A stirred tank has a capacity of \(V \mathrm{~m}^{3}\). Before time \(t=0\) the concentration of salt within the tank is \(c_{i}\) moles \(/ \mathrm{m}^{3}\). At time \(=0\), pure water is run in at ate of \(Q \mathrm{~m}^{3} / \mathrm{s}\) and brine is withdrawn from the bottom of the tank at the same rate. We assume negligible change in density of the fluid during dilution.

a. How long will it take for the concentration to be reduced to some final value, \(c_{f}\) ?

b. Consider incomplete mixing so that the average salt concentration within the tank, \(c\), is not the same as the outlet concentration \(c_{o}\). Assume that \(c\) and \(c_{o}\) may be related by the following simple function containing one parameter, \(b\), which is dependent on the stirrer speed and the geometry of the apparatus. How long will it take for \(c_{i}\) to be reduced to \(c_{f}\) in the tank?

\[\frac{c_{o}-c}{c_{i}-c}=e^{-b t}\]