Water Tank Level Problem
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Two tanks A and B are connected by a pipe at C. Once the connection at C is opened, water will flow from tank A towards tank B and the water will be level (at figure 2) at a height “h” which is ½ the distance AC. If the cross sectional area of pipe at C is 10square cm. the discharge at C will be equal to
Q = (10sq.cm) x (Velocity at the point.)
Velocity at point C —–>Vc = √2g(AC)
And the time it takes for the water to be level will be the volume of water to be transferred divided by the rate Q.
But Q here is variable, being maximum when the difference of water level between the two tanks is AC (while water just starts to flow from tank A) and zero when the water is at the same level in both tanks.
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Fig 1 (water still at tank A)
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Fig 2 (water when finally leveled)
The actual set-up, however, includes a return pipe which takes back the water to tank A which is powered by a pump which gives a nozzle velocity of 2m/sec to the water coming out of P.
How long would it take for the water to be level, given that the opening of the connection at C and the start of operation of the pump in the return pipe were done simultaneously?
…….P
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Particulars:
Bases of tank A and tank B = 2sq.m
Cross sectional area of connection at C and return pipe at P = 10sq.cm
Height of water level AC = 4m. (assume this to be from the center of the flow at C, to the upper water surface)
Nozzle velocity of water at P = 2m/sec
Update:
Zo Maar,
thanks for answering. That’s quite some analysis you have there. You are on track.
Al P,
I’m glad to see your diagram. I have some more diagrams in my answers which I go back to and click to COPY. If you know how to revise to suit the data it is there. I have told many others to just feel free to COPY and PASTE from my work. You and Zo Maar are on track.
Update 2:
Zo Maar,
Your solution is even much more than needed. You could have just answered that it would take “eternity” to have the water leveled.(lol)
Upon occular inspection, (that’s why I made the diagram to facilitate this) before the surface of the water at A reaches the equilibrium level (h = 1/2AC) there will be that specific height at which the velocity of the water at C equals that of the return pipe. Hence, at that level, water in = water out of tank B. So until the pump breaks, or the water got all evaporated or eternity comes, whichever comes first, those will be the only chances for the water level to be equal. hehe. But nice integration/differentiation, a true mark of a thorough mathematician. Needless to say, you more than deserve BA.
Update 3:
._▄♦★★★★★♦▄_.
▀▄.Great Stuff.▄▀
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.░░░░▀▄▀░░░░ …..Al P
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▀▄BestAnswer▄▀
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.░░░░▀▄▀░░░░ …..Zo Maar
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And thanks to all the starrers of this question. Thanks for the attention you have all shown.
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✅ Answers
? Favorite Answer
If I understand the question correctly, the water levels in tanks will never be the same. In the equlibrium, the nozzle velocity is equal to the velocity through the connection at C. This gives the equlibrium level difference h_{eq},
h_{eq} = v^2 / (2 g) = 20 cm.
Formally speaking, this equlibrium level difference will neither be achieved. The flux out of tank A is
Q_A = σ [ v – √ (2 g h) ].
Here, h is the difference between water levels in tanks A and B, v = 2m/sec is the nozzle velocity of water at P, σ=10 cm^2 is the cross-seciton of the nozzle of the connection at C. The flux into tank B is Q_B = – Q_A. Then
Q_A – Q_B = S dh/dt = 2 σ [ v – √ (2 g h) ],
where S=2 m^2 is the cross-section of tanks. We write this equation in the form
– dx / (√x – 1) = dt / τ.
Here, x = h / h_{eq} is the water level difference in tanks normalized on its equlibrium value and
τ = v S / (4 g σ ) = 10 sec
is the characteristic time of the process. Integration gives
f (x_0) – f(x) = t / τ,
f(x) = 2 [ √x + ln (√x – 1) ],
and x_0 = 4 m / h_{eq} = 20 is the initial values of x. The value of x can be arbitrarily close to 1, but it cannot be equal to 1, since f(1) is infinite. For example, after 20 τ, which is about five and a half hours, the value of x will be x=1.01.
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Questor, is there something missing in my solution?
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Did you ever see a fish tank
overflow pump at work?
Zo Maar is correct.
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Thank you, Questor.
Nice artwork!