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VinnyCee
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Homework Statement
Two vertical cylindrical tanks, each 10 meters high, are installed side-by-side. Their bottoms are at the same level. The tanks are connected at their bottoms by a horizontal pipe 2 meters long which has an internal diameter of 0.03 meters. The first tank is full of oil and the second tank is empty. Tank 1 has a cross-sectional area twice that of tank 2. Tank one has an outlet pipe (to the environment) at it's bottom as well. It is of the same dimension as the other pipe. Both of the valves for the horizontal pipes are opened simultaneously. What is the maximum oil level reached for tank 2 before the oil drains out of both tanks? Assume laminar flow in the pipes and neglect kinetic losses and pipe entrances and exits.
Homework Equations
The volume balance equations are as follows.
TANK 1: [tex]2A\,\frac{dy}{dt}\,=\,-q_1\,-\,q_2[/tex]
TANK 2: [tex]A\,\frac{dx}{dt}\,=\,q_2[/tex]
[tex]q_1\,=\,\frac{k\,y}{L}[/tex]
[tex]q_2\,=\,\frac{k\,\left(y\,-\,x\right)}{L}[/tex]
The Attempt at a Solution
[tex]\frac{dy}{dt}\,=\,\frac{1}{2A}\,\left(-q_1\,-\,q_2\right)[/tex]
[tex]\frac{dx}{dt}\,=\,\frac{1}{A}\,\left(q_2\right)[/tex]
[tex]\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\,=\,\frac{\frac{1}{2A}\,\left(-q_1\,-\,q_2\right)}{\frac{1}{A}\,\left(q_2\right)}[/tex]
[tex]\frac{dy}{dx}\,=\,\frac{-\left(q_1\,+\,q_2\right)}{2y}[/tex]
Substituting in for [itex]q_1[/itex] and [itex]q_2[/itex].
[tex]\frac{dy}{dx}\,=\,\frac{k\,\left(x\,-\,2y\right)}{4y}[/tex]
[tex]4\,\frac{dy}{dx}\,=\,k\,\frac{x}{y}\,-\,2\,k[/tex]
I don't know how to proceed, please help!
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