FIELD TESTING AND EVALUATION OF DUST DEPOSITION AND ...

At this point, it is possible to remove the dependence of c on x if it is assumed that
the wind speed u is nearly constant with height and by expressing u as dx/dt. This casts
the ADE in a Lagrangian frame where the concentration c is dependent only on the height
z and the time since emission, yielding the one-dimensional, time dependent ADE:
∂c
∂
= K
∂t
∂z
∂c
∂z
zz
.
Equation 4-4
Note that the time derivative in Equation 4-4 is not the same as that in Equation 4-1.
Specifically, in the latter equation δc/δt is evaluated at a stationary point in space,
whereas in the former, it is implied that the derivative is evaluated in a frame of reference
that moves downwind with the plume.
The downwind distance at a given time can be estimated by multiplying the time
with the wind speed at a 10 m height, obtained, for example, from a logarithmic profile
(e.g. Equation 2-9). Values of K zz as a function of height have been provided by a
number of investigators. The functional forms used in this model are
Unstable Conditions (Lamb and Duran, 1977):
K
zz
w z
*
i
4
1
3
4
z z z
2.5
κ
1
− 15 → 0 ≤ < 0.05
zi
L zi
2
z z
0.021+
0.408
+ 1.351
↵
zi
zi
3
4
z z z
= −
4.096
+ 2.560
→ 0.05 ≤ < 0.6
zi
zi
zi
z z
0.2exp6
−10
→ ≤ <
0.6 1.1
z
i
zi
z
0.0013 → ≥ 1.1
z
i
Equation 4-5
Neutral Conditions (Shir, 1973):
8 fz
K zz = κ u
−
*
z exp
u*
Equation 4-6
Stable conditions (Businger and Arya):
4-2