Solving a DE for Water Leaking from a Tank

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In summary: If you take the equation V = Aw * h and differentiate with respect to 't', what do you get?If you take the equation V = Aw * h and differentiate with respect to 't', you get dh/dt = 1/Aw * dV/dt.
  • #1
ehabmozart
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Homework Statement



Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to

c.Ah.(2gh)^.5

, where c (0 < c < 1) is a empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown. The radius of the hole is 2 in., and g = 32ft/s2

Homework Equations





The Attempt at a Solution



I am always confused in setting up a DE. Over here, the answer says that V= Aw(area of water). h . The next step written was that dh/dt = 1/Aw * dV/dt ... How does this make sense.. I am confused here. I understand that the "c.Ah.(2gh)^.5" given back in the question is dv/dt . But when you multiply it with 1/Aw, how do you get dh/dt ? Thanks for any help!
 
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  • #2
ehabmozart said:

Homework Statement



Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to

c.Ah.(2gh)^.5

, where c (0 < c < 1) is a empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown. The radius of the hole is 2 in., and g = 32ft/s2

Homework Equations





The Attempt at a Solution



I am always confused in setting up a DE. Over here, the answer says that V= Aw(area of water). h . The next step written was that dh/dt = 1/Aw * dV/dt ... How does this make sense.. I am confused here. I understand that the "c.Ah.(2gh)^.5" given back in the question is dv/dt . But when you multiply it with 1/Aw, how do you get dh/dt ? Thanks for any help!

The area of the hole is Ah? How can the hole's area depend on the height of water in the tank?
 
  • #3
Ray Vickson said:
The area of the hole is Ah? How can the hole's area depend on the height of water in the tank?
I think the OP means this as Ah, not A * h. Also, I think Aw means Aw.

ehabmozart,
You can make what you write clearer by using the features available on this site. For example, to write exponents and subscripts, click the Go Advanced button below the input area, which causes the advanced menu to open across the top. One button is X2, which you can use to write exponents. Another button is X2, which you can use to write subscripts.
 
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  • #4
ehabmozart said:
I am always confused in setting up a DE. Over here, the answer says that V= Aw(area of water). h . The next step written was that dh/dt = 1/Aw * dV/dt ... How does this make sense.. I am confused here. I understand that the "c.Ah.(2gh)^.5" given back in the question is dv/dt . But when you multiply it with 1/Aw, how do you get dh/dt ? Thanks for any help!

If you take the equation V = Aw * h and differentiate with respect to 't', what do you get?
 

FAQ: Solving a DE for Water Leaking from a Tank

1. How does a Differential Equation (DE) help in solving water leakage from a tank?

A DE is a mathematical tool that helps in modeling and solving various physical systems that involve rates of change. In the case of water leaking from a tank, a DE can be used to model the rate of change of water level in the tank and find a solution to stop the leakage.

2. What are the steps involved in solving a DE for water leaking from a tank?

The first step is to define the variables and parameters involved, such as the initial water level, leakage rate, and tank dimensions. Then, the DE can be formulated based on the physical laws governing the system. Next, the DE can be solved using various techniques such as separation of variables, substitution, or numerical methods.

3. What are the boundary conditions for solving a DE for water leaking from a tank?

The boundary conditions are the conditions that must be satisfied at the boundaries of the system, i.e. the initial and final states. In the case of water leaking from a tank, the initial condition would be the initial water level, and the final condition would be the desired water level, i.e. no leakage.

4. How can the solution of a DE for water leaking from a tank be verified?

The solution can be verified by plugging it back into the original DE and checking if it satisfies the equation. Additionally, the solution can be compared to real-world data or simulations to ensure its accuracy.

5. Are there any limitations to using a DE for solving water leakage from a tank?

Yes, there can be limitations depending on the complexity of the system. For example, if there are multiple leakage points or the tank has irregular shape, the DE may not accurately represent the system. In such cases, more advanced modeling techniques may be required.

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