Conservation of mass/flow through tanks

[takbq2]

1. Suppose that three tanks in series, connected by pipes of the same size are filled with water that is moved through the system with pumps at the rate of 5 gal/min. The first tank is 1000 gal. The second tank is 750 gal. And the third tank is 500 gal. The contents of each tank is vigorously stirred. The in-feed to Tank 1 is a constant flow of water in the same sized pipe from an external source at the same flow rate and the outflow from Tank 3 flows into a lake at the same flow rate. [Dye is added to the inflow at a constant rate for exactly one minute per hour over a long period of time (weeks, months,whatever)]

What is the concentration of the dye RELATIVE to its input concentration as a function of time over a 10-hour period after the "long-time" has passed.

Do you have to make further assumptions to solve the problem?
How do you deal with the "relative" concentration?



2. Okay so obviously this is a conservation of mass problem, so flow in - flow out = 0



3. I have drawn an accurate picture, and I have set up what I believe to be the correct system of equations:

x₁' = (Q_sA - Q₁₂x₁)*1/V₁
x₂' = (Q₁₂x₁ - Q₂₃x₂)*1/V₂
x₃' = (Q₂₃x₂ - Q_3Lx₃)*1/V₃

Where:
x₁ = Concentration in Tank 1, so on for x₂ and x₃
A = Entrance rate of substance with units kg*t/v
Q₁ = rate at which the substance is transported by water flow (same throughout)

V₁ = Volume of tank 1, so on for V₂, V₃.

Whatever you guys know what this stuff means these are straight forward normally but this one is throwing me off bad. I don't know if my system is right or what to solve I've been working on it and the rest of a huge problem set for too long.

It is due tomorrow morning. Please help.

 

[rude man]

Well, input concentration is 100% for 1 minute and 0% for the next 59 minutes. So presumably you have to compute the concentration in each of the three tanks over the 1 hr period, second by second, after the steady state is reached.

The concentration in each tank is probably represesnted by a first-order diff. eq. with constant coefficients. The forcing function is a square wave with 1/60 duty cycle. Sounds like a Fourier series application. If I think of any more brilliant suggestions I'll let you know, but probably too late ... :-)