 8221          i  I   r
 T  /--A. v      I  I   ^i^t  I   ,       •!•  I   I    I  / ^ '
2  t X J  -~   I  I   -    I  Q.Z  —  • •'  I   I   gr  (X '

 42       J0 L Lz  JH=O     Jn.  J0  L   22
                                                                           (49b)
                                                           H=°
Substituting  for  gi-(x,z)  from Eq. (46) and carrying  out the integration in


Eq. (49b), we obtain the following:
For V2l £ 0 and V22  £ 0,





           R2  T                ,               ?2                       S2
           P7                  (    *   -      £2                       P2
o ' TY'I — o     I   (n  in + v~\   !  ( V  IM   \ r*   erf r  t   - ( W /9V   } (*   erfc  S
g42(xj - e     ^   W2/Q1 -1- YJ   j  C V12/V22 ^ e   ertc  ^2  C VZV22; e   erIC  P2





                /v ,v\                f.2                        02         \
               -x/T  / *    *       ^.,                        P2         1

          -vo    CJfV   /V   1   *«   *»rfr ?  -  fW  /?  V   'i <>   prfc 6  I
            y  e      < IV13'V21J   e   er±C ^3    L 2/Z  21J          P2 j
          - Y W«  (V21  - V22) x - F2(x)                                   (50a)
where              =  2 V   x ,
                  = 2 V13 ^ '    V13 = Vll -  (W1  - W2)/2                 (5°b)
                         f00 r

               F  (x)  =     |F (x,z) "L
                         J0 L        JH=O
                                      37

-------
Substituting for F (x,z) from Eq. (46d) and integrating, we  obtain
       F/-*\   X  I   -1-      f ** M- /~ \  I  T    /  *•    ^    f
      2(x) = 27T I   ~7^-  exp(-xt/T )  j  1 - ^n £, e   erfc

               JQ  ^            C

              (V12/V22}  e   erfc ^5 "  (W2/2V22} 6    6rfc  ^     dt'
where          4  = |  Vt  ,  4  = 4  Vl^t  ,  l  =  P  Vl1^               (50d)
Equation (50) is applicable when the two pollutant  species  are  either gases



or small particles.  In the large-particle  limit of W —> V,  for one or



both of the species, three other forms of this  algorithm can  be derived as



follows:
For V21 # 0 and V22 = 0,
                         v)    (i + 2 c    «    erfc  C   -   2
              e"X/Tc   I  (V13/V21) e 3  erfc  ^  -  (W2/2V21)  e 2 erfc
                       x  • F2(x)                                         (51a)
where     £2 = 2 V12 x = Vd2  x  =  W2  x  = 02
                2 V13  x
                                      38

-------
                                             e   erfc
[
               (1 + 2 £) e   erfc ^  -  2
                                                n ]  dt,
(51b)
with
          4, = 4, Vt~ ,  4r = i
Equation (51) is applicable when species- 1 is a gaseous or small-particle



pollutant, and species-2 consists of large particles.
For V01 = 0 and V0_ ? 0,
     21
                 22
           ,2

           '3
 42

                                             erfC
                                                                  ^3
                                                                 e   erfc
                              2 43) e   erfc |3  -  2
               /  I  TT   ^   T^ f \
             V 4^ V22 X - F2(x)
                                                                        (52)
where
                  = 2 V12 x ,  43 = 2 V13 x = W2 x
and F?(x) is given by Eq. (50c).   This algorithm applies when species-1



consists of large particles,  and species-2 is a gaseous or small-particle



pollutant.
                                     39

-------
For V21 = 0 and V22 = 0,

               [ Vi+
                 ,2
                -4
               e  2     (1 + 2 £) e   erfc     _  2   /^              (53)
where               £2 = W2 x = P2-
This algorithm applies when both species consist only of large particles.

     The physical situations represented by Eqs. (52) and (53) are unlikely to
occur in reality, since any primary pollutant consisting only of large particles
would not generally reside in the atmosphere long enough to produce significant
concentrations of the secondary pollutant by chemical transformation (i.e.,
x   /U  « T ) .  These algorithms are included here primarily for completeness
 max        c
of the solutions.  Since the well-mixed region concentration algorithms are
                                                    A
independent of z, they apply to all heights 0 < z <[ L.

PLUME TRAPPING

     For x  < x < 2 x , where x  is the downwind distance at which the plume
          m          m         m
upper boundary (corresponding to an isopleth representing one-tenth of the plume
centerline concentration) extends to the height of the inversion lid, the mixing
depth L should be included in the concentration algorithms .  This is usually
done through calculation of multiple eddy reflections (Turner, 1970)  from both
the ground and the stable layer aloft, when the plume is trapped between these
two surfaces.  For the general case which includes deposition, sedimentation,
and chemical transformation, the concentration of the species at any height z
in the plume trapping region can be expressed as

-------
                                Ql   gl   S31
                    C.(x,y,z) s-i.-i.-4i                           (54a)
                     1          U    Jj    Jj
                                      y    z
                    C2(x,y,z) = -  . -
     One can write the equations for gl-(x,z) and g'  (x,z), following Rao



(1981).  These expressions look similar to those given for g'  (x,z) and

                                                               /\

g22(x>z), respectively, except that the effective plume height H will be

                 A    yv     /\,

replaced here by H- - H + 2NL and the equations are summed over N from - » to



In general, a maximum of N = ± 10 eddy reflections are adequate to obtain



convergence of the sum within a small tolerance.
     A simpler alternate procedure suggested by Turner (1970) may be adopted


if one is interested only in ground-level concentrations:  We may calculate


the ground-level centerline concentrations of each species at x  and 2x
                                                    r          m       m


using the appropriate algorithms given earlier in this section, and then


linearly interpolate between these values on a log-log plot of concentration


versus downwind distance to obtain the ground-level centerline concentration


at any x in the plume- trapping region.
                                                                              «.
                                     41

-------
                                    TABLE 1
               SUMMARY OF POINT SOURCE CONCENTRATION EQUATIONS:
                  Applicable Algorithms and Equation Numbers
                                   Primary
                                  Pollutant
                              Secondary
                              Pollutant
1.   Near-source region
      (0 < x < x )
             —  m
     U    L    L
           y    2
                                                            - -
                                                             2~ U    L    L
                                                                      y    2
    (a)  Elevated receptor
             (2 > 0)

    (b)  Ground-level receptor
             (z = 0)
     Eq.  (44)


     Eq.  (45)
                                                                 Eq. (46)


                                                                 Eq. (47)
2.  Plume trapping region
          (x  < x < 2x )
            m         m

    (a)  Elevated receptor
               (2 > 0)
                                    1 " U    L    L
                                              y
                                        Eq. (54a)
                              II   !i   _
                              U  " L  ' L
                                    y    2
                              Eq. (54b)
                                                                            32
    (b)  Ground-level receptor
               (2 = 0)
     Interpolation
                                                            Interpolation
3.  Well-mixed region
c  -    ..
 1 ~ U    L    L
           y
                                                             2   U    L    L
                                                                       y
          any 2 > 0
     Eq. (48)
                                                                 Eqs. (50 to 53)
                                     42

-------
SURFACE DEPOSITION FLUXES

     The surface deposition fluxes of the primary and the secondary pollutants

at ground-level receptors are calculated directly from Eqs. (Id) and (2d) as
                                                                        (55a)
                    D2(x,y) = Vd2 - C2(x,y,0)                           (55b)
D gives the amount of pollutant deposited per unit time per unit surface area,
                                  2
and is usually calculated as kg/km -hr, while seasonal estimates are expressed
        2
as kg/km -month.  The estimation of the monthly or yearly surface deposition

fluxes at a given downwind distance x from the source in a given wind-directional

sector requires the knowledge of the fraction of the time that a mean wind of a
given magnitude blows in that direction in a month or a year, respectively.  To
                 2                                         3
obtain D in kg/km -hr when V, is given in cm/s and C in g/m , the right-hand
                                                                    2
side of Eq. (55) should be multiplied by 36000.  To obtain D in Mg/m ~nr when
                                 3
V. is given in cm/s and C in (Jg/m , the corresponding multiplication factor is

36.  For D calculations, the ground-level receptor is generally defined as any

receptor which is not higher than 1 meter above the local ground-level
elevation.
AREA SOURCES


     Urban air pollution results from (a) elevated large point sources such

as tall stacks of electric power plants and industries, (b) isolated line
                                     43

-------
sources such as highways, and (c) distributed area sources such as industrial




parks, clustered highways, and busy interchanges,  parking lots, and airports.




Numerous small low-level point sources distributed over a broad area, such as




smoke from chimneys of dwellings in an urban residential area, can be treated




as an area source.  Therefore, an urban diffusion model should be able to




account for point, line, and area sources.








     The line and area source problems are generally treated by integrating




the point-source diffusion algorithms over a crosswind line or over an area.




Differences between urban air pollution models occur only in the details of




how the area source summation is carried out and in how various meteorolo-




gical paremeters are included.  The simple ATDL Area-Source Model described




by Gifford (1970), Gifford and Hanna (1970), and Hanna (1971) is widely used




in many of the current practical urban air pollution models, such as the




Texas Episodic Model (TAGS, 1979).  We briefly describe below the derivation




of the Gifford-Hanna algorithm for estimating the ground-level concentrations




due to urban area sources.








     Consider two equal grid squares with side Ax, one of them containing




the ground-level area source emissions (Q), assumed to be located at the




center of the square; the second square, also known as the "calculation




grid square," contains a ground-level receptor  (R) at its center.  The wind




(U) blows along the line from Q to R, as shown  in Figure l(a).  The distances




from the receptor R to the downwind and upwind  edges of the emission grid




square are denoted by x., and x~  , respectively.  Since the two grid squares




are equal in size, x_ and x_ are identical to the distances measured from Q




to the upwind and downwind edges, respectively, of the calculation grid square,
                                     44

-------An error occurred while trying to OCR this image.

-------
as shown in Figure l(a).
     Neglecting deposition, sedimentation, and chemical transformation, the

surface (z=0) concentration due to a ground-level (H=0) point source of

strength Q (units :  M T  ) is given, from Eq. (44), by
where
                    O    Or
     C(x,y,0) = § •  |7 * |7                              (56a)
                     y    z

               g2(x,0) = 2                               (56b)
The surface concentration due to a crosswind infinite line source can be

obtained by integrating Eq. (56a) with respect to y:
                    00
          C(x,0) = /  C(x,y,0) dy = § - ^ = Jl  £r                    (57)
                  -OB                     z
where Q now has units of M L   T   and represents the emission rate of the  line

source.
     The surface concentration at the receptor R due to an area source Q

can be obtained by integrating Eq. (57) with respect to x:
                      2
C,
                  =    C(x,0) dx = ,/F g      f                        C5B)
                                     46

-------
Assuming a  is given by a power law of the form,
          Z


                    a (x) = a xb                                         (59)
                     Z




where a and b are constants depending only on the atmospheric stability, we



obtain
CA = VF  ^TT^T  i  <"  ~  <~"  \                    (60)
                          	     1-b      1-b

                          Ua(l-b)  \  2    "  xl
                       -2  -1
Here Q has units of M L   T  , representing the emission rate of the area



source, and 0 < x. < x».  This equation gives the concentration of a single



area source Q located upwind of a single receptor R.  If the latter were located



at the center of the emission grid square itself, then x. =0, x~ = Ax/2, and



Eq. (60) becomes
                c   _        _
                CA  -  \H  Ua(l-b)l
     If N receptors are located downwind of a single area source Q , then



the concentration at the receptor R. in the i th grid square (for i = 0, 1,



- - -, N) is given by
                Ai





where x. .  and x« .  are the distances measured from Q  to the upwind and downwind



edges, respectively, of the calculation grid square with the i th receptor.



Figure 2 illustrates this for N = 4.
     If N area sources are located upwind of a single receptor R , then the



contribution to the concentration at the receptor by the source Q. in the i th
                                     47

-------
CO
^
to
00

2E
^ 
(0 3 co
3 §
a1 w J3
ca 4) *J •
S-l -H O
•O aj M O»
•H 3 o
W C71 00 4)
_ 00 CO r-4 U
CVl -^ 4) T3 3
00 M CJ W
d oo >-i
•H 3 41
co (4 O X5
O 00 4->
<0 -H 1
4J a B
00 (0 4) O
fl r-l M V-l
•H 3 «J "4-1
3 U
0 t-4 fl T3
^J « -i-l S
^, co o -i-i
Xx RJ O co 3
t-l -H 3 O
00 3 'O
nj fl w
•H 3 41 -O
•O O O 41
T3 0 S-l
U eo 3
•H S-l 4-> M
4-> 3 CO (0
«J O -H 4)
S «w -0 S
4)
O J3 T3 4) 4)
iri -^- u ca ^a >-i
1X1 •*• en co 4J (0


CN
2 00
J% rt


                                  48

-------
grid square (for i = 0, 1, 	, N) is given by
                          ?i 	    /  1-b .   1-b \                   „ .
                       n  Ua(l-b)      X2i     xli  j                   (63)
where x..  and x^.  are the distances measured from R  to the downwind and upwind

edges, respectively, of the i th emission grid square.  This is illustrated in

Figure 3 for N = 4.  As noted earlier, the distances x.. .  and x-.  in Eq. (63)

are identical to those used in Eq. (62), since all grid squares are equal in

size.  The total surface concentration at the receptor R  can be obtained by

summing up the individual contributions of all N area sources:
                                   i-0                       /


This algorithm for urban area sources was given by Gifford and Hanna (1970) in

a slightly different form by using




               xu = (2i - 1) Ax/2 , x2i = (2i + 1) Ax/2 ,

                                                                        (65)

                         ° = xli < X2i


A more general form of this algorithm was first derived by Gifford (1970),

assuming that the mean wind and vertical eddy diffusivity are functions of

height given by power laws.




     It should be noted that the Gifford - Hanna algorithm, Eq. (64), ignores

horizontal diffusion.  Gifford (1959) postulated that air pollutant concentration

at a receptor due to the distributed area sources depends only on sources located

in a rather narrow upwind sector.  The angular width of this sector, derived from
                                      49

-------
to
^
ro
00
2
o
S
•< 0
o
o

x
O 01
4-> H
O4 1 ^9
01 CO
O -H CO
^•4 CJ
oi a
K) ^- J3 j3 eo
i\i _ jj AJ jj
X X -i-l W
* S T3 -H
U CO
CO «• 09
3 to 01
W O fcH
•rl
*O 03
•H CO •
W -H CO •
oo s a o
OJ. ro ^ « .a «
X ** 00 4-» M O
a -i-» o *>
•r4 5 00 O»
CO rH 01
03 CO U
CO 0) 0)
M 0} M
00 CO U
a 3 n 01
•H O» 3 J3
3 tn O -U
0 «J
J3 T3 1 5
_ , Cfl -i-l CO O
T: cu M o) )-i
X x CO CO
oo a a a
CO -H -H -i-l
•H 3 S
3 Ol 3
U CO
•H 14 3 'O
4-> 3 01
CO O 03 >-l
S m 01 3
oi u co
Ojs *o a co
«•- U ft « O>
OJ v CO CO 4-) S
x " 	 «
2 00
x 	 2J
                                    50

-------
known values of the horizontal diffusion length scale L , is less than the

usual 22.5° resolution of observed wind directions.  Consequently, horizontal

diffusion can be ignored.  This assumption is referred to as the "narrow plume

hypothesis."  The crosswind variations in the source-strength patterns can be

similarly ignored, since the urban source-inventory box areas are quite large,

usually 5 x 5 km or more, and do not vary much in strength from box to box;

therefore, the contribution of the source box containing the receptor is generally

the dominant one, and the contributions of more remote upwind area sources to

this receptor concentration are comparatively small.  For this reason, it is

generally adequate to consider only four area sources immediately upwind of

each receptor grid square, i.e., N = 4 in Eq. (64).  For the same reason,

a and b are assumed to be independent of the upwind distance x.



     Gifford and Hanna (1973) noted that, for the usual case of receptors

within a city, the area source component of the urban air pollution is

strongly dominated by the source-strength pattern and by transport by the

mean wind.  Atmospheric diffusion conditions in cities tend to be of the

near-neutral type, without the strong diurnal variations found elsewhere.

For these reasons, they suggested a simple approximation of the area source

formula, Eq. (61), as

                         CA = k  Q/U                                    (66)


where k is given by
                                     1-b
                                    X                                   (67)
                                   a(l-bT
and x is the distance from a receptor to the upwind edge of the area

source.  The parameter k is a weak function of city size and should be
                                     51

-------
approximately constant.   Using a large quantity of air pollution data, average

annual emissions, and concentrations of particles for 44 U.S. cities and S0_

data for 20 cities, Gifford and Hanna (1973) found k = 225 for particles, and

50 for SCL, with standard deviations of roughly half their magnitude.  They

noted that Eq. (66) works better for longer averaging periods.




     None of the equations given above for area sources consider deposition,

sedimentation, and chemical transformation/decay of gaseous or particulate

pollutants.  However, removal and transformation processes can be important for

obtaining reasonable estimates of the pollutant deposition fluxes in urban

residential areas.  The area source concentration algorithms for the

general case that includes deposition, sedimentation, and chemical trans-

formation can be obtained by integrating the point-source concentration

algorithms for pollutant species-1 and 2 emitted at ground level, following

a procedure similar to that shown in Eqs. (56) to (60), as follows:
               C   =    2     C(x'y'0) dy
                Al    J    J  "1
                      x   -oo
                      Q,    f2
                   =  ~   J    qj(x,0) dx                               (68a)
                           Xl
                      X2
               CA2 =           C2(x,y,0) dy dx

                     xl    -°°
                                     52

-------
     In the above, we utilized Eqs. (3) and (4), and integrated with respect to y.




Though straightforward in principle, the x-integrations required in Eq. (68)




are very difficult to carry out in practice, especially for pollutant species-2.




This is due to the obvious fact that the integrand functions represented by




the probability densities q.(x,0) and q?(x,0) are complicated functions of x,




unlike the case in the derivation of the Gifford-Hanna algorithm.  Even if




one is successful in evaluating these difficult integrals analytically, one




finds it nearly impossible to physically interpret the terms of the resulting




complicated expressions.   Following this experience, we started exploring for




an alternative to this direct approach for deriving the area source concentra-




tion algorithms in the general case that includes deposition, sedimentation,




and chemical transformation.  These efforts were successful, culminating in the




derivation of an elegant alternate approach, which can be physically explained




in terms of mass balance considerations, as outlined below.








     We rewrite the differential equations and the deposition boundary conditions




of Eqs. (6) and (18) as follows:
               D 3qa/3x = 8(Kz Sq^Bz + V^ q^/Sz - q^T^               (69a)






                         + Wx qj     =  V   q(x,0)                    (69b)
               U 3q2/3x =
                                                                        (70b)
                                     53

-------
Integrating Eqs.  (69a)  and (70a) with respect to z  from 0 to », and substituting

Eqs. (69b) and  (70b), we obtain



                     /*oo                               s~ 00

                U  fj I   q  dz  = - V   q (x,0) -\       q  dz           (71a)
                     -*<1                *~            o Jn
           U IE J   %  dz   = - Vd2 12(X>0) + I  J
                •'o                             c •'o
                                                             dz           (71b)
For ground-level  sources (H = 0),
       J   ^  dZ = J
       Jf\           Jn
                                   ?   dz = g'  (x)                        (72a)
            \    q2   dz = I     22£X?Z) dz = g;  (x)
            -'n            Jn       7.
                                                                      (72b)
        'o   "
Substituting Eq.  (72)  in Eq. (71), and integrating  the latter with respect to x

from x.. to x^  and rearranging, we obtain
Vdl
 fX2                                      ,     ,    fX2
     q (x,0) dx = U     g'  (x ) - gl^x,,)    -  —      gin^) dx     (73a)
 »     A               I   ^ A  1     ^A  A>   I     v    I     "A
Jxl                   L                   J        Jxl
     rx2                                                 rx2
Vd        q2(x,0)  dx = U  [ g^ (xa) - g;2(x2)  1  + f      g;2U) dx      (73b)
     -'x,                   L                    J     r  -'x.
                                      54

-------
where V.. ^ 0 and V,0 ?t 0.
       ul          d2
Noting from Eq. (68) that
                    ft
                   J   qi(x,0) dx = U CM/Q1  ,     i =  1 or 2  ,          (74)
                   Xl


we can now write the area source concentration algorithms for pollutant



species-1 and 2 as follows:
                QI                          i    r2
          C   = -=—  f g* (x ) - g1 (x )	—   /   g*  (x) dx  |          (75a)

                 dl  ^                       c  xi
                                                 X2
          CA2 = V7  [ «42<*1> - *42(X2> + lit-  J   «il« dx J          <75a>

                 d2                          C  Xl



Here g' (x) and g!?(x) are the nondimensional algorithms derived for the point-



source concentrations (see Table 1) in the well-mixed region.  Thus the



alternate approach, outlined above, is unique in that it allows one to use



the same algorithms (and the same program subroutines) to compute the concentrations


due to both point and area sources.






Rao (1981) showed that, for H = 0,




                    g£00 = £00                                         (76a)




where



                      u f°° r°°
               £00 -Til  I   C(x,y,z) dy dz                            (76b)
                                     55

-------
is the suspension ratio, representing the proportion (fraction)  of the pollutant




released at a rate Q by a source located at (0,0,H)  that still remains air-



borne at downwind distance x.   Therefore, Q  *  g!(x) in Eq.  (75) represents



the flux of pollutant passing through an imaginary vertical  plane at downwind




distance x.  Referring to Figure l(b), we can now  physically interpret the




area source concentration algorithms, Eq. (75), as follows:   For each calcu-



lation grid square box formed by the ground surface  and two  imaginary vertical




planes at x = x. and x = x», the pollutant mass balance is given by








     Incoming flux - outgoing flux ± flux gain/loss  due to chemical



               transformation = surface deposition flux



where




          Incoming flux = Q.. • g].(x1),     i = 1  or 2






          Outgoing flux - Qj • g^Cxj),






          Flux loss due to chemical transformation (species-1)
                                    /"',,<.
          Flux gain due to chemical transformation (species-2)








                                YQ
                                U
                                  c
                                  1

                                                dx'
          Surface deposition flux = V,. • C..





The only unknowns in the above are the surface concentrations, CA- , which can be
                                     56

-------
calculated as shown in Eq. (75).  Thus the area source concentration algorithms



in the alternate approach, derived mathematically from the governing differential



equations and the deposition boundary conditions, can be explained in terms of



physically realistic mass budgets of the pollutant species.







     Note that the functions g! . are parameterized and given in terms of



x = x/V2cr  instead of x.  Further, we did not specify any particular form of
         z


variation for O (x) in the derivation of the area source algorithms using the
               z


alternate approach.  Therefore these algorithms should be valid for any specified



type of variation (e.g., power law, exponential, polynomial, etc.) of a (x).
                                                                       Z


For a power law of form a  = a x , used by Gifford and Hanna (1970), the
                         Z


algorithms given by Eq. (75) are valid when the value of b is not signifi-



cantly different from 0.5.  This latter value follows from the relations



given in Eq. (42), which allowed us to express the exact analytical solutions



in terms of the empirical Gaussian dispersion parameters.  For urban area



sources, however, Gifford and Hanna (1970) used values of b ranging from



0.91 to 0.71, depending on the atmospheric stability class.  These values



are based on extensive observational data summarized by Smith (1968) and



Slade (1968).  Thus when b is significantly different from 0.5, Eq. (75)



should be modified for use with a power law variation of a  as follows:
                                                          z
                                                   -X2
     C
      Al    2(l-b)Vdl   |_ S41^1'   641V~2'   UT(
                                                 (77a)
     JA2    2(l-b)Vd2
f"  g^Cx,)  -842(x2)+uH
                                                                        (77b)
                                     57

-------
For V..-
     di
0 and t —> «, the concentrations calculated by these algorithms
agree with corresponding values given by the Gifford-Hanna algorithm, Eq. (60)



The latter can be easily extended for two chemically-coupled pollutants as



follows:
     "Al
                      • -b   -X/UT   ,
                       x    e     c  dx
                                                            (78a)
                  	
             rt  Ua(l-b)
                              1-b    1-b1
                             X    — X
                            ,2     xl  ;
"Al
                                                                        (78b)
For V  -
     di
0 and any finite value of T ,  the concentrations evaluated by
Eqs. (77) and (78) show remarkable agreement, even though these two sets of



equations axe derived using different approaches.  This good agreement may



be considered as verification of the area source algorithms, Eq. (77), based



on the mass balance approach.








     Equations (77) can be easily extended, as shown before, to the case of



N receptors downwind of a single area source (see Fig. 2); the concentrations



at the receptor R. in the i th grid square can be obtained by using the



appropriate downwind distances x. . and x_. in Eq. (77).  If N area sources



are located upwind of a single receptor R , as shown in Fig. 3, then the total



surface concentrations at the receptor can be obtained by summing up the



individual contributions of all N area sources, as follows:
                                     58

-------
p — .
CA1
r. - .
1
1
v
dl I
1=0
N
V
             i=0
                                        tO-H
                                                       ]
In Eq. (79),  it should be noted that for i = 0,






                                   = Q2/Q1                  (80a)






specify the initial conditions.  Further, we note that for i > 0,
where j = i - 1.






In Eq. (79), therefore, one needs to compute the functions g! - (x) and g!2(x)




only at x = x? .  , i = 0, 1, --- , N, since their values at x = x- . are known.




The subroutines that compute g! 1 and g!_ are common to both point and area




sources.  Thus the area source algorithms, Eq. (79), are simple, accurate, and




computationally efficient.
                              59

-------
                                   SECTION 4







                            RESULTS AND DISCUSSION





     In this section, we consider the well-known problem of the atmospheric



transport and transformation of SO^ (species-1 or primary pollutant) to



SO, (species-2 or secondary pollutant).  The diffusion-deposition algorithms



developed in the previous section for various stability and mixing conditions



for an elevated rural continuous point source were tested using the following



nominal values for the model parameters:



     U = 5 m/s , H = 30 m , KST = 5 (P-G stability class E)



     Vdl = 1 cm/s ,  Wx = 0 , Qj = 1 g/s



     Vd2 = 0.1 cm/s , W2 = 0 , Q2 = 0



     k.  = 1% per hour (T  = 36xl06 s) , y = 1.5
      t                 c


Some of the important results, calculated up to a downwind distance of 20 km



from the source, are presented and discussed in this section.  Any variations



of these nominal values of the model parameters are clearly shown in the



figures and noted in the text.  The parameters a  and a  used in the calcu-



lations are the P-G values, which appear as graphs in Turner (1970) and in



Gifford (1976, Figure 2).  These values, which are widely used for continuous



point sources in rural areas, are most applicable to a surface roughness of



0.03 m (Pasquill, 1976).







     The diffusion over cities is enhanced, compared to that over rural areas,



due to increased mechanical and thermal turbulence resulting from the larger
                                     60

-------
surface roughness and heat capacity of the cities.  This is reflected in the




urban dispersion parameter curves based on interpolation formulas given by




Briggs (see Gifford, 1976, Figure 7).  Some urban air pollution models, such




as the Texas Episodic Model (TACB, 1979), simulate the increased surface




layer turbulence over urban areas by decreasing the P-G atmospheric stability




class index by one, for all classes except Class A.  In any case, the algo-




rithms given in the previous section can be applied to sources in urban as




well as rural areas by using the appropriate dispersion parameters.
SENSITIVITY ANALYSES








     The algorithm for the ground-level concentration of the secondary




pollutant given in Eq. (47a) can be written as




          g' (x,0) = Term 1 + Term 2 + Term 3




The physical interpretation of these terms was discussed in the previous




section.  Figure 4 shows the delicate balance that exists between the




three terms; Term 2 and Term 3 together nearly balance Term 1.  A non-zero




concentration for the secondary pollutant results from the small positive




imbalance of the three terms.  Figure 4 shows that Term 3, which accounts




for the differences in the deposition rates of the two pollutant species,




becomes increasingly important as x increases.
     Because of this tenuous balance between the three terms in g' (x,0),



the weighting function F-(x,0) in Term 3 must be evaluated by numerical



integration to a very high degree of accuracy.  An examination of F,(x,0)



in Eq. (47c) shows that the integrand function has singularities at the
                                     61

-------
               en     *•
                I       I
                 a   a
in
cn


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                                                        62

-------
end-points t = 0 and t = 1 of the integration domain in t.  A computer-




library subroutine named D01AJF, developed by the Numerical Algorithms




Group (NAG), was utilized in this study for the numerical integration.




This routine, which is capable of handling the singularities, has been




selected because of its accuracy and applicability.  It estimates the




value of a definite integral of an externally defined function over a




finite range to a specified absolute or relative accuracy, using Gauss-




Kronrod rules in an adaptive strategy with extrapolation.








     The behavior of F.(x,0) for four different effective plume heights is




explored next.   Noting that
                                    dt = 1,
we can easily see from Eq. (46d) that the value of the weighting function




F (x,z) is always between 0 and 1.  Figure 5 shows that the value of F-(x,0)




is close to 1 for ground-level non-buoyant sources (H=0), but decreases




rapidly as H increases, especially for small values of x.  For all non-zero




H values, F (x,0) increases with x as shown in Fig. 5.  This clearly illus-




trates that Term 3 becomes important as x increases.








     The behavior of F (x,z) for four different receptor heights is shown




in Fig. 6.  The weighting function decreases rapidly as z increases.  At




large x, the calculated values of F  are of the same order of magnitude for




all z values.  Figure 7 shows the variation of F..(x,0) for three different




values of the deposition parameter V - of the primary pollutant.  Though F..
                                     63

-------
                          1,1,1.1  1.,    ,      1,0,
                                                                                                u
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                                                                                                14  (8

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                                                                                               60
                                                                                              •H
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                                  64

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                         65

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                           66

-------
increases with x initially for all V_1 values, as shown in previous plots,




F- reaches a maximum at a certain value of x, and then starts decreasing as




x increases.  This is illustrated by an order of magnitude increase in the

                     /s.

deposition parameter V.. 1 in Fig. 7.  A similar behavior is shown in Fig. 8




for the secondary pollutant.  This plot suggests that the effects of the




deposition parameter in the evaluation of F..(x,0) may be ignored if




V   < 2 x 10"4, i = 1 or 2.
GROUND-LEVEL CONCENTRATIONS (GLC)
     The variation of ground-level (z=0) plume-centerline (y=0) relative



concentrations, UC1(x,0,0)/Q , of the primary pollutant are shown in Fig. 9



for Vdl = 0, 2 x 10"3, and 2 x 10"2; Wj = 0 in all three cases.  The



V,- = 0 value corresponds to the zero-deposition case in which the new



concentration algorithms reduce to the well-known Gaussian plume algorithms



but with a first-order chemical decay of pollutant.  This case is included here



for comparison.  As V,- increases, the peak GLC's and the downwind distances


                                            -2
where they occur decrease.  For V,  = 2 x 10  ,  representing moderately



strong deposition, the concentration at x = 20 km is about an order of magni-


                                              *            -3
tude smaller than the corresponding value for V,- = 2 x 10
     Figure 10 shows the GLC variation for the secondary pollutant.  Since



the direct emission of species-2 is zero, the concentrations shown are



entirely due to the chemical transformation.  For k  = 1% per hour, the



peak concentration of species-2 is about three orders of magnitude smaller



than the corresponding value of species-1, and the downwind-distances of
                                      67

-------
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                                                       69

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                                                            70

-------
these peak values are larger than those of species-1.  It should be noted




that C« depends on the deposition parameters of both species as well as on




the chemical transformation rate.








     Figure 11 shows a comparison of calculated ground-level centerline concen-




trations of the secondary pollutant, when the latter is composed of (a) particu-



                                                -3
late matter with V,_ = W_ = 1 cm/s (V..- = 5 x 10  ), or (b) gaseous species with


                      *•       _o

Vd- = 1 cm/s, W  = 0 (V - =10  ).  The wind speed U is identical in both




cases.  The peak GLC for case (a) exceeds the corresponding value for the




non-deposition case (V,? = VL = 0) and occurs closer to the source.  This is




a result of gravitational settling of particles which tends to move the plume




toward the ground as it travels downwind.  The effect of deposition in this




case can be seen as an increase of GLC near the source, and a compensating




decrease farther downwind.  The GLC variation in case (b) for the gaseous




pollutant is similar to that shown in the previous figure.  These results




are consistent with those given by Rao (1981).
CHEMICAL TRANSFORMATION RATE








     The effect of varying the chemical transformation rate by two orders of




magnitude on the secondary pollutant concentration is shown in Fig. 12.




Since Q_ = 0 here, chemical transformation accounts for the entire concen-




tration of species-2.  For all k  values shown, the peak GLC's occur at the




same downwind location, since the meteorology and the deposition parameters




are kept constant.  An order of magnitude increase in k  increases the
                                      71

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concentration by about the same magnitude.  The GLC of the secondary pollu-


tant increases sharply with x until its peak value is attained, and then it


gradually decreases primarily due to a decrease in the reactant concentration.


At small x, the rate of increase in the GLC is directly proportional to k .




     The corresponding plot for the primary pollutant concentrations calcu-


lated over the same range of k  values is shown in Figure 13.  The results


shows that C. is not significantly altered by even a value of k  = 10% per


hour, which corresponds to I  = 36 x 10  s.  The pollutant-transport time

                              3
scale, given by x/U, is 4 x 10  s at x = 20 km; this is much smaller than


t , and so exp(-x/UT ) ~ 1.  Even for a value of k  = 100% per hour, GI


decreases by only about 1.1% at x = 20 km.
EFFECTS OF ATMOSPHERIC STABILITY





     Figure 14 shows the primary pollutant concentrations calculated for


P-G stability classes A to F (KST = 1 to 6).  The peak GLC's decrease slightly


with increasing stability, and their downwind locations move farther away from


the source.  The effects of deposition on the GLC generally increase markedly


as the stability increases.  However, under convective conditions, atmos-


pheric turbulence enhances the mixing of the plume and, in the well-mixed


region, distributes the pollutant uniformly through the entire mixing depth.


Therefore, as the stability decreases, the GLC's decrease as shown.  For KST = 1,


the interpolation for concentration in the plume-trapping region  (1 < x < 2 km)


can be clearly seen in the figure.
                                     74

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                                       76

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     The corresponding variations of the secondary pollutant concentrations




are shown in Fig. 15.  It can be seen that the peak GLC's of species-2




increase with increasing stability.   Their values are generally about three




orders of magnitude smaller than the corresponding values for species-1.




The downwind distances of the peak GLC's of the secondary pollutant are




generally larger than those for the  primary pollutant.








     The concentration curves shown in Figs. 14 and 15 do not appear smooth




for x < 0.5 km because of the finite steps in x used in the calculations.  Smoother




curves can be obtained by decreasing the step size Ax and increasing the number




of steps in this region.
                                     77

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                                                                                               O -H
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                                                                                               a  u
                                                                                               o  d
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                                                                                               4-1 "+-»
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                                                      78

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                                   SECTION 5








                            SUMMARY AND CONCLUSIONS








     A gradient-transfer (K-theory) model that accounts for the atmospheric




diffusion and deposition of two gaseous or particulate pollutant species,




which are coupled through a first-order chemical transformation, has been




formulated.  This model, which includes a deposition boundary condition for




each species, treats the pollutant removal mechanisms in a physically realistic




and straightforward manner.  The exact analytical solutions for concentrations




of the two pollutants released from an elevated continuous point source have




been derived.  For practical application of the model to a variety of atmospheric




stability conditions, the eddy-diffusivity coefficients in the analytical




solutions have been expressed in terms of the widely-used Gaussian plume disper-




sion parameters, which are functions of downwind distance and stability class.




This approximation allows one to utilize the vast amount of empirical data on




these parameters, for a variety of diffusion conditions, within the framework




of the standard turbulence-typing schemes.








     In order to facilitate comparison, the new diffusion-deposition-transforma-




tion algorithms for various stability and mixing conditions have been parame-




terized and presented as analytical extensions of the well-known Gaussian plume




dispersion algorithms presently used in EPA air quality models.  In the limit




when the deposition and settling velocities and the chemical transformation rate




are zero, the new algorithms reduce to the standard Gaussian plume equations.
                                     79

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Thus the model outlined here retains the ease of application--and is subject



to the same basic assumptions and limitations—associated with Gaussian plume-



type models.







     The formulations and the solutions are general enough to permit either



(or both) of the two pollutant species to be a gas or particulate matter (of



a known size).  Direct emission of the secondary (reaction product) pollutant



from the source is permitted.  Simplifications of the new algorithms for



ground-level sources and/or receptors, and very slow chemical reactions,



are indicated.  Limiting expressions of the algorithms for large particles



are derived.







     An innovative mathematical approach has been outlined to derive elegant



expressions for ground-level concentrations of the two species due to emissions



from area sources.  These simple expressions, derived from the governing



equations and the deposition boundary conditions, involve only the point source



algorithms for the well-mixed region.  This permits one to use the same program



subroutines to compute the concentrations due to both point and area sources.







     The new area source concentration algorithms are physically explained in



terms of the mass budgets for the pollutant species in each calculation grid



square.  For practical application to urban air pollution models, these algo-



rithms are extended to multiple area sources and receptors.  For power-law



variation of a (x), simple modifications of the algorithms are suggested to
              z


establish agreement with the Gifford-Hanna area-source algorithms in the limit



when the deposition loss of species is negligible.  Thus the area-source
                                     80

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algorithms, derived here through a new mathematical approach are physically




realistic, simple, efficient, and accurate.








     A sensitivity analysis of the concentration algorithm for the secondary




pollutant is given to illustrate the delicate balance of its terms and the




behavior of the numerically-evaluated weighting function in one of the




terms.  For an elevated point source in a rural area, the variations of




the ground-level concentrations of the two species are calculated by varying




the assigned nominal values of each of the model parameters.  The results




are presented and discussed with reference to the Gaussian plume concentra-




tions .








     The point and area source concentration algorithms developed in this




report may have wide applicability in practical rural and urban air pollu-




tion and particulate models, such as the Pollution Episodic Model (Rao




and Stevens, 1982), for use in research and regulation.
                                     81

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                                  REFERENCES
Budak, B. M.,  A. A. Samarskii and A.  N.  Tikhoiiov,  1964:   A Collection of
     Problems  on Mathematical Physics,  Pergamon Press,  The MacMillan Co.,
     New York, 770 pp.

Calder, K. L,  1961:  Atmospheric diffusion of particulate material consi-
     sidered as a boundary value problem.   J. Meteorol.  18, 413-416.

Carslaw, H. S., and J. C. Jaeger, 1959:   Conduction of Heat in Solids,
     2nd ed. Oxford University Press, London, 510  pp.

	, 	, 1963:  Operational Methods in Applied Mathematics,
     Dover Publications, New York, 360 pp.

Ermak, D. L., 1977:  An analytical model for air pollutant transport and
     deposition from a point source.  Atmos. Environ.  11, 231-237.

          , 1978:  Multiple Source Dispersion Model.  Report No.  UCRL-52592,
     NTIS, Springfield, VA, 68 pp.

Gifford, 1959:  Computation of pollution from several sources.   Int.  J.
     Air Pollut. 2, 109-110.

          , 1968:  An outline of theories of diffusion in the lower layers
     of the atmosphere.  Chapter 3, Meteorology and Atomic Energy 1968, D. H.
     Slade, ed.; available as TID-24190. NTIS, Springfield, VA, 65-116.

    	, 1970:  Atmospheric diffusion in an urban area.  2nd IRPA Con-
     ference, Brighton, UK.  ATDL Contribution File No. 33, 7 pp.

          , and S. R. Hanna, 1970:  Urban air pollution modelling.  In Proc.
     of 2nd Int. Clean Air Congress, Washington, D.C.  ATDL Contribution
     File No. 37, 6 pp.

    	, 	, 1973:  Modelling urban air pollution.  Atmos. Environ.
     7, 131-136.
          , 1976:  Turbulent diffusion-typing schemes:  A review.  Nuclear
     Safety 17, 68-86.

Hanna, S. R., 1971:  A simple method of calculating dispersion from urban
     area sources.  APCA Journal 21, 774-777.

Heines, T. S., and L. K. Peters, 1973:  An analytical investigation of the
     effect of a first-order chemical reaction on the dispersion of pollutants
     in the atmosphere.  Atmos. Environ. _7,  153-162.
                                      82

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Izrael, Y. A., J. E. Mikhailova and A. J.  Pressman, 1979:   A model for
     operative evaluation of transboundary flows of pollutants.  WMO
     Symposium on the Long-Range Transport of Pollutants and its Relation
     to General Circulation Including Stratospheric/Tropospheric Exchange
     Processes.  WMO No. 538, Geneva, Switzerland, 271-279.

Lee, H. N., 1980:  A study of analytical mesoscale nodel for atmospheric
     transport, diffusion and removal of pollutants.  Proc. Symp. on Interme-
     diate Range Atmospheric Transport Processes and Technology Assessment,
     Gatlinburg, TN, 289-297.  CONF-801064, NTIS, Springfield, VA.

Monin, A. S., 1959:  On the boundary condition on the earth surface for
     diffusing pollution.  Adv. Geophys. 6_, 435-436.

Pasquill, F. 1976:  Atmospheric dispersion parameters in Gaussian plume
     modeling.  Part II.  Possible requirements for change in the Turner
     Workbook values.  EPA-6QO/4-76-030b,  U.S. Environmental Protection
     Agency,  Research Triangle Park, NC,  44 pp.

Rao, K. S., 1975:  Models for sulfur oxide dispersion from the Northport
     power station.  The LILCO/Town of Huntington Sulfates Program, Project
     Report P-1336, Environmental Research & Technology, Inc., Concord, MA.

	, 1981:  Analytical solutions of a gradient-transfer model for
     plume deposition and sedimentation. EPA-600/3-82-079, U.S. Environmental
     Protection Agency, Research Triangle Park, NC; NOAA Tech. Memo.
     ERL ARL-109. 75 pp.  ATDL Contribution File No. 81/14.

          , and M. M. Stevens, 1982:  Pollution Episodic Model User's Guide.
     EPA-	,  U.S.  Environmental Protection Agency, Research Triangle
     Park, NC;  NOAA Tech. Memo. ERL ARL-      ATDL Contribution File No.
     82/28, 186 pp.

Slade, D. H., 1968:  Ed., Meteorology and Atomic Energy,  USAEC;  available
     as TID-24190, NTIS, Springfield, VA, 455 pp.

Smith, M. E., 1968:  Ed., Recommended Guide for the Prediction of the Dispersion
     of Airborne Effluents.   ASME, New York, 1st ed., 94 pp.

Texas Air Control Board, 1979:  User's Guide:  Texas Episodic Model.   Permits
     Section, Austin, TX, 215 pp.

Turner, D. B., 1970:  Workbook of Atmospheric Dispersion Estimates.   Public
     Health Service Publication No. 999-AP-26, U.S. Environmental Protection
     Agency, Research Triangle Park, NC,  84 pp.

Tychonov, A. N., and A. A. Samarskii, 1964:  Partial Differential Equations
     of Mathematical Physics. Vol. I.  Holden-Day, Inc.,  San Francisco, 380 pp.

Van der Hoven, I., 1968:  Deposition of particles and gases.  Section 5-3,  in
     Meteorology and Atomic Energy 1968,  D. H. Slade, ed.; available  as
     TID-24190. NTIS, Springfield, VA, 202-208.
                                     83

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                                   APPENDIX








                      SETTLING AND DEPOSITION VELOCITIES








     For a monodisperse particulate cloud, the individual particles have a con




stant gravitational settling velocity.  This terminal velocity is given by




Stokes' equation (Fuchs, 1964):
                                   18
where d is the diameter of the particle, g is acceleration due to gravity, p is




the particle density, and p is the dynamic viscosity of air.  However, for




d > 100 |Jm, the terminal fall velocity is sufficiently great that turbulence




in the wake of the particle cannot be neglected, and the drag force on the




particle is greater than the viscous drag force given by the Stokes' law,




F, = 3nd|jW.  For a particle with d = 400 pm, the actual value of W is about




one-third the value given by Eq. (A-l) .   Furthermore, Stokes' expression for




the drag force describes the effects of collisions between air molecules




and a particle, assuming air to be a continuum.  This assumption is not valid




for very small particles, since the mean free path between molecular collisions




is comparable to the particle size; under these conditions "slippage" occurs,




and the particles undergo Brownian motion and diffusion, which give a terminal




velocity greater than that predicted by Eq.  (A-l).  A discussion of the slip




correction factor for the Stokes' equation can be found in Fuchs (1964) and




Cadle (1975).
                                      84

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     The values for the terminal gravitational settling velocities for different




particulate materials are given in a tabular form by Lapple (1961) based on




particle diameter and Reynolds number.  These values, which account for




the deviations from Stokes1  equation discussed above, are given for spherical




particles with a specific gravity of 2.0 in air at 25°C and 1 atm. pressure.




This table has been reprinted in Sheehy et al (1969) and Stern (1976).








     The dry deposition pollutant-removal mechanisms at the earth's surface




include gravitational settling, turbulent and Brownian diffusion, chemical




absorption, inertial impaction, thermal, and electrical effects.  Some of the




deposited particles may be re-emitted into the atmosphere by mechanical resus-




pension.  Following the concept introduced by Chamberlain (1953), particle




removal rates from a polluted atmosphere to the surface are usually described




by dry deposition velocities which vary with particle size, surface properties




(including surface roughness z  and moisture), and meteorological conditions.




The latter include wind speed and direction, friction velocity lu, and thermal




stratification of the atmosphere.  Deposition velocities for a wide variety




of substances and surface and atmospheric conditions may be obtained directly




from the literature (e.g., McMahon and Denison, 1979; Sehmel, 1980).  Sehmel and




Hodgson (1974) give plots relating deposition velocity V, to d, z , u^., and the




Monin-Obukhov stability length, on the basis of wind tunnel studies.








     Considerable care needs to be exercised in choosing a representative




deposition velocity, since it is a function of many factors and can vary by




two orders of magnitude for particles.  Generally, V, should be defined




relative to the height above the surface at which the concentration measurement




is made.  The particle deposition velocity is approximately a linear function
                                     85

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of wind speed and friction velocity, and its minimum value occurs in the particle




diameter range 0.1-1 fjm.








     In the trivial case of W = V, = 0, settling and deposition effects are negligible




For very samll particles (d < 0.1 }Jm),  gravitational settling can be neglected,




and dry deposition occurs primarily due to the nongravitational effects mentioned




above.  In this case, W = 0 and V, > 0.  For small to medium-sized particles




(d = 0.1~50 [Jim), 0 < W < V.; deposition is enhanced here beyond that due to




gravitational settling, primarily because of increased turbulent transfer




resulting from surface roughness.  For larger particles (d > 50 |jm), it is




generally assumed that V, = W > 0, re-entrainment of the deposited particles




from the surface into the atmosphere is implied as, for example, in a dust




storm.  The first four sets of model parameters given above are widely used




in atmospheric dispersion and deposition of particulate material.  The deposition




of gases is a special case of the particulte problem with W = 0.
                                     86

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                                  REFERENCES
Cadle, R. D.,  1975:  The Measurement of Airborne Particles.   John Wiley & Sons,
     New York, 342 pp.

Chamberlain, A. C., 1953:  Aspects of travel and deposition of aerosol and vapor
     clouds.  A.E.R.E.  Report H.P.-1261, Atomic Energy Research Estab., Harwell,
     Berks., U.K., 32 pp.

Fuchs, N. A.,  1964:  The Mechanics of Aerosols.  The Macmillan Co., New York,
     408 pp.

Lapple, C. E., 1961:  J. Stanford Res. last. 5, p.  95.

McMahon, T. A., and P.  J. Denison, 1979:  Empirical atmospheric deposition
     parameters - a survey.  Atmos. Environ. 13, 571-585.

Sehmel, G. A., and W. H. Hodgson, 1974:  Predicted dry deposition velocities.
     Atmosphere-Surface Exchange of Particulate and Gaseous Pollutants.
     Available as CONF-740921 from NTIS, Springfield, VA,  399-423.

Sehmel, G. A., 1980:  Particle and gas dry deposition:  a  review.  Atmos.
     Environ.  14, 983-1011.

Sheehy, J. P., W. C. Achinger, and R. A. Simon, 1969:  Handbook of Air
     Pollution.  Public Health Service Publication No. 999-AP-44, U.S.
     Environmental Protection Agency, Research Triangle Park, NC.

Stern, A. C.,  1976:  Air Pollution.  Academic Press, New York, Vol. I, 3rd ed.,
     715 pp.
                                     87

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TECHNICAL REPORT DATA
- - - (Please read Instructions on the reverse before completing)
1. REPORT NO. 2.
4. TITLE ANO SUBTITLE
PLUME CONCENTRATION ALGORITHMS WITH DEPOSITION,
SEDIMENTATION, AND CHEMICAL TRANSFORMATION
7. AUTHOR(S)
K. Shankar Rao
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Atmospheric Turbulence and Diffusion Division
National Oceanic and Atmospheric Administration
Oak Ridge, Tennessee 37830
12. SPONSORING AGENCY NAME ANO ADDRESS
Environmental Sciences Research Laboratory - RTP, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
3. RECIPIENT'S ACCESSION-NO.
5. REPORT DATE
6. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT NO.
10. PROGRAM ELEMENT NO.
CDTA1D/02-1606 FY-84
11. CONTRACT/GRANT NO.
IAG-AD-13-F-1-707-0
13. TYPE OF REPORT AND PERIOD COVERED
Final 9/81-3/83
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
16. ABSTRACT

      A gradient-transfer  model  for the  atmospheric  transport, diffusion, deposition,
 and first-order chemical  transformation  of gaseous and particulate pollutants emitted
 from an elevated  continuous point source  is  formulated and analytically solved using
 Green's functions.  This  analytical  plume model  treats gravitational settling and dry
 deposition  in  a physcally realistic  and  straightforward manner.  For practical appli-
 cation of the  model, the  eddy diffusivity coefficients in the  analytical  solutions
 are expressed  in  terms of the widely-used Gaussian plume dispersion parameters.  The
 latter can  be  specified  as functions of the downwind distance  and  the atmospheric
  stability class  within  the  framework  of the   standard  turbulence-typing  schemes.

      The analytical  plume  algorithms  for  the primary  (reactant) and  the  secondary
  (product) pollutants  are presented  for  various  stability  and mixing conditions.   In
 the limit  when deposition and  settling velocities  and the  chemical   transformation
  rate are  zero, these equations  reduce  to the  well-known Gaussian  plume diffusion
  algorithms  presently  used  in  EPA dispersion models  for assessment of air quality.
  Thus the  analytical  model  for  estimating  deposition  and  chemical   transformation
 described here retains the ease of application associated with Gaussian plume models,
  and is  subject to the same basic assumptions and limitations  as  the latter.
17.
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