Transport with Decay and Degradation - Environmental Modeling - page 97

Table 5.1 Half-lifes of selected radionuclides
Radionuclide Half-life Radionuclide Half-life

10 9 years U-238 4.5 H-3 12.35 years

U-235 32,500 years Ra-228 5.8 years

Ra-226 1,600 years Th-228 1.91 years

Am-241 432.2 years Gd-153 242 days

Pu-238 87.74 years Po-210 138 days

Cs-137 30 years Sr-89 50.5 days

Pb-210 22.3 years Th-234 24.1 days

From mathematical point of view, the major difference between radioactive
decay and other forms of first order losses lies in the decay constant. The half-life
of radionuclides is constant under all conditions known. The rate of exponential
decay of a nuclide is not influenced by any environmental condition, neither by
temperature, nor by pressure, nor by the biogeochemical surrounding. Table 5.1
lists some radionuclides and their half-lifes. The simulation of a chain of
radionuclides is described in Chap. 18.2.
In contrast, chemical and biochemical rates are strongly affected by environ-
mental variables. In fact, the decay law can often be understood as a most simplified
rule, in which the interaction of several complex processes are gathered and where l

is a lumped parameter. Clearly, in a changed environment the parameter is differ-
ent. More complex degradation rules, using the Michaelis-Menten or Monod
kinetics, are treated below (see Chap. 7).

5.2 1D Steady State Solution

As mentioned above, the differential equation for the steady state is obtained by
setting the time derivatives in the transport ( 5.2 ) to zero. The right hand side of the
1D transport equation has thus been set to zero:

@x D @ c

@

@x v @c @x l c ¼

0 (5.7)

This is an ordinary differential equation for the independent variable x .With
MATLAB ordinary differential equations can be solved numerically (see Chap. 9).

Here, an analytical solution, which provides the solution in an explicit formula, is an
alternative if the coefficients are constants, i.e. independent of x and t .Inorderto
solve differential ( 5.7 ) analytically, it is appropriate to note it in a different form:

®

c ¼

@
@x m 1

@
@x m 2

0 (5.8)