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Chapter 15

Q. 15.12

A 12 in. diameter, round cast iron drain pipe is laid on a 0.2° slope. Assuming uniform flow, find the flowrate if the pipe is one-quarter, one-half, three-quarters, and completely full. What are the velocity and Froude number for each of these flows?

Step-by-Step

Verified Solution

The pipe is sketched in Figure 15.37. From Table 15.1 the Manning coefficient for cast iron is n = 0.013. Since we know that the slope of the channel is 0.2°, we can use Eq. 15.34 to calculate the bed slope as SB = tan θ = tan 0.2° = 0.00349.

SB = tan θ = \frac{h_1-h_2}{L}              (15.34)

Using the formulas in Figure 15.6 for a round pipe we can first calculate θ = 2 cos−1(1 − 2y/D), where y is the depth, and D = 1 ft. Next we calculate the flow area A(y) = (D2/8)(θ – sin θ), wetted perimeter P(y) = Dθ/2, and hydraulic radius RH(y) = (D/4)(1 – sin θ/θ). The results are shown in Table 15.2. The volume flowrate is calculated next, from Eq. 15.53b: Q = (C0/n)AR^{2/3}_ H S^{1/2} _B , with C0 = 1.49, since we are working in BG units. Velocity can then be obtained from V = Q/A, and we can calculate Fr = V/\sqrt{gy} . The results are included in Table 15.2.

TABLE 15.1 Values of the Manning Roughness Coefficient, n

Wetted Perimeter n
A. Natural channels
Clean and straight 0.030
Sluggish with deep pools 0.040
Major rivers 0.035
B. Floodplains
Pasture, farmland 0.035
Light brush 0.050
Heavy brush 0.075
Trees 0.15
C. Excavated earth channels
Clean 0.022
Gravelly 0.025
Weedy 0.030
Stony, cobbles 0.035
D. Artificially lined channels
Glass 0.010
Brass 0.011
Steel, smooth 0.012
Steel, painted 0.014
Steel, riveted 0.015
Cast iron 0.013
Concrete, finished 0.012
Concrete, unfinished 0.014
Planed wood 0.012
Clay tile 0.014
Brick work 0.015
Asphalt 0.016
Corrugated metal 0.022
Rubble masonry 0.025

TABLE 15.2 Solution Table for Example 15.12

yN (ft) θ^{rad}_N A(y) ft2 P(y) ft RH(y) ft R^{2/3}_H Q ft³/s V ft/s Fr
0.25 2.094 0.153 1.05 0.146 0.277 0.29 1.90 0.67
0.5 3.14 0.393 1.57 0.250 0.397 1.06 2.70 0.67
0.75 4.19 0.632 2.10 0.302 0.450 1.93 3.05 0.62
1.0 6.28 0.785 3.14 0.250 0.397 2.11 2.69 0.47
15.37
15.6