*5.1. Mathematical Formulation of Mixing Problem*

The mixing process described in the previous section is presented mathematically by the following simultaneous differential equations:

$$\begin{aligned} \dot{\mathbf{x}} &= \eta \mathbf{a} - \frac{\beta}{A} \mathbf{x} + \frac{\gamma}{B} \mathbf{y} \\ \dot{\mathbf{y}} &= \frac{\beta}{A} \mathbf{x} - \frac{\gamma + \delta}{B} \mathbf{y} \end{aligned} \tag{1}$$

Subject to the initial conditions *x*(0) = 0, *y*(0) = *kB*, where *δ* < *γ* < *β* and 0 < *k* < 1. Moreover, from the principle of flow, we get

$$
\beta = \kappa + \gamma = \gamma + \delta \tag{2}
$$

Solving the system (1), one can obtain the concentrations of the liquids in container-I and container-II as follows:

$$\mathbf{x}(t) = \exp\left(-\frac{k\_1}{2}t\right) \left\{ c\_1 \exp(k\_4 t) + c\_2 \exp(-k\_4 t) \right\} + \frac{k\_3}{k\_2} \tag{3}$$

$$y(t) = \frac{B}{\gamma} \left[ c\_1 \left( k\_4 - \frac{k\_1}{2} + \frac{\beta}{A} \right) \exp\left\{ \left( k\_4 - \frac{k\_1}{2} \right) t \right\} + c\_2 \left( -k\_4 - \frac{k\_1}{2} + \frac{\beta}{A} \right) \exp\left\{ \left( -k\_4 - \frac{k\_1}{2} \right) t \right\} \right] - \frac{\eta}{\gamma} uB + \frac{\beta Bk\_3}{\gamma A k\_2} \tag{4}$$

where

$$\begin{aligned} k\_1 &= \frac{\beta}{A} + \frac{\gamma + \delta}{B}, \\ k\_2 &= \frac{\beta \delta}{AB} \\ k\_3 &= \eta \frac{a(\gamma + \delta)}{B} \\ k\_4 &= \frac{\sqrt{k\_1^2 - 4k\_2}}{2} \\ \varepsilon\_1 &= -\frac{k\_3}{2k\_2} + \frac{1}{2k\_4} \left[ k\gamma + \eta a + \frac{k\_3}{k\_2} \left( \frac{\beta}{A} - \frac{k\_1}{2} \right) - \frac{\beta k\_3}{A k\_2} \right] \\\\ c\_2 &= -\frac{k\_3}{2k\_2} - \frac{1}{2k\_4} \left[ k\gamma + \eta a + \frac{k\_3}{k\_2} \left( \frac{\beta}{A} - \frac{k\_1}{2} \right) - \frac{\beta k\_3}{A k\_2} \right]. \end{aligned}$$

and

$$=-\frac{k\_3}{2k\_2} - \frac{1}{2k\_4} \left[ k\gamma + \eta\alpha + \frac{k\_3}{k\_2} \left( \frac{\beta}{A} - \frac{k\_1}{2} \right) - 1 \right]$$
