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December 25th, 2013, 11:53 AM  #1 
Member Joined: Sep 2013 Posts: 43 Thanks: 0  Water remaining in inner cubic tank bounded by outer tank
A cubic tank initially consist of water at height h, width x and length x as well is bounded by another cubic tank of width y and length y, the outer tank is initially empty. given y^2=2x^2 The water is allowed to flow to the outer tank from the inner tank to the outer tank through a hole at the bottom with surface area A. I want to compute the water remaining in the inner tank after time t. since, pressure P=pgh and pressure=0.5pv^2 as well velocity of water flow through velocity, v=(2gh)^0.5 dv/dt=velocity*area=A(2gh)^0.5 the velocity of water should be depend on the difference in pressure between both tank so PinnerPouter=pg(h1h2) so dv/dt=(2g(h1h2))^0.5 now i got h1 and h2. Not sure how to proceed to the integration. Please give me some idea on this question.Thank you. 
December 26th, 2013, 08:32 PM  #2  
Senior Member Joined: Jul 2012 From: DFW Area Posts: 642 Thanks: 99 Math Focus: Electrical Engineering Applications  Re: Water remaining in inner cubic tank bounded by outer tan
Hi chuackl, First, be careful with your variables. I believe that you use v for both velocity and volume which may contribute to the variable A being left out of the equation for dV/dt in your second set of equations: Quote:
Quote:
which gives: where is the initial height of the water in the inner tank. Now we have: so substitute that into the equation. You should be able to separate the variables V, dV from dt and integrate. Use the initial condition that at t=0, V is to find the constant of integration (it is not 0). For V, I get: I have run a numerical simulation with and the results match the equation well. Note: This analysis assumes the tanks are sitting on top of one another to be consistent with the equations given. The height dimension of the area A is assumed to be small and the difference in pressure across the height is neglected.  
December 27th, 2013, 05:06 PM  #3 
Member Joined: Sep 2013 Posts: 43 Thanks: 0  Re: Water remaining in inner cubic tank bounded by outer tan
Yes! that does make sense. Irrespect on how h1 and h2 change, the sum of h1+h2 is always equal to h0. Thank you! 

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