My Math Forum water remaining in tank

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November 3rd, 2013, 07:55 PM   #11
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Re: water remaining in tank

Quote:
 anyway the approximation sequence should look something like this correct? since dv/dt=f(h),dv=f(h)*dt lwh-f(h)dt=lw(h2) lw(h2)-f(h2)dt=lw(h3) and so on
Yes. That is correct.

Quote:
 Just noticed that the software u use t_step=0.0001. In order to calculate the h at time t=100second i need to repeat the sequence 100/0.0001=1000k times. I couldnt done the approximation in excel as well.
Actually, it can be done in Excel, and I encourage you to do so because:

1) In my previous post, I said that for each approximation you would need to do an integration. That was a mistake. In my simulation program I was doing an integration over the slot height and checking it against the formula. But if you just use the formula:

$\frac{dV}{dt}=\left{ {-0.1w \cdot \frac{2}{3}\cdot \sqrt{2g}\left(h^{\frac{3}{2}}-(h-H)^{\frac{3}{2}}\right), \ h \gt H} \atop {-0.1w \cdot \frac{2}{3}\cdot \sqrt{2g}\cdot h^{\frac{3}{2}}, \ h \leq H}$

your procedure can easily be implemented in Excel. You will still need the IF statement, but that is easy to use. Sorry about my confusion (I will edit my previous post, mentioning the error).

2) I did use t_step = 0.0001 in my previous simulation, but that was only to check it against the formula to high accuracy. I ran a simulation with the same parameters as in the first simulation, but with H=1 and using the $\ \frac{dV}{dt} \$ formula given above. After 100s, these are my results:

t_step=0.1, volume = 6.248
t_step=0.01, volume = 6.277
t_step=0.001, volume = 6.28
t_step=0.0001, volume = 6.281

So even a t_step of 0.1s is fairly accurate. I made an Excel workbook with a t_step of 0.1 a got the same result. I'll bet Excel would have no problem with a t_step of 0.01 (or 0.05 or whatever) but I did not try it.

So I know your problem can be done in Excel with decent accuracy. Have fun, and feel free to reply if you have any more questions.

 November 3rd, 2013, 11:30 PM #12 Member   Joined: Sep 2013 Posts: 43 Thanks: 0 Re: water remaining in tank I have checked some mathematic book with list of integration table for this integral and couldnt recognize any pattern. So in mathematic, there is still equations that not possible to do the integral and need to find it using series? Since is an approximation method, the curve is not smooth compare to the actual formula( if it exist) and i wonder what the result using approximation to get set of value than apply least square method or newton-gauss method to smoothen the curve. I couldn't done this now due to lack of knowledge in this method and having difficulty in understand it.

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