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Chapter 8: Problem 80

Water flows in a horizontal constant-area pipe; the pipe diameter is \(75 \mathrm{mm}\) and the average flow speed is \(5 \mathrm{m} / \mathrm{s}\). At the pipe inlet, the gage pressure is \(275 \mathrm{kPa}\), and the outlet is at atmospheric pressure. Determine the head loss in the pipe. If the pipe is now aligned so that the outlet is \(15 \mathrm{m}\) above the inlet, what will the inlet pressure need to be to maintain the same flow rate? If the pipe is now aligned so that the outlet is \(15 \mathrm{m}\) below the inlet, what will the inlet pressure need to be to maintain the same flow rate? Finally, how much lower than the inlet must the outlet be so that the same flow rate is maintained if both ends of the pipe are at atmospheric pressure (i.e., gravity feed)?

Short Answer

Expert verified
The head loss in the pipe is calculated in the first step. The required inlet pressures to maintain the same flow rate with the outlet 15m above and 15m below the inlet are then computed. Finally, the height the outlet must be below the inlet for gravity feed is calculated.

Step by step solution

01

Calculate Head Loss

The Bernoulli equation can be used to determine the head loss, \( h_L \), by equating the inlet and the outlet. Given that the pipe is horizontal and the flow speed is constant, the Bernoulli equation simplifies to: \( h_L = (P_1 - P_2)/(蟻*g) \). Given data include \( P_1 = 275 kPa = 275000 Pa \), \( P_2 = 0 kPa = 0 Pa \) (atmospheric pressure), \( 蟻 = 1000 kg/m^3 \) (density of water), and \( g = 9.81 m/s^2 \). Substituting these values into the simplified equation yields the head loss.
02

Calculate Inlet Pressure For Raised Outlet

The Bernoulli equation can once again be used in the changed scenario where the outlet is 15m above the inlet. Now, \( z_2 = 15m \), and the Bernoulli equation is used to find the required inlet pressure \( P_1 \) to maintain the same flow rate.
03

Calculate Inlet Pressure For Lowered Outlet

In this scenario, the outlet is 15m below the inlet, i.e., \( z_2 = -15m \). The Bernoulli equation is again used to calculate the new inlet pressure \( P_1 \) to maintain the same flow rate.
04

Calculate Outlet Height For Gravity Feed

In the final scenario, both ends of the pipe are at atmospheric pressure. Again, the Bernoulli equation is employed to find the value of \( z_2 \) that would maintain the same flow rate. Since the pressures are equal and the velocity is constant, the height difference must be equal to the previously calculated head loss to equate the Bernoulli equation and maintain the flow rate.

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