If we have a cylindrical tank of area $A$ which has a small outlet of area $a$ at the bottom of the surface . And the container starts filling with a constant rate $k [m^3/sec]$ . Now the max. level of water will be $$h = k^2/(2a^2g)$$ as by using Bernoulli's equation . But how will find the time after which level of water become $h$ ? I got it till $$\frac{dM}{dt} = pk -pa\sqrt{2gh}$$
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$\begingroup$ So you are close, as you can see at steady-state $d_tM=0$ you get your maximum height back from the rate equation. Now to get the time; like i said in my answer, $M=\rho V\left(t\right)$ where the volume $V$ of liquid in the tank is related to the tank area $A$ and liquid height $h\left(t\right)$. You then have a rate equation for the liquid height; solve it as a function of time, plug in the maximum height and solve for the time to reach it. $\endgroup$– nluigiDec 18, 2015 at 10:45
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$\begingroup$ I just realized something and am going to spoil the answer for you; so the maximum height you calculate is the height at steady-state $d_tM=0$. But it mathematically takes an infinite amount of time to reach a steady-state, i.e. $\Delta t\rightarrow\infty$. So unfortunately the mathematical answer to your question is an infinite amount of time! You could however solve for the case $0.99h_{max}$ which would give you the finite time until it has reached $99\%$ of the steady-state height. $\endgroup$– nluigiDec 18, 2015 at 10:49
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Relate the rate of change of fluid mass in the tank to the amount flowing in and the amount flowing out: $$\frac{dM}{dt} = \phi_{in} - \phi_{out}\left(t\right)$$ where $\phi_{in}=\rho k$ and $\phi_{out}\left(t\right)=\rho a v_{out}$ needs the outflow velocity $v_{out}$ to be determined from the Bernoulli equation in terms of the height $h\left(t\right)$ of the fluid. The mass in the tank: $$M\left(t\right) = \rho V\left(t\right)$$ where the volume $V\left(t\right)$ of the liquid in the tank is related to the area of the tank and the height $h\left(t\right)$ of the water. From there you should be able to get a rate equation for the height of the liquid in the tank.
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$\begingroup$ How the mass in the tank is related to area and height $\endgroup$ Dec 16, 2015 at 17:02
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$\begingroup$ @user101522: $M=\rho V$, where $V$ is the volume of the liquid in the tank, can you guess how the volume is related to the tank area and the height of the fluid? $\endgroup$– nluigiDec 16, 2015 at 17:04
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$\begingroup$ dm/pK= dt then integrate it so I am getting hA/K = t . Is it correct $\endgroup$ Dec 16, 2015 at 17:16
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$\begingroup$ @user101522 in your first equation you are not accounting for the fluid flow out of the tank, $\rho a v_{out}$, just an inflow. If there is only inflow the level in the tank will increase indefinitely right? can you think of a way to calculate $v_{out}$? Hint: the name of the equation is in your question $\endgroup$– nluigiDec 16, 2015 at 19:00
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