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Chapter 8: Problem 101

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.

Short Answer

Expert verified
The pipe diameter necessary would be found from iterative solving of Bernoulli's equation taking into account the friction factor from the Darcy-Weisbach formula and the equivalent length concept for fittings.

Step by step solution

01

Set up Bernoulli's equation between the two points

The Bernoulli's equation for this situation, neglecting velocity heads because the diameters aren't changing significantly and height differences because they're stated negligible, would be: \[ P1 = P2 + f_{D}(L/D)(v^{2}/2g) \] where \( P1, P2 \) are the initial and final pressures, \( f_D \) is the Darcy friction factor, \( L \) is the pipe length, \( D \) is the pipe diameter we're looking to find, \( v \) is the average flow velocity, and \( g \) is the acceleration due to gravity.
02

Convert units and calculate average velocity

The units of pressure need to be converted to a consistent system, and the average velocity from the given flow rate needs to be calculated. The equation to determine the velocity from the flow rate is \( Q = vA \), where \( A \) is the cross-sectional area of the pipe \( A = \pi(D/2)^2 \), which would give us the relationship \( v = Q/\(\pi(D/2)^2). The pressures would be converted from psi to lbf/ft^2: 1 psi = 144 lbf/ft^2.
03

Use the equivalent length concept for the fittings

To account for the minor losses due to the fittings (elbows), the equivalent length concept can be used, where loss due to fittings are put in terms of an equivalent length of straight pipe. The total length in the equation thus becomes \( L_{total} = L + L_{eq}\). Typical equivalent lengths for different fittings can be found in fluid mechanics texts; in this case, it's for the six 90-degree, threaded, standard elbows.
04

Iteratively solve for the diameter

The Darcy friction factor depends on the Reynolds' number, which in turn depends on the diameter. So, an initial guess for the diameter is to be made, and corrected iteratively (using the Moody chart or the Colebrook equation for example) until a consistent set of values for the friction factor, Reynolds number, and the velocity is obtained. Once that is done, these values are substituted back into Bernoulli's equation to solve for the diameter.

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Most popular questions from this chapter

Water at \(10^{\circ} \mathrm{C}\) flows through a smooth 60 -mm-diameter pipe with an average velocity of \(8 \mathrm{m} / \mathrm{s}\). Would a layer of rust of height \(0.005 \mathrm{mm}\) on the pipe wall protrude through the viscous sublayer? Justify your answer with appropriate calculations.

Asphalt at \(120^{\circ} \mathrm{F}\), considered to be a Newtonian fluid with a viscosity 80,000 times that of water and a specific gravity of 1.09 flows through a pipe of diameter 2.0 in. If the pressure gradient is 1.6 psi/ft determine the flowrate assuming the pipe is (a) horizontal; (b) vertical with flow up.

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