Page 370 - Fundamentals of Air Pollution 3E

II. Modeling Techniques 325

8 X/8t = [A* ^ + M2j(Xb - X) + A*(A 2i/AO fa - *)]/A* z, (20-16)
This forms a basis for an urban photochemical box model (6) discussed
later in this chapter.

B. Narrow Plume Hypothesis
By assuming that the principal contributors to the concentration at a
receptor are the sources directly upwind, especially those nearby, the con-
centration due to area sources can be calculated using the vertical growth
rate rather than uniform vertical mixing and considering the specific area
emission rate of each area upwind of the receptor. Area emission rate
changes in the crosswind direction are neglected as being relatively unim-
portant. The expansion in the vertical is usually considered using the
Gaussian vertical growth (7, 8).

C. Gradient Transport Models

The mean turbulent flux of concentration in the vertical direction z is
w'x'- Assuming that this turbulent flux is proportional to the gradient of
concentration, and in the direction from higher to lower concentrations,
an overall diffusivity K can be defined:

The change in concentration with respect to time can be written as

where the term in parentheses on the left accounts for advection. The terms
on the right account for diffusivities in three directions, K x, K y, and K z
(where K x is in the direction of the wind, K y is horizontally crosswind, and
K z is vertically crosswind), and S represents emissions. This equation is
the basis for the gradient transport model, which can handle varying wind
and diffusivity fields. The vector speeds u, v, and w (where u is in the
direction of the wind, v is horizontally crosswind, and w is vertically cross-
wind) and concentrations imply both time and space scales. Fluctuations
over times and distances less than these scales are considered as turbulence
and are included in the diffusivities.
The gradient transport model is most appropriate when the turbulence
is confined to scales that are small relative to the pollutant volume. It is
therefore most applicable to continuous line and area sources at ground