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Chapter 8: Problem 65

Assume a car's exhaust system can be approximated as \(14 \mathrm{ft}\) of 0.125 -ft-diameter cast-iron pipe with the equivalent of six \(90^{\circ}\) flanged elbows and a muffler. (See Video V8.14.) The muffler acts as a resistor with a loss coefficient of \(K_{t}=8.5 .\) Determine the pressure at the beginning of the exhaust system if the flowrate is \(0.10 \mathrm{cfs},\) the temperature is \(250^{\circ} \mathrm{F}\), and the exhaust has the same properties as air.

Short Answer

Expert verified
The pressure at the beginning of the exhaust system can be found by adding the pressure losses due to major and minor components to the atmospheric pressure. This requires applying Bernoulli's equation and the concept of friction and loss coefficients.

Step by step solution

01

Identify Given Variables

Identify and list all the given variables from the problem statement. Length of pipe \(L = 14 \text{ ft}\), Diameter \(D = 0.125 \text{ ft}\), Number of elbows \(n = 6\), Loss coefficient of muffler \(K_t = 8.5\), Flowrate \(Q = 0.1 \text{ cfs}\), Temperature \(T = 250^{\circ} F\). Assume air properties for exhaust.
02

Compute Reynolds Number and Friction Factor

The Reynolds number is calculated using the formula \(Re=\frac{4Q}{\pi D\nu}\) and the friction factor can be computed using the Colebrook-White formula or Moody's diagram. But given the complexity of these methods, it can be approximated for full turbulent flow as \(f=0.0791 Re^{-0.25}\), where \(\nu\) is the kinematic viscosity of air at the given temperature.
03

Calculate the Major Loss

The major loss is due to pipe friction and can be computed using the formula \(h_f=\frac{4fLQ^2}{g \pi^2 D^5}\), where \(g\) is the acceleration due to gravity. It is given in units of length, and signifies how much head (or pressure) is lost due to friction.
04

Calculate the Minor Loss

The minor loss is due to bends, fittings and the muffler. From the data, there are seven fittings - six elbows and one muffler. The loss coefficient for elbows is usually taken as 0.3. Summing up the losses for all components, we calculate the minor loss as \(h_m = \left(n \times \text{K (elbow)} + K_t \right) \times \frac{v^2}{2g}\). The velocity \(v\) can be calculated by re-arranging the continuity equation \(Q=vA\).
05

Apply Bernoulli's Equation

Using Bernoulli's equation, we can relate the total head at the beginning and at the end of the exhaust system, which leads to the formula for pressure at the start of the system as \(P_{start} = P_{end} + \rho g (h_f + h_m)\). Here, \(\rho\) represents the density of air at the given temperature. Since the exhaust is open to the atmosphere, \(P_{end}\) is atmospheric pressure. The pressure is given by the formula \(P = \rho gh\), where \(h\) is the total head loss in the system.

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Most popular questions from this chapter

A 2.5 -in.-diameter flow nozzle meter is installed in a 3.8 -in.- diameter pipe that carries water at \(160^{\circ} \mathrm{F}\). If the air-water manometer used te measure the pressure difference across the meter indicates a reading of \(3.1 \mathrm{ft}\), determine the flowrate

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

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

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 \(0.5 \mathrm{m} / \mathrm{s}\)

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