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Chapter 8: Problem 114

Water discharges to atmosphere from a large reservoir through a moderately rounded horizontal nozzle of \(25 \mathrm{mm}\) diameter. The free surface is \(2.5 \mathrm{m}\) above the nozzle exit plane. Calculate the change in thow rate when a short section of 50 -mm-diameter pipe is attached to the end of the nozzle to form a sudden expansion. Determine the location and estimate the magnitude of the minimum pressure with the sudden expansion in place. If the flow were frictionless (with the sudden expansion in place), would the minimum pressure be higher, lower, or the same? Would the flow rate be higher, lower, or the same?

Short Answer

Expert verified
In conclusion, attaching a section of 50mm diameter pipe to the end of the nozzle causes a reduction in outflow rate and decrease in pressure within the system. On the other hand, in a frictionless flow with the sudden expansion in place, the flow rate would remain the same and the minimum pressure location would have higher pressure than with friction.

Step by step solution

01

Initial Outflow Rate Calculation

Calculate the initial outflow rate without any expansion in place using Bernoulli's equation. \[v_1=\sqrt{2gh}\] where, \(v_1\) is the velocity of water exiting the nozzle, g is the gravitational acceleration and h is the height above the nozzle. The discharge rate \(Q_1\) can be calculated by \[Q_1 = A_1 v_1\] where, \(A_1\) is the area of the nozzle.
02

Outflow Post Expansion

Now, calculate the effect of sudden expansion on outflow rate. For a sudden expansion, the outlet velocity decreases and becomes \[v_2 = \frac{A_1}{A_2} v_1\] where, \(v_2\) is the new velocity after expansion and \(A_2\) is the area after expansion. The new discharge rate after expansion \(Q_2\) becomes \[Q_2 = A_2 v_2\]
03

Minimum Pressure Calculation

For the minimum pressure, energy conservation equation can be used, which leads to Bernoulli鈥檚 equation being set up between the free surface and the point right after the expansion. We can estimate minimum pressure as the hydrostatic pressure at the location which is less than the atmospheric pressure.
04

Comparative Study with Frictionless Flow

With the sudden expansion in place, if the flow is frictionless, the momentum equation between points immediately before and after the sudden expansion yields \(v_2 = v_1\). As such, the flow rate remains same but the minimum pressure would likely be higher as there is no loss of energy due to friction.

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