Chapter 2.P2, Problem 5P

Effects of Partial Source Remediation.

(a) Assume that a source remediation process results in a $90\%$ reduction in the initial amount of DNAPL mass in the source region. Repeat Problem $4$ with $m_0$ and $c_0$ respectively. Compare the graphs of $c_s\left(t\right)$ in this case with the graphs obtained in Problem $4$.

(b) Assume that the $90\%$ efficient source remediation process is not applied until $t_1=10$ years have elapsed following the initial deposition of the containment. Under this scenario, plot the graphs of $c_s\left(t\right)$ using the parameters and initial conditions of Problem $4$. In this case, use Eq. $\left(2\right)$ to compute concentration for $0\le t<t_1$. Following remediation, use the initial condition $m\left(t_1\right)=m_1=0.1m\left(t_1-0\right)=0.1{\lim }_{t\to t_1}m\left(t\right)$ for Eq. (i) and use the following modification of Eq. (2):

$\begin{array}{ccc}\frac{c_s\left(t\right)}{c_1}={\left[\frac{m\left(t\right)}{m_1}\right]}^{\gamma },& & t>t_1\end{array}$, (ii)

where, $c_1={\left(0.1\right)}^{\gamma }c\left(t_1-0\right)={\left(0.1\right)}^{\gamma }{\lim }_{t\to t_1}c\left(t\right)$ to compute concentrations for times $t>t_1$. Compare the graphs of $c_s\left(t\right)$ in this case with the graphs obtained in Problems $4$ and $\text{5}\left(\text{a}\right)$. Can you draw any conclusions about the possible effective ness of source remediation? If so, what are they ?

Assume the following values for the parameters; $m_0=1,620kg,c_0=100mg/L,A_s=30m^2,v_d=20m/year,\lambda =0$. Use the solutions obtained in Problem $2$ to plot the graphs of $c_s\left(t\right)$ for each of the following cases:

(i) $\gamma =0.5$ for $0\le t\le t_f$, where $c_s\left(t_f\right)=0$ and (ii) $\gamma =2$ for $0\le t\le 100$ years.

Whereas an algebraic relationship between $c_s\left(t\right)$ and $m\left(t\right)$ is postulated in the form of a power law,

$\frac{c_s(t)}{c_0}={\left[\frac{m(t)}{m_0}\right]}^{\gamma }$, $\left(2\right)$