91Ó°ÊÓ

A cylindrical holding water tank has a \(3 \mathrm{m}\) ID and a height of \(3 \mathrm{m}\). There is one inlet of diameter \(10 \mathrm{cm},\) an exit of diameter \(8 \mathrm{cm},\) and a drain. The tank is initially empty when the inlet pump is turned on, producing an average inlet velocity of \(5 \mathrm{m} / \mathrm{s}\). When the level in the tank reaches \(0.7 \mathrm{m}\) the exit pump turns on, causing flow out of the exit; the exit average velocity is \(3 \mathrm{m} / \mathrm{s}\). When the water level reaches \(2 \mathrm{m}\) the drain is opened such that the level remains at \(2 \mathrm{m}\). Find (a) the time at which the exit pump is switched on, (b) the time at which the drain is opened, and (c) the flow rate into the drain \(\left(\mathrm{m}^{3} / \mathrm{min}\right)\)

Step by step solution

01

Calculate the time at which the exit pump is switched on

The time at which the exit pump is switched on can be calculated using the formula for volume, \(V=\pi r^{2} h\), where \(r\) is the radius of the cylinder and \(h\) is the height of the fluid. The flow rate into the tank when only the inlet pump is on is given by the formula \(Q=vA\), where \(v\) is the velocity and \(A\) is the area through which the fluid flows. Therefore, the time it takes for the tank to fill to a height of 0.7 m can be calculated by dividing the volume of the tank by the flow rate into the tank. Hence, \(t_{\text {pump start }}=\frac{V}{Q}=\frac{\pi\left(\frac{1.5}{2}\right)^{2} \cdot 0.7}{5 \cdot \pi\left(\frac{0.10}{2}\right)^{2}}\)

02

Calculate the time at which the drain is opened

To calculate the time at which the drain is opened, consider the time when both the inlet pump and the exit pump are on. The total flow rate into the tank is the difference between the flow rate from the inlet pump and the flow rate from the exit pump. The time it takes for the tank to fill from a height of 0.7 m to a height of 2 m can be calculated by dividing the change in volume of the tank during this period by the total flow rate into the tank. Hence, \(t_{\text {drain start }}=\frac{V_{2}-V_{1}}{Q_{\text {in}}-Q_{\text {out}}}=\frac{\pi\left(\frac{3}{2}\right)^{2} \cdot 2-\pi\left(\frac{3}{2}\right)^{2} \cdot 0.7}{5 \cdot \pi\left(\frac{0.10}{2}\right)^{2}-3 \cdot \pi\left(\frac{0.08}{2}\right)^{2}}\) where \(V_{2}\) and \(V_{1}\) are the final and initial volumes of the fluid in the tank, respectively.

03

Calculate the flow rate into the drain

The flow rate into the drain can be calculated by considering the equilibrium condition when the water level remains at 2 m, i.e. when the total flow rate into the tank equals the total flow rate out from the tank. The total flow rates are the summation of the flow rates of the inlet, the exit, and the drain. Therefore, the flow rate into the drain, \(Q_\text{drain}\), can be calculated as the difference between the total flow rate into the tank and the total flow rate out from the tank when the water level remains at 2 m. Hence, \(Q_{\text {drain }}=Q_{\text {in}}-Q_{\text {out}}=5 \cdot \pi\left(\frac{0.10}{2}\right)^{2}-3 \cdot \pi\left(\frac{0.08}{2}\right)^{2}\)

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