surface is zero.
This allows one to obtain an expression for K 1’?
since, when [ = 0,
c.
(3)
Solving equation (3) gives
c= Cc, exp
Ky
KS)
(4)
Furthermore, it is reasonable to assume for the equilibrium situation that the concentration of the impurity has
a decreasing exponential dependence upon the height:
c= c¢. exp ( Mer.
(5)
Thus, by comparing equations (4) and (5), an expression for K, is obtained:
K,-
_ Mg
yp Kp
(6)
where g is the acceleration due to gravity.
Knowing K) and K,
in equation (1) allows one to calculate the flow rate at time t = 0, provided the con-
centration and concentration gradient are known for t = 0.
flow rate at all points for t = 0 may be calculated.
may be calculated for t = At.
Thus given an initial concentration distribution, the
Then, using these flow rates, the concentration distribution
This process can then be repeated again and again.
It is thus possible to numeri-
cally obtain the concentration profile as a function of time and height for any given initial concentration profile;
i.e., one may numerically solve the following differential equation:
K
ac 2 | a
at
tft
K
=2) ac]
2{()..(
ex]
~
7)
Equation (7) has been numerically solved on Sandia's 704 Computer for several different initial concentration profiles.
Two of the cases which have been considered will be discussed herein.
In Figure 1 the time
development of particles of mass 150 atomic units is plotted for an initial square distribution centered at
350 kilofeet.
From this illustration it is seen that the injected profile rapidly goes into a profile which remains
constant as it falls.
This has been true for any initial profile which we have considered in our calculations.
Figure 2, the time development of particles of mass 1500 atomic units has been plotted.
taken as our initial profile a square distribution centered at 305 kilofeet.
In
Here again, we have
The same qualitative behavior exists
for these heavier particles as for the lighter particles of Figure 1; i.e., the initial profile rapidly goes into a
constant profile which does not change as it falls.
From these illustrations and our other calculations which are not reported here, some semiquantitative
conclusions can be obtained.
First, as mentioned before, regardless of the shape of the injected profile, a con-
stant profile is rapidly obtained.
the impurity particles.
The half-width of this constant profile depends only on the mass and size of
Also, as will be immediately shown, once the constant profile is obtained, the descent
of the profile is governed only by the gravitational term of the flow rate equation.
This is true, since for the
profile to remain constant, the entire profile must fall at the same rate as the maximum point of the profile.
18