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Chapter 8: Problem 14

Water flows in a constant-diameter pipe with the following conditions measured: At section (a) \(p_{a}=32.4\) psi and \(z_{a}=56.8 \mathrm{ft}\) at section (b) \(p_{b}=29.7\) psi and \(z_{b}=68.2 \mathrm{ft}\). Is the flow from (a) to (b) or from (b) to (a)? Explain.

Short Answer

Expert verified
The flow of water is from section (a) to section (b).

Step by step solution

01

Understand Bernoulli's equation

Bernoulli's equation for a fluid flowing through a pipe is given by \[ P + \frac{1}{2} \rho v^{2} + \rho gh = \text{constant} \], where \( P \) is the pressure of the fluid, \( \rho \) is the density of the fluid, \( v \) is the velocity of the fluid, \( g \) is the acceleration due to gravity and \( h \) is the height of the fluid above some reference point. The key point to remember is that for a streamline flow in a non-viscous liquid, the Bernoulli's equation remains constant.
02

Formulate Bernoulli's equation for points (a) and (b)

The Bernoulli's equations for point (a) and point (b) can be given as: \[ p_{a} + \frac{1}{2} \rho v^{2}_{a} + \rho g z_{a} = C \] and \[ p_{b} + \frac{1}{2} \rho v^{2}_{b} + \rho g z_{b} = C \] respectively, where \( p \) is pressure, \( z \) is height, \( v \) is velocity, \( C \) is the constant along the streamline and \( \rho g \) is the weight density of the fluid.
03

Consider constant velocity

As there is no information on the changes in the velocities at points (a) and (b) and since the pipe has a constant diameter, it can be assumed that the velocities at both points remain the same. So, \( v_{a} = v_{b}\). This simplifies the Bernoulli's equation to: \[ p_{a} + \rho g z_{a} = p_{b} + \rho g z_{b} \]
04

Compare the Bernoullis equation at points a and b

Rearrange the simplified Bernoullis equation from the previous step by bringing \( p_{b} + \rho g z_{b} \) to the left side: \[ p_{a} + \rho g z_{a} - p_{b} - \rho g z_{b} = 0 \] Simplifying this equation gives: \[ p_{a} - p_{b} = \rho g (z_{b} - z_{a}) \] From the given problem, the pressure at point a is greater than the pressure at point b and the z-coordinate of point b is greater than the z-coordinate of point a. Therefore, the flow is from point (a) to point (b).

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Most popular questions from this chapter

The vented storage tank shown in Fig. \(\mathrm{P} 8.90\) is used to refuel race cars at a race track. A total of \(40 \mathrm{ft}\) of steel pipe \((\mathrm{I.D},=0.957 \text { in. })\) two \(90^{\circ}\) regular elbows, and a globe valve make up the system. Calculate the time needed to put 20 gal of fuel in a car tank. The pressures, \(p_{2}\) and \(p_{1},\) are aqual, the connections are threaded, and the fuel has the properties of \(68^{\circ} \mathrm{F}\) normal octane \(\left(\nu=8.31 \times 10^{-6} \mathrm{ft}^{2} / \mathrm{s}\right)\)

A 10 -m-long \(, 5.042\) -cm I.D. copper pipe has two fully open gate valves, a swing check valve, and a sudden enlargement to a \(9.919-\mathrm{cm}\) I.D. copper pipe. The \(9.919 \mathrm{cm}\) copper pipe is \(5.0 \mathrm{m}\) Iong and then has a sudden contraction to another 5.042 -cm copper pipe. Find the head loss for a \(20^{\circ} \mathrm{C}\) water flow rate of \(0.05 \mathrm{m}^{3} / \mathrm{s}\)

A person with no experience in fluid mechanics wants to estimate the friction factor for 1 -in.-diameter galvanized iron pipe at a Reynolds number of 8,000 . The person stumbles across the simple equation of \(f=64 / \mathrm{Re}\) and uses this to calculate the friction factor. Explain the problem with this approach and estimate the error.

A certain process requires 2.3 cfs of water to be delivered at a pressure of 30 psi. This water comes from a large-diameter supply main in which the pressure remains at 60 psi. If the galvanized iron pipe connecting the two locations is 200 ft long and contains six threaded \(90^{\circ}\) elbows, determine the pipe diameter. Elevation differences are negligible.

Given two rectangular ducts with equal cross-sectional area but different aspect ratios (width/height) of 2 and \(4,\) which will have the greater frictional losses? Explain your answer
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