## 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 S_{B} = tan θ = tan 0.2° = 0.00349.

S_{B} = 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) = (D^{2}/8)(θ – sin θ), wetted perimeter P(y) = Dθ/2, and hydraulic radius R_{H}(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 = (C_{0}/n)AR^{2/3}_ H S^{1/2} _B , with C_{0} = 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**

y_{N} (ft) |
θ^{rad}_N | A(y) ft^{2} |
P(y) ft | R_{H}(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 |