Period

Equation

Period I.

(Constant extraction rate)

$e=y_{r}q\left[1-\exp \left(-Z\right)\right]$ (a)

$Z=\frac{k_f{a}_{0\rho }}{\left(1-\epsilon \right)\dot{q}{\rho }_s}$ (b)

Period II.

(Decay of extraction)

$e=y_r\left[q-q_m\exp \left(Z_w-Z\right)\right]$ (c)

$q_m=x_0-\frac{x_k}{y_{r}Z}$ (d)

$\frac{Z_W}{Z}=\frac{y_r}{W{x}_0}\ln \frac{x_0\exp \left[\left[W\left(q-q_m\right)\right]\right]-x_k}{x_0-x_k}$ (e)

Period III.

(Extraction controlled by diffusion)

$e=x_0-\frac{y_r}{W}\ln \left\{1+\frac{\left[\exp \left(\frac{W{x}_0}{y_r}\right)-1\right]\exp \left[W\left(q-q_m\right)\right]x_k}{x_0}\right\}$ (f)

$W=\frac{k_s{a}_0}{\dot{q}\left(1-\epsilon \right)}$ (g)