Consider the mixing process shown in the figure A mixing cha

Consider the mixing process shown in the figure. A mixing chamber initially contains 5 liters of a clear liquid. Clear liquid flows into the chamber at a rate of 10 liters per minute. A dye solution having a concentration of 0.75 kilograms per liter is injected into the mixing chamber at a constant rate of r liters per minute. When the mixing process is started, the well-stirred mixture is pumped from the chamber at a rate of 10 + r liters per minute. Develop a mathematical model for the mixing process. Let Q represent the amount of dye in kilograms in the mixture. The objective is to obtain a dye concentration in the outflow mixture of 0.1 kilograms per liter. What injection rate r is required to achieve this equilibrium solution? Would this equilibrium value of r be different if the fluid in the chamber at time t = 0 contained some dye? Assume the mixing chamber contains 5 liters of clear liquid at time t = 0. How many minutes will it take for the outflow concentration to rise to within 3% of the desired concentration of 0.1 kilograms per liter?

Solution

r litres has 0.75 * r kgs of dye

so we have

0.75r kgs dye inserted per minute

so dQ/dt = 0.75r

2 ) the amount of dye 0.75r is in 10+r litre solution

10 litres is clear solution and r litres dye solution

we want 0.75r/(10+r ) =0.1

i.e. 10+r = 7.5r

=>6.5r = 10

r = 10/6.5