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Chapter 8: Problem 24

A 3 -in. schedule 40 commercial steel pipe (with an actual inside diameter of 3.068 in.) carries \(210^{\circ} \mathrm{F}\) SAE 40 crankcase oil at the rate of 6.0 gal/min. The oil specific gravity is \(0.89,\) and the absolute viscosity is \(6.6 \times 10^{-7} 1 \mathrm{b} \cdot \mathrm{sec} / \mathrm{ft}^{2} .\) Calculate the pipe size required to carry the same flow rate at approximately one-half the pressure drop of the 3 -in. pipe. Both pipes are horizontal.

Short Answer

Expert verified
The required pipe size is approximately 2.73 inches in diameter.

Step by step solution

01

Convert the Flow Rate from Gallons/Minute to Cubic Feet/Second

To start with, we first convert the flow rate from gallons per minute to cubic feet per second since our other measurements are in imperial system. There are 7.48052 gallons in one cubic foot and there are 60 seconds in one minute. Thus, \( Q = 6.0 \ \text{gal/min} = 6.0 \ \text{gal/min} \times \frac{1 \ \text{cubic foot}}{7.48052 \ \text{gallons}} \times \frac{1 \ \text{minute}}{60 \ \text{seconds}} = 0.00223 \ \text{cubic feet/second} \).
02

Use Hagen-Poiseuille’s Law to Derive the Relationship between the Diameters of the Pipes and the Pressure Drops

Next, we apply the Hagen-Poiseuille’s law, which states that the volume flow rate \( Q \) through a pipe is proportional to the fourth power of the pipe’s diameter, the pressure drop \( \Delta P \), and inversely proportional to the pipe’s length \( L \), and the fluid’s dynamic viscosity \( \mu \). Mathematically this means that, \( Q = \frac{\pi \Delta P D^{4}}{128 \mu L} \) Thus, if we keep everything constant and just alter \( \Delta P \) and \( D \), we get \( D1^{4} / D2^{4}= \Delta P2 / \Delta P1 \) or \( D2 = D1 \times \left( \frac{\Delta P2}{\Delta P1} \right)^{1/4} \)
03

Substitute the Known Values to Calculate the Required Diameter

From the problem, \( \Delta P2 = \frac{1}{2} \Delta P1 \) and \( D1 = 3.068 \ \text{in} \). Hence, \( D2 = 3.068 \ \text{in} \times \left( \frac{1/2}{1} \right)^{1/4} = 2.73 \ \text{in} \). So, the size of the pipe required to transport the oil at the same flow rate and approximately half the pressure drop of the 3-inch pipe is around 2.73 inch.

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Most popular questions from this chapter

A 0.064 -m-diameter nozzle meter is installed in a 0.097 -m-diameter pipe that carries water at \(60^{\circ} \mathrm{C}\). If the inverted air-water U-tube manometer used to measure the pressure difference across the meter indicates a reading of \(1 \mathrm{m}\), determine the flowrate.

Water flows downward through a vertical 10 -mm-diameter galvanized iron pipe with an average velocity of \(5.0 \mathrm{m} / \mathrm{s}\) and exits as a free jet. There is a small hole in the pipe \(4 \mathrm{m}\) above the outlet. Will water leak out of the pipe through this hole, or will air enter into the pipe through the hole? Repeat the problem if the average velocity is \(0.5 \mathrm{m} / \mathrm{s}\)

For fully developed laminar pipe flow in a circular pipe, the velocity profile is given by \(u(r)=2\left(1-r^{2} / R^{2}\right)\) in \(\mathrm{m} / \mathrm{s},\) where \(R\) is the inner radius of the pipe. Assuming that the pipe diameter is \(4 \mathrm{cm},\) find the maximum and average velocities in the pipe as well as the volume flow rate.

When soup is stirred in a bowl, there is considerable turbulence in the resulting motion (see Video \(\mathrm{V} 8.7\) ). From a very simplistic standpoint, this turbulerce consists of numerous intertwined swirls, each involving a characteristic diameter and velocity. As time goes by, the smaller swirls (the fine scale structure) die out relatively quickly, leaving the large swirls that continue for quite some time. Explain why this is to be expected.

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 \(\left(\nu=8.31 \times 10^{-6} \mathrm{ft}^{2} / \mathrm{s}\right)\)
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