Conservation of mass/flow through tanks

In summary, the problem at hand involves three tanks in series, filled with water that is pumped through the system at a rate of 5 gal/min. The first tank has a volume of 1000 gal, the second tank has a volume of 750 gal, and the third tank has a volume of 500 gal. The inflow to Tank 1 is a constant flow of water from an external source at the same rate, with dye added at a constant rate for one minute per hour over a long period of time. The outflow from Tank 3 goes into a lake at the same flow rate. The problem is to determine the concentration of the dye relative to its input concentration over a 10-hour period after the long-time
  • #1
takbq2
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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' = (QsA - Q12x1)*1/V1
x2' = (Q12x1 - Q23x2)*1/V2
x3' = (Q23x2 - Q3Lx3)*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.

 
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  • #2
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 ... :-)
 

1. What is the law of conservation of mass?

The law of conservation of mass states that the total mass of a closed system remains constant over time, regardless of any physical or chemical changes that may occur within the system.

2. How does the law of conservation of mass apply to flow through tanks?

In flow through tanks, the law of conservation of mass is used to ensure that the amount of substance entering the tank is equal to the amount leaving the tank, thus maintaining a constant mass within the tank.

3. What factors can affect the conservation of mass in a tank?

Factors such as leaks, evaporation, and chemical reactions can all affect the conservation of mass in a tank. It is important to monitor and account for these factors in order to accurately maintain mass balance in the tank.

4. How do you calculate mass balance in a flow through tank?

To calculate mass balance in a flow through tank, you must measure and record the mass of substance entering and exiting the tank, as well as any changes in mass within the tank. The net change in mass should be zero if the law of conservation of mass is being followed.

5. Why is the law of conservation of mass important in tank systems?

The law of conservation of mass is important in tank systems because it ensures that the system is operating properly and efficiently. Any discrepancies in mass balance can indicate a problem in the system, such as a leak or malfunction, which can be addressed to prevent potential hazards or losses. Additionally, maintaining mass balance is crucial in industries where precise measurements and quantities are necessary for production or experimentation.

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