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Chapter 4: Problem 40

Viscous liquid from a circular tank, \(D=300 \mathrm{mm}\) in diameter, drains through a long circular tube of radius \(R=50 \mathrm{mm}\) The velocity profile at the tube discharge is \\[ u=u_{\max }\left[1-\left(\frac{r}{R}\right)^{2}\right] \\] Show that the average speed of flow in the drain tube is \(\bar{V}=\frac{1}{2} u_{\max } .\) Evaluate the rate of change of liquid level in the tank at the instant when \(u_{\max }=0.155 \mathrm{m} / \mathrm{s}\)

Short Answer

Expert verified
The average speed of flow in the drain tube is \(\bar{V}=\frac{1}{2} u_{\max }\). The rate of change of the liquid level in the tank when \(u_{\max }=0.155 \mathrm{m} / \mathrm{s}\) can be calculated using the derived formula from step 2.

Step by step solution

01

Derive Average Velocity

Begin by integrating the velocity profile \(u=u_{\max }\left[1-\left(\frac{r}{R}\right)^{2}\right]\) over the area of the tube in order to obtain the total flow rate, \(Q\). As we know, the flow rate \(Q\) is equal to \(Q = \int u \, dA = \int_0^R u_{\max }\left[1-\left(\frac{r}{R}\right)^{2}\right] 2 \pi r \, dr = \frac{1}{2} \pi R^2 u_{\max}\). Here, \(dA=2\pi r \, dr\) is an annulus at radius \(r\) and width \(dr\).\n\nThe average speed of the liquid is the total flow rate divided by the cross-sectional area of the tube, \(\bar{V} = \frac{Q}{A}\). Setting the cross-sectional area \(A=\pi R^2\), the average speed becomes \(\bar{V}=\frac{Q}{A}=\frac{1}{2} u_{\max}\).
02

Evaluate Rate of Change of Liquid Level

The rate of change of the liquid level in the tank is the total flow rate divided by the cross-sectional area of the tank. From step 1, we know that \(Q = \frac{1}{2} \pi R^2 u_{\max}\). The cross-sectional area of the tank is \(A = \frac{\pi}{4} D^2\). Therefore, we can determine the rate of change as follows: \(dh/dt = -Q/A = -\frac{1}{2} \frac{R^2}{D^2} u_{\max}\), setting \(u_{\max} = 0.155 \, m/s\), we can now evaluate \(dh/dt\). The negative sign indicates that the liquid level in the tank is decreasing.

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Most popular questions from this chapter

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