Chapter 1.1, Problem 12P

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 pond initially contains $1,000,000$ gal of water and an unknown amount of an undesirable chemical. Water containing $0.01$ g of this chemical per gallon flows into the pond at a rate of $300$ gal/h. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond.

Write a differential equation for the amount of chemical in the pond at any time.

How much of the chemical will be in the pond after avery long time? Does this limiting amount depend on theamount that was present initially?