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Chapter 10: Problem 52

Four sewer pipes of 0.5 -m diameter join to form one pipe of dianeter \(D\). If the Manning coefficient, \(n\), and the slope are the same for all of the pipes, and if each pipe flows half full, determine \(D\)

Short Answer

Expert verified
The diameter, \(D\), of the larger pipe can be found by setting the Manning's equation for the total inflow volume equal to the Manning's equation for the outflow volume and solving for \(D\).

Step by step solution

01

Realize flow conditions for each small pipe

Given that each small pipe of diameter 0.5m is flowing half full. This means a cross-sectional area (\(A\)) of 0.25\(\pi\)m² (half of a circle), a hydraulic radius (\(R\)) of 0.25m (radius of the half-circle), and a wetted perimeter (\(P\)) of 0.5\(\pi\)m (half the circumference of the full circle).
02

Apply Manning's equation for small pipes

Manning's equation is \(Q = \frac{1}{n} AR^{2/3}S^{1/2}\) where \(Q\) represents the discharge or flow rate, \(n\) is the Manning coefficient, \(A\) is the cross-sectional area, \(R\) is the hydraulic radius (area/wetted perimeter), and \(S\) is the slope. Apply this equation for one small pipe to find the flow rate for each small pipe.
03

Calculate total inflow from all small pipes

By assuming flow conditions are similar in all four small pipes, the total inflow can be calculated as four times the flow rate in one small pipe.
04

Realize flow conditions for the big pipe

For the bigger pipe of diameter \(D\) that is flowing half full, \(A\) will be 0.5\(\pi D^2\), \(R\) will be \(D/2\), and \(P\) will be \(\pi D\).
05

Apply conservation of mass principle and Manning's equation

According to the principle of conservation of mass, the total inflow from the four small pipes must equal the outflow from the bigger pipe. Thus, applying Manning's equation to the bigger pipe for \(Q\) equal to the total inflow volume, one can solve for the unknown diameter \(D\).

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Most popular questions from this chapter

The depth downstream of a sluice gate in a rectangular wooden channel of width \(5 \mathrm{m}\) is \(0.60 \mathrm{m}\). If the flowrate is \(18 \mathrm{m}^{3} / \mathrm{s}\) determine the channel slope needed to maintain this depth. Will the depth increase or decrease in the flow direction if the slope is (a) \(0.02 ;\) (b) \(0.01 ?\)

A rectangular channel \(3.0 \mathrm{m}\) wide has a flow rate of 5.0 \(\mathrm{m}^{3} / \mathrm{s}\) with a normal depth of \(0.50 \mathrm{m} .\) The flow then encounters a dan that rises \(0.25 \mathrm{m}\) above the channel bottom. Will a hydraulic jump occur? Justify your answer.

Water flows in a horizontal rectangular channel with a flowrate per unit width of \(q=10 \mathrm{ft}^{2} / \mathrm{s}\) and a depth of \(1.0 \mathrm{ft}\) at the downstream section \((2) .\) The head loss between section upstream and section (2) is 0.2 ft. Plot the specific energy diagram for this flow and locate states (1) and (2) on this diagram.

Find the diameter required for reinforced concrete pipe laid at a slope of 0.001 and required to carry a uniform flow of \(19.3 \mathrm{ft}^{3} / \mathrm{sec}\) when the depth is \(75 \%\) of the diameter.

The following data are obtained for a particular reach of the Provo River in Utah: \(A=183 \mathrm{ft}^{2}\), frec-surface width \(=55 \mathrm{ft}\) average depth \(=33 \mathrm{ft}, R_{h}=3.32 \mathrm{ft}, V=6.56 \mathrm{ft} / \mathrm{s},\) length of reach \(=116 \mathrm{ft},\) and elevation drop of reach \(=1.04 \mathrm{ft}\). Determine (a) the average shear stress on the wetted perimeter, (b) the Manning coefficien. \(n,\) and (c) the Froude number of the flow.
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