McKay, Donald. "Front matter" Multimedia Environmental Models ...

it to decay in concentration with time, but maintain equilibrium between all phases
at the same time. This is analogous to a batch chemical reaction system. Although
it is possible to include emissions or advective inflow, we prefer to treat first the
case in which only reaction occurs to an initial mass M. We assume that all volumes
and Z values are constant with time.
But,
Solving gives
©2001 CRC Press LLC
dM/dt = –SV iC ik i = –fSV iZ ik i = –fSD Ri
M = SV iZ if = fSV iZ i
df/dt = –fSV iZ ik i/SV iZ i = –fSD Ri/SV iZ i
f = f O exp(–k Ot)
where k O = SV iZ ik i/SV iZ i = SD Ri/SV iZ i, and f O is the initial fugacity. Note that k O,
the overall rate constant, is the reciprocal of the overall residence time.
Worked Example 6.6
Calculate the time necessary for the environment in Example 6.3 to recover to
50%, 36.7%, 10%, and 1% of the steady-state level of contamination after all
emissions cease.
Here, SVZ is 24 and SD is 0.2587. Thus,
f = f O exp (–0.2587t/24) = f O exp (–0.01078t)
Since M is proportional to f, and f O is 96.6 Pa, we wish to calculate t at which f is
48.3, 35.4, 9.66, and 0.966 Pa. Substituting and rearranging gives t = –1/0.01078 ln
(48.3/96.6), etc., or t is, respectively, 64 h, 93 h, 214 h, and 427 h. The 93-hour time
is significant as both the steady-state residence time and the time of decay to 36.7%
or exp(–1) of the initial concentration.
It is possible to include advection and emissions with only slight complications
to the integration. The input terms may no longer be zero.
This example raises an important point, which we will address later in more
detail. The steady-state situations in the Level II calculations are somewhat artificial
and contrived. Rarely is the environment at a steady state; things are usually getting
worse or better. A valid criticism of Level II calculations is that steady-state analysis
does not convey information about the rate at which systems will respond to changes.
For example, a steady-state analysis of salt emission into Lake Superior may demonstrate
what the ultimate concentration of salt will be, but it will take 200 years
for this steady state to be achieved. In a much smaller lake, this steady state may