Two cylindrical tanks, connecting pipe and outlet pipe - Differential Equation

In summary: In the problem, q is changed to y when it should be the other way around.The problem says that q is changed to y, but the equation for y should be q is changed to 2y.
  • #1
VinnyCee
489
0

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!
 
Last edited:
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  • #2
background: solving systems of 1st order ODE. for the homogeneous NxN linear system:
[tex]\dot{\textbf{x}}(t)=\mathbb{A} \textbf{x}(t)[/tex]
where [tex]\dot{\textbf{x}}(t), \textbf{x}(t)[/tex] are N-vectors while [tex]\mathbb{A}[/tex] is a NxN matrix
(with constant coefficients assumed) then the general solution takes the form:
[tex]\textbf{x}(t) = c_1\textbf{V}_1\,e^{\lambda_1 t} + c_2\textbf{V}_2\,e^{\lambda_2 t}+\cdots+c_N\textbf{V}_N\,e^{\lambda_N t}[/tex]
where [tex]\lambda_i[/tex] are the eigenvalues of [tex]\mathbb{A}[/tex] and [tex]c_i[/tex] are arbitrary integration constants,
[tex]\textbf{V}_i[/tex] are linearly independent eigenvectors of [tex]\mathbb{A}[/tex].

For your system, it would look something like this:
[tex]\begin{pmatrix}\dot{y} \\ \dot{x}\end{pmatrix}=
\begin{pmatrix}-k/(AL) & k/(2AL)\\ k/(AL) & -k/(AL)\end{pmatrix}\;
\begin{pmatrix}y \\ x\end{pmatrix}[/tex]
 
  • #3
can you explain what q, k, x, y and L are? I kinda can't follow what you were doing up there.
 
  • #4
yes, i want to know too
 
  • #5
The problem doesn't say what those values are. I know some are constant and some are variables.

Here is the picture that came with the problem

http://img505.imageshack.us/img505/8951/tankproblemdg2.jpg [Broken]
 
Last edited by a moderator:
  • #6
VinnyCee, why 2q changed to 2y?
 

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1. What is the purpose of a connecting pipe in a system of two cylindrical tanks?

The connecting pipe serves as a pathway for the fluid to flow between the two cylindrical tanks. It allows for the transfer of fluid from one tank to the other, creating a continuous system.

2. How does the outlet pipe affect the fluid flow in the system of two cylindrical tanks?

The outlet pipe acts as a control for the fluid flow in the system. By adjusting the size of the outlet pipe, the rate of fluid flow can be regulated. This can also affect the pressure and volume of the fluid in the tanks.

3. What is the role of differential equations in analyzing this system?

Differential equations are used to model the behavior of the fluid in the system. By considering factors such as the rate of change of fluid volume and the flow rate through the pipes, differential equations can help predict the behavior of the system over time.

4. How do the dimensions of the tanks and pipes impact the differential equations used in this system?

The dimensions of the tanks and pipes will affect the coefficients in the differential equations. The shape and size of the tanks will determine the initial conditions and boundary conditions, while the diameter and length of the pipes will impact the flow rate and pressure of the fluid.

5. What are some real-world applications of differential equations in systems with cylindrical tanks and connecting pipes?

Differential equations are commonly used in industries such as chemical engineering, where tanks and pipes are used for fluid storage and transportation. They can also be applied in hydraulic systems, water treatment plants, and oil pipelines to analyze and optimize the behavior of the fluid within the system.

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