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## 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?

## 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)A$R^{2/3}_ H S^{1/2} _B$ , 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