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OCR for page 194

Appendix C
A Simple Box Mode! with Recirculation
One can formulate a simple mode] to follow the evolution of pollutant
concentrations based on looking at the air mass above an area of interest as
a well-mixed box. in this case, the time varying volume of the box, V(t),
is given as the area over which the model is being applied, multiplied by the
effective height into which the pollutants are mixed (i.e., the mixing height,
H(t)), noting that it can vary with time.
V(t) = L ~v7I(t)
where L is the length of the box (in the direction of the wind) and His the
width of the box perpendicular to the wind, as shown in Figure C-1 .
Applying the principle of conservation of mass, one then gets
dLWHc
dt = Min - MOut + E
where Min is the mass flux in (mass/time) due to the wind carrying the
contaminant (CO) in from outside and Moot is the mass flux out; c is the
time dependent CO concentration (mass/volume) in the box, and E the time
dependent emission rate from all sources in the box.
The mass flux in is given by the wind velocity, U. multiplied by the
area of the side of the box (OH) and background CO concentration Cb,
Min = UW~6t9Cb.
194

OCR for page 194

Appendix C 195
/
UCb ~
H(t)
~ EU
c(t)
T
W
Uc
FIGURE C-1 Diagram of a simple box model win a box of width W. length L,
and time dependent height H(t) and wind speed U. E is the total mass rate of
pollutant emissions within the box, which is assumed to be well mixed, with CO
concentration c throughout; cb is the background concentration.
If the background concentration is small it can be neglected. As noted
below, the assumption that the mass flow into the box depends only on the
background concentration may be wrong if there is CO that leaves the box
but is recirculated. The mass flux out is given by
Mout= UVVH(t)c
where the assumption that the box is well mixed leads to having the con-
taminant concentration leaving the box the same as that within it. Using the
above two expressions for Min and Moot leads to the classical box model
formulation for the evolution of pollutant concentration c,
TIC UCb Uc E 1 dH~
At L
:L 1,WH(~t) H(t~a7tl
dH
·—>0
dt
.
The last term causes the concentration to decrease if the mixing height
increases with time, because the CO mixes in an expanding volume. If the
mixing height decreases with time, the change in height has no effect on the
concentration in the box (the last term is zero).

OCR for page 194

196 Appendix C
if the emission rate is much greater than the flux in due to the wind (E
>> Mind, the first term can be dropped. This equation can account for the
time variation in both the mixing height and emission rate, but assumes that
the length and width of the box are fixed. Further, it assumes that the pol-
lutant concentrations do not vary spatially within the box, which implies
that the emissions do not vary significantly spatially. This limits the size
of the mode] application area, and means that it should not be used to esti-
mate hot spot concentrations. Also, it assumes that none of the contami-
nant that leaves the box returns, except after being added to the background
levels. This assumption can be wrong. In that case, the mode! should be
modified to account for the fraction of contaminant that leaves the box and
is recirculated back into it. This can be done by adding a recirculation
coefficient ~ that is the Faction of contaminant that returns to the box after
being advected out. Physically, or must be between O and 1. The resulting
mode! becomes
TIC UCb U(~-a~c E c do Ucb E ~U(~-~)
= — + _ _ =—+ _ + C'.
At L L LWH(t) H(t) dt dH>o L LWH(t) L H(t)
If or is zero, the original box model is obtained. As cc approaches one,
virtually all of the air and contaminant leaving the box re-enters, allowing
the contaminant concentration to build up dramatically.