Rate of flow of water fluid mechanics

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 $$\frac{dM}{dt} = pk -pa\sqrt{2gh}$$

• 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. Dec 18, 2015 at 10:45
• 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. Dec 18, 2015 at 10:49

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.
• @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? Dec 16, 2015 at 17:04
• @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 Dec 16, 2015 at 19:00