Answer to Question #289621 in Differential Equations for Emjay

Question

A tank with a horizontal sectional area constant at 10 square meter and 4 m high contains water to a depth of 3.5 m. the tank has a circular orifice 5 cm in diameter and located at its side 0.5 m above the bottom. if the coefficient of discharge of the orifice is 0.60, find the duration of flow though the orifice.

Answer 1

Given $A=10 m^2, h_1=3.5m, h_2=3.5m-0.5m=3m,$

$r=5cm/2=0.025cm,C_d=0.60$

After time $dt$

$$Qdt=-Adh$$

$$C_d(\pi r^2)\sqrt{2gh}dt=-Adh$$

$$dt=-\dfrac{A}{\pi C_dr^2\sqrt{2g}}h^{-1/2}dh$$

Integrate

$$\displaystyle\int_{0}^{T}dt=-\displaystyle\int_{h_1}^{h_2}\dfrac{A}{\pi C_dr^2\sqrt{2g}}h^{-1/2}dh$$

$$T=-\dfrac{A}{\pi C_dr^2\sqrt{2g}}[2\sqrt{h}]\begin{matrix} h_2 \\ h_1 \end{matrix}$$

$$T=\dfrac{2A(\sqrt{h_1}-\sqrt{h_2})}{\pi C_dr^2\sqrt{2g}}$$

$$T=\dfrac{2(10m^2)(\sqrt{3.5m}-\sqrt{3m})}{\pi (0.6)(0.025m)^2\sqrt{2(9.81m/s^2)}}$$

$$=532 s$$

The duration of flow though the orifice is $532s.$