Encyclopedia of Environmental Science and Engineering, Volume I and II

URBAN AIR POLLUTION MODELING 1169

to represent the actual conditions. Various authors have

proposed mathematical algorithms that include appropriate

modifications of Equation (1). In addition, a source-oriented

model developed by Roberts et al. (1970) to allow for time-

varying sources of emission is discussed below; see the section

“Time-Dependent Emissions (the Roberts Model).”

Chemical Kinetics: Removal or Transformation

of Pollutants

In the chemical-kinetics portion of the model, many differ-

ent approaches, ranging in order from the extremely simple

to the very complex, have been tried. Obviously the simplest

approach is to assume no chemical reactions are occurring at

all. Although this assumption may seem contradictory to our

intent and an oversimplification, it applies to any pollutant

that has a long residence time in the atmosphere. For exam-

ple, the reaction of carbon monoxide with other constituents

of the urban atmosphere is so small that it can be considered

inert over the time scale of the dispersion process, for which

the model is valid (at most a few hours).

Considerable simplification of the general problem can

be effected if chemical reactions are not included and all vari-

ables and parameters are assumed to be time-independent

(steady-state solution). In this instance, a solution is obtained

that forms the basis for most diffusion models: the use of the

normal bivariate or Gaussian distribution for the downwind

diffusion of effluents from a continuous point source. Its use

allows steady-state concentrations to be calculated both at

the ground and at any altitude. Many modifications to the

basic equation to account for plume rise, elevated sources,

area sources, inversion layers, and variations in chimney

heights have been proposed and used. Further discussion of

these topics is deferred to the following four sections.

The second level of pseudo-kinetic complexity assumes

first-order or pseudo-first-order reactions are responsible for

the removal of a particular pollutant; as a result, its concentra-

tion decays exponentially with time. In this case, a characteris-

tic residence time or half-life describes the temporal behavior

of the pollutant. Often, the removal of pollutants by chemical

reaction is included in the Gaussian diffusion model by simply

multiplying the appropriate diffusion equation by an exponen-

tial term of the form exp(− t / T ), where T represents the half-life

of the pollutant under consideration. Equations employing this

procedure are developed below. The interaction of sulfur diox-

ide with other atmospheric constituents has been treated in this

way by many investigators; for examples, see Roberts et al.

(1970) and Martin (1971). Chemical reactions are not the only

removal mechanism for pollutant. Some other processes con-

tributing to their disappearance may be absorption by plants,

soil-bacteria action, impact or adsorption on surfaces, and

washout (for example, see Figure 2^ ). To the extent that these

processes are simulated by or can be fitted to an exponential

decay, the above approximation proves useful and valid.

These three reactions appear in almost every chemical-

kinetic model. On the other hand, many different sets of equa-

tions describing the subsequent reactions have been proposed.

For example, Hecht and Seinfeld (1972) recently studied the

propylene-NO-air system and list some 81 reactions that can

occur. Any attempt to find an analytical solution for a model

utilizing all these reactions and even a simple diffusion sub-

model will almost certainly fail. Consequently, the number of

equations in the chemical-kinetic subroutine is often reduced

by resorting to a “lumped parameter” stratagem. Here, three

general types of chemical processes are identified: (1) a chain-

initiating process involving the inorganic reactions shown

above as well as subsequent interactions of product oxidants

with source and product hydrocarbons, to yield (2) chain-

propagating reactions in which free radicals are produced;

these free radicals in turn react with the hydrocarbon mix to

produce other free radicals and organic compounds to oxide

NO to NO 2 , and to participate in (3) chain-terminating reac-

tions; here, nonreactive end products (for example, peroxy-

acetylnitrate) and aerosol production serve to terminate the

chain. In the lumped-parameter representation, reaction-

rate equations typical of these three categories (and usually

selected from the rate-determining reactions of each category)

are employed, with adjusted rate constants determined from

appropriate smog-chamber data. An attempt is usually made

to minimize the number of equations needed to fit well a large

sample of smog-chamber data. See, for examples, the studies

of Friedlander^ and Seinfeld (1969) and Hecht and Seinfeld

(1972). Lumped parameter subroutines are primarily designed

to simulate atmospheric conditions with a simplified chemical-

kinetic scheme in order to reduce computing time when used

with an atmospheric diffusion model.

Elevated Sources and Plume Rise

When hot gases leave a stack, the plume rises to a certain

height dependent upon its exit velocity, temperature, wind

speed at the stack height, and atmospheric stability. There

are several equations used to determine the total or virtual

height at which the model considers the pollutants to be

emitted. The most commonly used is Holland’s equation:

H

v

u

P

TT

T

ssad

a





15 ..268 10−^2 ()

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

where

∆ H: plume rise

v s : stack velocity (m/sec)

d: stack diameter (m)

u: wind speed (m/sec)^

P: pressure (kPa)

T s : gas exit temperature (K)

T a : air temperature (K)

The virtual or effective stack height is

H 
 h  ∆ H

where

H: effective stack height

h: physical stack height

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