Question 2.9: Tank Drainage Consider the efflux of a liquid through a smoo......
Tank Drainage
Consider the efflux of a liquid through a smooth orifice of area \rm A_0 at the bottom of covered tank (\rm A_T). The depth of the liquid is y. The pressure in the tank exerted on the liquid is \rm p_T while the pressure outside of the orifice is \rm p_a. Develop an expression for \mathrm{v_0(y)}.
| Concept | Assumptions | Sketch |
| • Bernoulli’s equation applied to points ➀ and ➁ | • Steady frictionless flow |  |
| • Mass conservation | • Streamline represents all fluid element trajectories | |
| • Averaged velocities |
At descending point ➀ and exit point ➁, most of the information is given. Thus, Eq. (2.28) now reads:
{\frac{\mathrm{v}_{1}^{2}}{2}}+{\frac{\mathrm{p}_{1}}{\mathrm{\rho }}}+{\mathrm{g}}\mathrm{z_1}={\frac{\mathrm{v}_{2}^{2}}{2}}+{\frac{\mathrm{p}_{2}}{\mathrm{\rho }}}+{{{\mathrm{gz_2}}}} (2.28)
{\frac{\mathrm{v}_{1}^{2}}{2}}+{\frac{\mathrm{p}_{T}}{\mathrm{\rho }}}+{\mathrm{g}}\mathrm{y}={\frac{\mathrm{v}_{0}^{2}}{2}}+{\frac{\mathrm{p}_{a}}{\mathrm{\rho }}}+{{{0}}}The velocities are related via continuity (see Eq. (2.7)):
\Sigma{\rm Q}_{\mathrm{in}}=\Sigma{\rm Q}_{\mathrm{out}} (2.7)
\mathrm{v_1\,A_T=v_0\,A_0}so that with \mathrm{v_1=v_0\,A_0/A_T},
\mathrm{v_0=\left[\frac{2/ρ(p_T-p_a)+2gy}{1-\left\lgroup A_0/A_T\right\rgroup^2 } \right] ^{1/2}}Comment: Toricelli’s law, \mathrm{v=\sqrt{2gh} } can be directly recovered when Δp ≈ 0 and \mathrm{A_T >> A_O} . Clearly, the liquid level y and pressure drop \mathrm{p_T − p_A} are the key driving forces. The solution breaks down when \mathrm{A_O → A_T}.