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Chapter 8: Problem 11

The wall shear stress in a fully developed flow portion of a 12-in.-diameler pipe carrying water is 1.85 Ib/ft? Determine the pressure gracient, \(\partial p / \partial x,\) where \(x\) is in the flow direction, if the pipe is (a) horizontal, (b) vertical with flow up, or (c) vertical with flow down.

Short Answer

Expert verified
Therefore, for a horizontal pipe (a), the pressure gradient is calculated in step 3. For a vertical pipe with flow upwards (b), there is additional pressure gradient due to gravity. For a vertical pipe with flow downwards (c), the pressure gradient reduces due to gravity.

Step by step solution

01

Identification of Parameters

Firstly, identify all the known parameters. Given the wall shear stress \( \tau = 1.85 Ib/ft^2 \) and the pipe diameter \(D = 12 in = 1 ft\). The density of water \(\rho\) is approximately 62.4 lbm/ft³ and the viscosity of water \(\mu\) is about 0.00034 lb/ft.s. All measurements should be in a consistent set of units. The unknown parameter is the pressure gradient \(\partial p / \partial x\).
02

Use the Shear Stress and Velocity Gradient Relation

The shear stress at the wall of a pipe can be given by \( \tau = \mu(\partial u / \partial y) \). In a fully developed flow of a pipe, the velocity gradient \( \partial u / \partial y \) at the wall can be simplified to \(U/ \delta \), where \( U \) is the average velocity and \( \delta \) is radius of the pipe. Substituting this into the shear stress wall formula, we get \( \tau = \mu * U / \delta \). From this, we can solve for the average velocity \( U \) as \( U = (\tau * \delta)/ \mu \).
03

Use the Pressure Gradient Equation

The pressure gradient can be calculated by using the formula \( \partial p / \partial x = 2* \mu * U / \delta^2 \). Substituting the expression for \( U \) found in Step 2, we get \( \partial p / \partial x = 2* \mu * \tau * \delta / \mu* \delta^2 = 2 * \tau / \delta \). Calculate the pressure gradient using this formula.
04

Adjust for Pipe Orientation

To account for different pipe orientations, we can add or subtract the hydrostatic pressure change due to gravity. For a horizontal pipe (a), the pressure gradient remains unchanged as \( \partial p / \partial x \). For a vertical pipe with flow upwards (b), the gravitational effects add to the pressure gradient, so the pressure gradient becomes \( \partial p / \partial x + \rho * g \). For a vertical pipe with flow downwards (c), the gravitational effects reduce the pressure gradient, so the pressure gradient becomes \( \partial p / \partial x - \rho * g \).

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Most popular questions from this chapter

A 250 ft-high building has a 6.065 -in.-diamcter steel standpipe and a 100 -ft-long \(2 \frac{9}{16}\) -in. diameter fire hose on each floor. The nearest fireplag is \(100 \mathrm{ft}\) from the standpipe's ground-level connection. Assume that fire-fighters connect a 6 -in.- -diameter, 50 -ft-long fire hose from the fireplug to the fire truck and a 4-in.- -diameter. 50-ft-long fire hose from the fire truck to the standpipe's groundlevel connection. The National Fire Protection Association (NFPA) requires that a minimum pressure of 65 psig be meintained at the connection of the \(2 \frac{9}{16}\) -in.- -diameter hose and the standpipe while maintaining a flow rate of 500 gal/min through the fire hose. What pressure rise must the pump on the fire engine supply to satisfy the NFPA requirement for this building? The fire hydrant water pressure is 80 psiz and the water temperature is \(60^{\circ} \mathrm{F}\). The connections are threaded.

For a given head loss per unit length, what effect on the flowrate does doubling the pipe diameter have if the flow is (a) laminar, or (b) completely turbulent?

A motor-driven centrifugal pump delivers \(15^{\circ} \mathrm{C}\) water at the rate of \(10 \mathrm{m}^{3} / \mathrm{min}\) from a reservoir, through a 2500 -m-long, \(30-\mathrm{cm} 1 . \mathrm{D} .\) plastic pipe, to a second reservoir. The water level in the second reservoir is \(40 \mathrm{m}\) above the water level in the first reservoir. The pump efficiency is \(75 \% .\) Find the motor output power. The pipe entrance is square edged.

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.

A company markets ethylene glycol antifreeze in halfgal bottles. A machine fills and caps the bottles at a rate of 60 per minute. The \(68^{\circ} \mathrm{F}\) ethylene glycol \(\left(\rho=69.3 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}, v=1.93 \times\right.\) \(\left.10^{-4} \mathrm{ft}^{2} / \mathrm{s}\right)\) is pumped to the machine from a tank \(36 \mathrm{ft}\) away. The pressure at the discharge of the pump is 35 psig, and the pressure at the inlet to the filling machine must be at least 15 psig. Determine the diameter necessary for drawn copper pipe. The pipe has no elevation change.
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