Bernoullis Equation for Unsteady Flows

Example 5.13. Ifithe tank in Fig. 5.5 is cylindrical with a diameter of 10 m, and. if the outlet hole is 1 m in diameter, how long does it take the fluid level to drop from 30 m above the tank outlet to 1 m above the tank outlet  

Here we assume that the dVIdi) is small we check that assumption later. Then the instantaneous flow rate is assumed to be given by Bernoulli s equation, which here takes the form of Torricelli s equation  

Where Fj is the rate at which the free surface of the tank is moving downward, which is equal tp —dh/dt. So 

The maximum velocity in this tank is at the outlet, and all other velocities are proportional to it therefore, the maximum value of dVIdt) must occur at the outlet. Differentiating Torricelli s equation with respect to time yields 

Thus in this example the maximum value of dVIdt) is the acceleration of gravity, and the unsteady-flow aspect of the problem can be neglected safely.  

---