Question #289621

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.

1
Expert's answer
2022-01-25T04:43:52-0500

Given A=10m2,h1=3.5m,h2=3.5m0.5m=3m,A=10 m^2, h_1=3.5m, h_2=3.5m-0.5m=3m,

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

After time dtdt


Qdt=AdhQdt=-Adh

Cd(πr2)2ghdt=AdhC_d(\pi r^2)\sqrt{2gh}dt=-Adh

dt=AπCdr22gh1/2dhdt=-\dfrac{A}{\pi C_dr^2\sqrt{2g}}h^{-1/2}dh

Integrate


0Tdt=h1h2AπCdr22gh1/2dh\displaystyle\int_{0}^{T}dt=-\displaystyle\int_{h_1}^{h_2}\dfrac{A}{\pi C_dr^2\sqrt{2g}}h^{-1/2}dh

T=AπCdr22g[2h]h2h1T=-\dfrac{A}{\pi C_dr^2\sqrt{2g}}[2\sqrt{h}]\begin{matrix} h_2 \\ h_1 \end{matrix}

T=2A(h1h2)πCdr22gT=\dfrac{2A(\sqrt{h_1}-\sqrt{h_2})}{\pi C_dr^2\sqrt{2g}}

T=2(10m2)(3.5m3m)π(0.6)(0.025m)22(9.81m/s2)T=\dfrac{2(10m^2)(\sqrt{3.5m}-\sqrt{3m})}{\pi (0.6)(0.025m)^2\sqrt{2(9.81m/s^2)}}

=532s=532 s

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


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