Chapter 10, Problem 10.33P

10.33. A 2000-liter tank initially contains 400 liters of pure water. Beginning at t = 0, an aqueous solution containing 1.00 g/L of potassium chloride flows into the tank at a rate of 8.00 L/s and an outlet stream simultaneously starts flowing at a rate of 4.00 L/s. The contents of the tank are perfectly mixed, and the densities of the feed stream and of the tank solution, $\rho \left(g/L\right)$, may be considered equal and constant. Let V(t)(L) denote the volume of the tank contents and C(t)(g/L) the concentration of potassium chloride in the tank contents and outlet stream.

1. Write a balance on total mass of the tank contents, convert it to an equation for dV/dt, and provide an initial condition. Then write a potassium chloride balance, show that it reduces to

   $\frac{dC}{dt}=\frac{8-8C}{V}$

and provide an initial condition. (Hint: You will need to use the mass balance expression in your derivation.)

Without solving either equation, sketch the plots you expect to obtain for V versus t and C versus t. If the plot of C versus t has an asymptotic limit as t ? 8, determine what it is and explain why it makes sense.

Solve the mass balance to obtain an expression for V(t). Then substitute for V in the potassium chloride balance and solve for C(t) up to the point when the tank overflows. Calculate the KCl concentration in the tank at that point.