📂Fluid Mechanics

Proof of Torricelli's theorem

Theorem ¹

Consider a tank containing a fluid in steady flow (steady-state flow) (../2773), where the inflow equals the outflow so that the fluid surface height $h$ remains constant. Assume the fluid (../1768) is inviscid (../2783) and incompressible (../2785). For a tank with constant cross-sectional area $A_{1}$ and an outlet at the bottom consisting of a pipe with cross-sectional area $A_{2}$ that is very small compared to the tank, the volumetric flow rate (../2801) $Q$ and the exit velocity $u_{2}$ are expressed in terms of the density (../2800) $\rho$ and the gravitational acceleration $g$ as follows. $$ \begin{align*} u_{2} =& \sqrt{2 g h} \\ Q =& A_{2} \sqrt{2 g h} \end{align*} $$ Together these are called Torricelli’s theorem.

Explanation

The velocity formula in Torricelli’s theorem is the same as the speed $v = \sqrt{2 g h}$ of a body that has fallen through a distance $h$ in classical mechanics (../13).

Proof

Bernoulli equation: The energy per unit volume is given by the following and is constant. $$ {\frac{ \rho u^{2} }{ 2 }} + \rho g z + p $$

The energy at the tank free surface $1$ and at the outlet $2$ is the same, and by Bernoulli’s equation we have: $$ {\frac{ \rho u_{1}^{2} }{ 2 }} + \rho g z_{1} + p_{1} = {\frac{ \rho u_{2}^{2} }{ 2 }} + \rho g z_{2} + p_{2} $$ Here $p_{1}$ and $p_{2}$ are equal to atmospheric pressure and therefore cancel. $$ {\frac{ \rho u_{1}^{2} }{ 2 }} + \rho g z_{1} = {\frac{ \rho u_{2}^{2} }{ 2 }} + \rho g z_{2} $$ Dividing both sides by $\rho$ and multiplying by $2$ yields: $$ u_{1}^{2} + 2 g z_{1} = u_{2}^{2} + 2 g z_{2} $$

Continuity equation $$ Q = A_{1} u_{1} = A_{2} u_{2} $$

By the continuity equation $u_{1} = (A_{2} / A_{1}) u_{2}$ holds; since the outlet cross-sectional area is very small compared to the tank’s cross-sectional area, $A_{2} / A_{1} \approx 0$, and we can set $u_{1}^{2} \approx 0$. Rearranging this for $u_{2}$ gives: $$ u_{2} = \sqrt{ 2 g \left( z_{1} - z_{2} \right)} = \sqrt{ 2 g h } $$ And once again, from the continuity equation we see that $Q = A_{2} u_{2}$.

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