Chapter 1.1, Problem 11P

Mixing Problems. Many physical systems can be cast in the form of a mixing tank problem. Consider a tank containing a solution—a mixture of solute and solvent–such as salt dissolved in water. Assume that the solution at concentration $c_i(t)$ flows into a tank at a volume flow rate $r_i(t)$ and is simultaneously pumped out at a volume flow rate $r_o(t)$. If the solution in the tank is well mixed, then the concentration of the outflow is $Q(t)/V(t)$, where $Q(t)$ is the amount of solute at time $t$ and $V(t)$ is the volume of solution in the tank. The differential equation that models the changing amount of solute in the tank is based on the principle of conservation of mass,

$\underset{\text{rate}\text{of}\text{change}\text{of}Q(t)}{\underset{︸}{\frac{dQ}{dt}}}=\underset{\text{rate}\text{in}}{\underset{︸}{c_i(t)r_i(t)}}-\underset{\text{rate}\text{out}}{\underset{︸}{\{Q(t)/V(t)\}{r}_0(t)}}$, (i)

where $V(t)$ also satisfies a mass conservation equation,

$\frac{dV}{dt}=r_i(t)-r_0(t).$ (ii)

If the tank initially contains an amount of solute $Q_0$ in a volume of solution, $V_0$, then initial conditions for Eqs. (i) and (ii) are $Q(0)=Q_0$ and $V(0)=V_0$, respectively.

A tank contains $100$ gallons (gal) of water and $50$ ounces (oz) of salt. Water containing a salt concentration of $\frac{1}{4}\left(1+\frac{1}{2}\sin t\right)\text{oz/gal}$ flows into the tank at a rate of $2\text{gal/min}$, and the mixture flows out at same rate. Write initial value problem for the amount of salt in the tank at any time $t$.