Answer to Question #108471 in Physics for Sarah

Question

A large reservoir is drained through a pipe with an internal diameter of 15 cm.
If the opening of the pipe is 17 m below the surface of the water in the reservoir then at what speed is the water exiting the pipe?
(treat water as an incompressible non-viscous fluid)

Answer 1

Solution: For an incompressible non-viscous fluid, the Bernoulli's equation is written as

(1)$P+\rho g h+\rho\frac {V^2}{2}=const$ . This means that in any cross section of the current ideal fluid, the sum of the static and dynamic pressures remains constant. We'll write this equation in our case.

We will take the first section inside the tank not far from the pipe entry. Pascal's pressure is

(2)$P_1=\rho g h$ Here we took into account that the specific gravity of water $\rho=10^3kg/m^3$, the acceleration of gravity $g=9.8 m/s^2$ and the depth of the pipe $h=17m$. Since the cross section of the tank is significantly greater than the cross section of the pipe, we can assume that the liquid in it is practically at rest $V_1=0$ and therefore the dynamic pressure inside the tank is zero. The pressure of the external environment $P$ in (1) is the same, both on the surface of the tank and around the flowing stream, so it can be omitted from the equation.

We will take the second section of the fluid flow immediately after the fluid leaves the pipe. In this section, the liquid has a free surface not limited by the pipe so the pressure in it is equal to the outside $P$ . The desired flow rate $V_2$ leads to a dynamic pressure equal to $\rho\frac{V_2^2}{2}$ . We write the resulting Bernoulli's equation

(3) $\rho g h=\rho\frac{V_2^2}{2}$

From this equation for the flow velocity $V_2$ we obtain the Torricelli's law [1]

(4) $V_2=\sqrt{2gh}$ . A calculation using this formula gives $V_2=\sqrt{2\cdot9.8\cdot 17}=18.3 m/s$ . Thus, the cross section of the drain pipe does not affect the speed of the flowing water (if treat water as an incompressible non-viscous fluid).

Answer: The speed of the water exiting the pipe is $18.3 m/s$

[1] https://en.wikipedia.org/wiki/Torricelli%27s_law