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Chapter 3: Problem 73

Carbon dioxide flows at a rate of \(1.5 \mathrm{ft}^{3} / \mathrm{s}\) from a 3 -in. pipe in which the pressure and temperature are 20 psi (gage) and \(120 \%\) into a 1.5 -in. pipe. If viscous effects are neglected and incompressible conditions are assumed, determine the pressure in the smaller pipe.

Short Answer

Expert verified
The calculation involves a mix of fluid mechanics principles, mainly Bernoulli's equation and the continuity equation. It requires finding the velocities in both sections of the pipe and then using Bernoulli's equation to find the final pressure \(P_{2}\). Thus, the pressure in the smaller pipe can be calculated. The exact numerical value will depend on the referenced density of carbon dioxide.

Step by step solution

01

Applying the Continuity Equation

Initial flow rate \(Q_{1}\) from 3-inch pipe is given as \(1.5 \, \mathrm{ft^{3}/s}\). Under the assumption of incompressible fluid, the flow rates must be equal in both sections of the pipe. So, the flow rate \(Q_{2}\) in the 1.5-inch pipe is also \(1.5 \, \mathrm{ft^{3}/s}\). Next, using the continuity equation \(Q = vA\), where \(v\) is the velocity and \(A\) is the area, the velocities in the pipes can be calculated. The cross-sectional area \(A\) of a pipe is given by \(\pi D^{2}/4\), where \(D\) is diameter of the pipe.
02

Calculating velocities in the pipes

The diameters are 3 inches and 1.5 inches respectively. Convert these values to feet (since other given values are in feet). With \(D_{1} = 3/12 \, \mathrm{ft}\) and \(D_{2} = 1.5/12 \, \mathrm{ft}\), calculate the velocities \(v_{1}\) and \(v_{2}\) using formula \(v = Q/A\).
03

Applying Bernoulli's Equation

Now, apply the Bernoulli's equation \(P_{1}+1/2 \rho v_{1}^{2} = P_{2}+1/2 \rho v_{2}^{2}\) where \(P_{1}\) and \(P_{2}\) are pressures at the two points, \(v_{1}\) and \(v_{2}\) are their respective velocities, and \(\rho\) is the density of carbon dioxide. The initial gauge pressure \(P_{1}\) is given as 20 psi, and the density of carbon dioxide at provided conditions can be found in a standard table.
04

Calculating \(P_{2}\)

Finally, after calculating the velocities and knowing the density of carbon dioxide, calculate the required gauge pressure in the smaller pipe, \(P_{2}\), by rearranging the Bernoulli's equation and solving for \(P_{2}\).

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