Assignment 0: Conical tank emptying time

Let’s calculate how long it takes to empty a conical tank

The simplification of the continuity equation leads, for a generic \(h\), to

\[ u_1 D_h^2 = u_1 \left(d + 2h \tan \theta \right)^2 = u_2 d^2 \tag{1} \]

\[ \Rightarrow \; u_2 = u_1\left(1 + 2\frac{h}{d} \tan \theta \right)^2 \tag{2} \]

On the other hand, Bernoulli’s equation, considering quasi-steady flow and without viscosity,

\[ u_1^2 + 2gh = u_2^2 = u_1^2 \left(1 + 2\frac{h}{d} \tan \theta \right)^4 \tag{3} \]

\[ \Rightarrow \; u_1^2 \left[ \left(1 + 2\frac{h}{d} \tan \theta \right)^4 -1 \right] = 2gh \tag{4} \]

\[ \Rightarrow \; u_1 = \left[ \frac{2gh}{\left(1 + 2\frac{h}{d} \tan \theta \right)^4 -1} \right]^\frac{1}{2} = -\frac{\textrm{d} h}{\textrm{d} t} \tag{5} \]

\[ \Rightarrow \; \boxed{dt = - \left[ \frac{\left(1 + 2\frac{h}{d} \tan \theta \right)^4 -1}{2gh}\right]^\frac{1} {2} \textrm{d} h} \label{eq:int} \tag{6} \]

This can be written dimensionless with a time characteristic \(\tau = \sqrt{\frac{d}{2g}}\), a change to dimensionless variable \(\xi = \frac{h}{d}\), and a parameter \(A = 2\tan \theta\), becoming

\[ \frac{\textrm{d} t}{\tau} = - \left[ \frac{\left(1 + A \xi \right)^4 -1}{\xi}\right]^\frac{1 }{2} \textrm{d} \xi \tag{7} \]

Use sympy to analytically compute this time, making an estimation with \(d \ll h\) so that it can be supposed that \(\xi \gg 1\).