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

What is the absolute minimum piscicide usage every 30 days to maintain 1 mg/m 3 ?
Answer
A total of 1.08 kg added continuously over a 30 day period
These unsteady-state solutions usually contain exponential terms such as
exp(–kt). The term k is a characteristic rate constant with units of reciprocal time.
It is thus somewhat difficult to grasp and remember. A quantity of 0.01 h –1 does not
convey an impression of rapidity. It is convenient to calculate its reciprocal 1/k or
100 h, which is a characteristic time. This is the time required for the process to
move exp(–1) or to within 37% of the final value, i.e., it is 63% completed. Those
working with radioisotopes prefer to use half-lives rather than k, i.e., the time for
half completion. This occurs when the term exp(–kt) is 0.5 or kt is ln2 or 0.693,
thus the half-time t is 0.693/k. Another useful time is the 90% completion value,
which is 2.303/k.
Two common mistakes are made if rate constants are manipulated as times rather
than frequencies. A rate constant of 1 day –1 is 0.042 h –1 , not 24 h –1 —a common
mistake. If there are two first-order reactions, the total rate constant is the sum of
the individual rate constants. This has the effect of giving a total half-time or halflife
that is less than either individual half-time. It is a disastrous mistake to add halflives.
Their reciprocals add.
In some cases, the differential equation can become quite complex, and there
may be several of them applying simultaneously. Setting up these equations requires
practice and care. There is a common misconception that solving the equations is
the difficult task. On the contrary, it is setting them up that is most difficult and
requires the most skill. If the equation is difficult to solve, tables of integrals can
be consulted, computer programs such as Mathematica or Matlabs can be used, or
an obliging mathematician can be sought. For many differential equations, an analytical
solution is not feasible, and numerical methods must be used to generate a
solution. We discuss techniques for doing this later.
2.5 EULERIAN AND LAGRANGIAN COORDINATE SYSTEMS
It is usually best to define the mass balance envelope as being fixed in space.
This can be called the Eulerian coordinate system. When there is appreciable flow
through the envelope, it may be better to define the envelope as being around a certain
amount of material and allow that envelope of material to change position. This “fix
a parcel of material then follow it in time as it moves” approach is often applied to
rivers when we wish to examine the changing condition of a volume of water as it
flows downstream and undergoes various reactions. This can be called the Lagrangian
coordinate system. It is also applied to “parcels” of air emitted from a stack and
subject to wind drift. Both systems must give the same results, but it may be easier
to write the equations in one system than the other. The following example is an
illustration. It also demonstrates the need to convert units to the SI system.
©2001 CRC Press LLC